Cold Formed2

7/27/2019 Cold Formed2

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PARTICULAR FEATURES FOR THE DESIGN OF THE COLD-FORMED THIN GAUGE

SECTIONS

Cold-formed (thin gauge) sections may buckle under normal stresses smaller than the yield limitof the steel.

The instability of the thin gauge flat sheets subjected to in-plane loading is due to imperfections.The following assumptions are demonstrated to be inconsistent:

(1) The perfect planarity - the initial deformations of the sheets due to faults of fabrication

must be between certain limits. Still, the real plane elements do have initial geometricalimperfections- an initial deflection w0, which grows along with the increase of loading. Due to the

effect of membrane behavior, the ultimate strength of the sheet is bigger than the critical elastic

force of buckling,Ncr. This reserve of strength clearly insures a post-critical behavior.

Fig. 1. Plate in compression: conditions of supports and post-critical reserve

(2) Reduced deformations out of the plane of the plate this assumption is normallyavailable in the theory of linear buckling in elastic domain. In reality, the ultimate strength of the

plate exceeds the critical stress, the deformations being rather important;

(3) Axial loads - this assumption is impossible from the practical point of view, theplanarity of the plate being an ideal assumption;

Fig. 2. Local buckling in compression and bending of the thin walled elements(4) Linear elastic behavior of the material this condition is satisfied up to the yield limit.

Still, due to residual stresses caused by rolling, welding, cutting etc, in some fibers the plastic

stresses are reached for applied stresses lower thanfy.


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two distinct stages in the post-critical domain of the behavior of a plate are:

Elastic- uniformly distributed stresses, under the critical force;

Post-critic- below the critical force, the plate is deformed more and more, the stresses

are not anymore uniform.

Fig. 3. Consecutive stages of stress distribution in stiffened compressed elements

Buckling is reached for a critical value of the normal stress: c cr where the critical stress

is determined with the known relationship:

( )3

22

2

2

10190112

=

=

pp

crb

tk

b

tEk

[N/mm2] (1)

The coefficient kdepends on the nature and the distribution of the stress on the width of the

wall, on the boundary conditions, on the ratio between the dimensions of this wall.

- non - stiffened walls: k =0.425;

- stiffened walls: k=4.0, the supports are considered articulated.

!It is important to observe that: in the case of a wall under compression in its plane, the lost of strength capacity will not

happen as long as the longitudinal edges will remain rectilinear;

the limits of strength capacity are much increased for certain types of walls. This remark

leads to the theory ofeffective width of the wall.

The design concept the grid model proposed by Winter (1959) for the instabilityphenomenon. The cross section for these profiles is made up from flat elements (walls) withconstant thickness inter-connected, generating a grid.

Fig. 4. Winters model (grid)

In the post critical stage (post buckling strength) the central griddo not work anymore whilethe extreme grids, where the strains are smaller, are able to take over stresses that may reach the

design value of strength. At the moment when the maximum strength value of the material R, is

reached in the extreme zones, a bigger portion in the internal part of the wall already isnt working

anymore (where = 0), the deformations being very important.The width of the wall reaches its minimum value, called the effective width beff.


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Fig. 5. Effective width of compression plate

From the point of view of the local buckling:-stiffened compressed elements (walls) -flat elements in compression with both edges parallel

to the direction of stress, which are stiffened by web elements, flanges or edge stiffeners of

sufficient rigidity-non-stiffened compressed elements (walls) -flat elements in compression which are stiffened

only at one edge parallel to the direction of the stress.

Considering that in the situation of buckling in elastic of a wall having its effective width,beff, the stress cr,eff reaches the maximum stress in the plate in post-critical domain, that is: max = fy.Then the relationship (1) becomes:

( )

22

2

2

,112

=

=

eff

p

cr

eff

effcrb

b

b

tEk

(2)

From this relationship it results that the effective width of the wall depends on the ratio

cr/max :

max

crpeff bb = (3)

where:

cr the critical stress of buckling in elastic of the plate, considering its total width;max - maximum stress on the edges of the plate.

Considering that in the phase of buckling the averaged stress on the whole width of the wallis u, the equivalence between the stresses will impose the following equation:

upeffupyeff bbbfb == max (4)

Von Karman determined the following relationship for the effective wall:

( )max

2

2

21

112

=

p

peffb

tEkbb (5)

In the case of the plate articulated all around and uniformly compressed, k = 4.0 and:

max

9.1

E

tbeff=

(6)

EC3 uses the following relationships in order to simplify the further design specifications:

relative slenderness (of the plate) referred to bp:

k

t

b

fp

cr

y

p

==4.28

(7)

reduction factor:


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y

u

p

eff

fb

b == (8)

influence of the elastic limit:

yf

240= (9)

Based on von Karmans relationship it will result that:1

1=

and

p

(10)

The slenderness of a wall,pis the ratio between the flat width of the wall, bpand its thickness, t.

Winter proposed a semi - empirical relationship, derived from that of von Karmans that takesinto account the imperfections:

=

maxmax

415.019.1

E

t

b

Etb

p

eff (11)

This is used by EC3 in the design of the strength of very slender sections. The following

annotations are used: for: 673.0p we have: 1= (12)

for: 673.0>p we have:

=

pp

22.01

1 (13)

Specifications:

1) The effective width of a flat wall in compression and/or in bending is determined

considering the relative slenderness p referred to the width of the flat wall, bpand also, the

limit of yield strength,fyb.2) In order to identify the way the cross section of a wall is working we have to compare the

effective slenderness with the limit slenderness.

The recommended values of the maximum slenderness (limit slenderness) for different types

of cold-formed sections are presented in table 1. The common experience and the tests in laboratoryimpose these values.

The limit slenderness is defined as the ratio between the width and the thickness of the wallin the case when the normal stresses are uniformly distributed on the whole cross section and equal

with the design strength of the material. The values of the limit slenderness depend on the kind of

the wall and the grade of the steel. The presence of the imperfections reduces the theoretical values

of these limits over which buckling may occur anytime, see table 2.


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Table 1. Maximum values of the slenderness of the walls at the cold formed thin gauge

sections. Modeling the static behavior

Tab.2. Values for the limits of slenderness of the walls of the cold formed thin gauge sections


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The effective width and effective area of the walls in buckling

Von Karmans theory mentions that the maximum stress in the wall max systematicallyreaches the elastic limitfy, so a pattern of the determination of the effective widths comprises:

Determination of the stress ratio that shows the distribution of the stresses in the wall

considered with its effective width (tab. 2 and 3).For doubly supported elements the stress ratio

may be based on the properties of the gross cross section; Considering the supports (internal wall or end wall as cantilever) and again the value ofthe

buckling coefficient is determined k;

Relative slenderness p is determined;

Reduction factor is determined;

The effective width is calculated with the help of tables 2 and 3.

! Specification:In the case when the initial stress applied to the wall is small enough the amplified stress due to

the lost of the efficacy max may reach a value much lower than the elastic limit fy. It is rational inthis case to determine the effective width on the basis of the compression stress and not based on the

limit of elasticity. For that, the parameter is computed by replacingfywith com as a firstapproximation of the max value.

A new, altered value ofmax is determined for the effective width based on the reiteration of the

method and starting from the determination of the relative slenderness of the wall. A procedure of

convergence for the stress max , until is reaches the recommended values is based on calculation ofthe relative slenderness of the effective wall and then using this value in the expression of.

Tab.4. Computation of the effective width of the intermediary (internal) walls in compression


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Tab.5. The computation of the effective width for the external walls in compression

Elements without stiffeners (plane elements)

I step

The reduction factor for the determination of the effective widths according to tab. 4. for

doubly supported or 5., for singly supported elements shall be obtained as we have already seen. Thevalue of relative slenderness is determined with:

kEt

bcomp

p = 052.1 (14)

where:

com effective stress of compression on the extremities of the wall, 1, determined with

respect to the effective area of the transversal section and multiplied with the safety factor,

M1;


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k buckling coefficient according to tab. 4 and 5..

II step

The design for the limit state of serviceability, 1-fy:

a) the value of the reduction factor is determined with the relative slenderness

obtained as in the I step, where com = 1 M1 and the effective stress calculated is 1 < fy/M1.

b) The following relationships are used: For: 673.0pd we take:=1;

For : 673.0>pd we take:0.1

6.018.0

22.01

+

=pu

pdpu

pd

pd

(15)

After determining the values:

kEt

bcomp

pd = 052.1 and :

kE

f

t

byp

pu

= 052.1 (16)

III stepIn tables 6 and 7 the geometrical width of the flat wall is bp. In the case of the lateral webs

without intermediate stiffeners (the folders of the sheeting), the annotation sw is equivalent with bp.

Tab. 6. Sequences of computation for the design of the end stiffeners


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Tab. 7. Sequence of computation of the intermediary stiffeners

Elements with edge or intermediate stiffeners

The design of the stiffened elements is based on the assumption that the stiffener itself works

as a beam on elastic foundation represented by a spring stiffness, which depends on the bending

stiffness of adjacent parts of plane elements and on the boundary conditions of the element.

The determination of the spring stiffness is illustrated in figure 6 for intermediate and edgestiffeners respectively, where: Cs = 1/fs and Cr= 1/fr.

The significance of the terms are:

f- the deflection of the stiffener due to a force equal with 1;fs and fr are taken as in the figure 5.5.b.

In the determination of the rotational stiffness in the supports C o , C01 and C02 , the effects of

other stiffeners is considered if there is the case for any element that forms the cross section incompression.

For an edge stiffener, the deflection fy is determined with the relationship:

( )3

22

112

3 tE

bbf

p

py

+=

(17)

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