Simplified design approach for cold-formed stainless steel compression members subjected to flexural buckling

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Simplified design approach for cold-formed stainless

steel compression members subjected to

flexural buckling

S.H. Lin

a,

*, S.I. Yen

b

, C.C. Weng

c a

Department of Civil Engineering, Minghsin University of Science and Technology, No. 1, Hsinsin Rd, Hsinfeng, Hsinchu 304, Taiwan, ROC

b_{Power Project Group, E&C Engineering Corporation, Taipei 222, Taiwan, ROC} c_{Department of Civil Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, ROC}

Received 8 December 2004; accepted 26 August 2005

Abstract

The design criteria of stainless steel compression member are more complicated than those of carbon steels due to the nonlinear stress strain behavior of the material. In general, the tangent modulus theory is used for the design of cold-formed stainless steel columns. The modified Ramberg–Osgood equation given in the ASCE Standard can be used to determine the tangent modulus at specified level of stresses. However, it is often tedious and time-consuming to determine the column buckling stress because several iterations are usually needed in the calculation. This paper presents new formulations to simplify the determination of flexural buckling stress without iterative process. Taylor series expansion theory is utilized in the study for numerical approximations. The proposed design formulas are presented herein and can be alternatively used to calculate the flexural buckling stress for austenitic type of cold-formed stainless steel columns. It is shown that the column strengths determined by using the proposed design formulas have good agreement with those calculated by using the ASCE Standard Specification. A design example is also included in the paper for cold-formed stainless steel column designed by using the ASCE Standard equations and the proposed design formulas.

Keywords: Cold-formed stainless steels; Compression members; Specification; Tangent modulus; Flexural buckling; Numerical approximation

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* Corresponding author. Tel.: C886 3 5593142x3302; fax: C886 3 5573718. E-mail address: [email protected] (S.H. Lin).

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1. Introduction

For the design of cold-formed stainless steel compression members, the ASCE Standard Specification can be used to determine the design axial strength. The current

specification was published in 2002 by ASCE as SEI/ASCE 8-02[1]. This updated edition

of the Specification is a revision of the ASCE Standard published in 1991 [2]. Due to

Notation

A Area of the full, unreduced cross section

Ae Effective area of the cross section

b Effective design width of compression element

C Slenderness ratio, KL/r

Co Specified slenderness ratio at FnZFy

C1 Limiting slenderness ratio at FnZF1

Eo Initial modulus of elasticity

Et Tangent modulus

f Nominal stress

Fy Specified yield strength

F1 Specified buckling stress with respect to C1

Fn Nominal buckling stress

Fn,ASCE Nominal buckling stress determined from ASCE Standard Specification

Fn,prop Nominal buckling stress determined from the proposed design formulas

Ix Moment of inertia of x-axis

Iy Moment of inertia of y-axis

k Plate buckling coefficient

K Effective length factor

L Unbraced length of member

n Coefficient used for determining the tangent modulus

Pn Nominal axial strength of member

r Radius of gyration

t Thickness of the section

w Flat width of element exclusive of radii

a Eo/EtK1

b Constant

fc Resistance factor for axial strength

lo Parameter used for determining buckling stress

l1 1Klo

r Reduction factor

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the differences in mechanical behavior of stainless steels as compared with carbon steels as

shown inFig. 1, the design of stainless steel compression members is more complex than

those of carbon steels. It is noted that stainless steels have gradually yielding type of

stress–strain curves with relatively low proportional limits[3,4]. Because of the nonlinear

stress-strain behavior, the design of stainless steel compression members is based on the

tangent modulus theory[5,6].

Stainless steel has different mechanical properties in longitudinal and transverse directions for tension and compression modes of stress. As a result, different tangent

modulus, Et, are provided for different types of stainless steels in the design tables and

figures of the ASCE Specification. Tangent modulus is used to account for the inelastic buckling of stainless steel compression components. Alternatively, it can be determined by

using the modified Ramberg–Osgood equation[7,8]given in Appendix B of the ASCE

Standard for specified types of stainless steels. Because of the nonlinear nature of tangent modulus, the column buckling stress is determined through an iterative process until the

satisfied tolerance is reached[9,10]. This type of calculation is often tedious and

time-consuming as compared with that of hot-rolled steel column design.

To simplify the design process, this paper presents a new approach to obtain the design formulas for determining the flexural buckling stress without successive iterations. These formulas are developed from the tangent modulus equation obtained from the modified Ramberg–Osgood equation. A mathematical approximating method utilizing the Taylor series expansion theory is formulated for numerical approximation. New design formulas are proposed for austenitic type of cold-formed stainless steel columns subjected to flexural buckling. It is shown that the approximated solutions have a good agreement as compared with the ASCE Standard solutions. To illustrate the simplified design approach, a design example of cold-formed stainless steel column subjected to flexural buckling is included in Appendix of the paper.

a b c d Stress Strain

a Hot rolled carbon or low alloy steel

b Cold rolled carbon low alloy steel

c Annealed and flattened stainless steel

(longitudinal tension)

d Annealed and flattened stainless steel

(longitudinal compression)

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2. Review of ASCE design criteria

Section 3.4 of the ASCE Standard provides the design requirements to determine the design axial strength for concentrically loaded cold-formed stainless steel compression members. It specifies that the resultant of all loads acting on the members is an axial loads

passing through the centroid of the effective section calculated at the stress Fn. The design

axial strength, fcPn, is calculated as follows:

fcZ 0:85

PnZ AeFn (1)

where:

AeZeffective area calculated at stress Fn

FnZthe least of the flexural, torsional, and torsional-flexural buckling stress

determined according to Sections 3.4.1–3.4.3 of the ASCE Standard, respectively. Section 3.4.1 of the ASCE Standard specifies that, for doubly symmetric sections, closed cross sections, and any other sections which are not subjected to torsional or

torsional–flexural buckling, the flexural buckling stress, Fn, is determined as follows:

FnZ

p2Et

ðKL=rÞ2%Fy (2)

where:

EtZtangent modulus in compression corresponding to buckling stress Fn

KL/rZslenderness ratio

FyZspecified yield strength, as given inTable 1for austenitic type stainless steels

To calculate the flexural buckling stress in Eq. (2), it is necessary to have a proper value

of Et, which can be obtained from design tables or figures in the ASCE Standard for the

assumed stress. Alternatively, it can be determined by using analytical expression, which is based on the modified Ramberg–Osgood equation

EtZ

EoFy

FyC0:002 nEoðFn=FyÞnK1

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Table 1

ASCE specified yield strength Fyfor austenitic type stainless steels [1] Types of stress Fy,Mpa

Types 201, 301, 304, 316

Annealed 1/16 Hard 1/4 Hard 1/2Hard

Longitudinal tension 206.9 310.3 517.1 758.5 Transverse tension 206.9 310.3 517.1 758.5 Transverse compression 206.9 310.3 620.6 827.4 Longitudinal compression 193.1 282.7 344.8 448.2

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in which Eois the initial modulus of elasticity and n is the coefficient used for determining

tangent modulus. Table 2 gives values of Eoand n for austenitic type stainless steels,

which are obtained from design tables in the ASCE Standard.

Because the buckling stress, Fn, and the tangent modulus, Et, are interdependent as

self-explanatory in Eqs. (2) and (3), the determination of flexural buckling stress, Fn, requires

iterative process. The buckling stress, Fn, is needed to determine Etin Eq. (3), but it is not

known until it has been obtained from Eq. (2). Therefore, to determine a proper value of Et,

a buckling stress is first to be assumed in Eq. (3). Then, this calculated value of Etis used to

determine the buckling stress, Fn, in Eq. (2). In view of the fact that the calculated buckling

stress is seldom equal to the first assumed buckling stress, further successive iterations are needed to obtain the final buckling stress. This buckling stress can be achieved when a satisfied convergence of error is reached.

For details of determining the buckling stress by using ASCE Standard design provisions, see the design example in the appendix of the paper.

3. Simplified approach

As discussed above, for the design of cold-formed stainless steel columns, the determination of flexural buckling stress is tiresome due to its iterative process of calculation. As a result, a simplified approach is proposed herein to determine the flexural buckling stress without having iterative calculations. New formulations are developed as targets of the numerical approximation. In order to simplify the calculations, Taylor series expansion is applied to the approximating equation. The following section gives detailed presentations on this subject.

3.1. Initial concept

The flexural buckling stresses determined by using the ASCE design equations for Type 304 stainless steel columns in longitudinal compression (LC) and transverse

compression (TC) are shown in Figs. 2 and 3, respectively. Nonlinear buckling stress

curves are found typical for those columns when the flexural buckling stresses are less than

Fy, which is the upper bond of Eq. (2). Thus, a linear equation is initialized in this study to

simplify these nonlinear curves.

Table 2

ASCE specified initial modulus of elasticity Eoand coefficient n for austenitic type stainless steels[1] Types of stress Types 201, 301, 304, 316

Annealed and1/16 Hard 1/4Hard 1/2Hard

Eo(MPa) n Eo(MPa) n Eo(MPa) n

Longitudinal tension 193100 8.31 186200 4.58 186200 4.21 Transverse tension 193100 7.78 193100 5.38 193100 6.71 Transverse compression 193100 8.63 193100 4.76 193100 4.54 Longitudinal compression 193100 4.10 186200 4.58 186200 4.22

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0 50 100 150 200 Slenderness Ratio, KL/r 0 100 200 300 400 500

Flexural Buckling Stress, F

n

(MPa)

Type 304 Stainless Steel Columns (Longitudinal Compression)

Annealed 1/16 Hard 1/4 Hard 1/2 Hard

Fig. 2. ASCE Standard flexural buckling stresses for Type 304 stainless steel columns in longitudinal compression. 0 50 100 150 200 Slenderness Ratio, KL/r 0 200 400 600 800 1000

n

(MPa)

Type 304 Stainless Steel Columns (Transverse Compression)

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As shown inFig. 4, it is observed that when applying logarithm to the flexural buckling

stress curves, i.e. log(Fn), a portion of the nonlinear buckling stress curve becomes a

straight line segment between points A(C0, logFy) and B(C1, logF1). The linearized

portion of the curve can be defined by these two specified points as follows:

log F₁Klog F_y C1KC0 Z log F_nKlog F_y CKC0 (4)

in which CZKL/rZslenderness ratio, and C0and C1are two specified slenderness ratios

with their corresponding buckling stresses at Fyand F1in logarithmic scale, respectively.

This linear approximation is beneficial to simplify the procedure of finding the nonlinear buckling stresses.

3.2. Development of formulations

The simplified approach can be achieved by taking the advantage of this linear approximation. New formulations are developed for the corresponding parameters used in the simplified scheme. Eq. (4) can be rearranged as the following simplified form:

FnZ F ðCKC0=C1KC₀Þ 1 !F ðC1KC=C1KC0Þ y (5) Slenderness Ratio, KL/r Flexural Buckling

Stress inLog Scale, log F

n Eq. (5) Eq. (13) A (C₀, logF_y) B ( C₁, log F₁ ) log F1 C₁ C₀ log F_y

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The parameters in the above equation can be obtained from the flexural buckling stress

equation in Eq. (2) and the tangent modulus equation in Eq. (3). As shown inFig. 4, the

slenderness ratio of C0in Eq. (5) is determined when Fnis equal to Fy. That is

FnZ FyZ p2Ey ðKL=rÞ2 (6) r ðKL=rÞ2Z p2_E y Fy Let C0Z KL=r Z p ffiffiffiffiffiffi Ey Fy s ðfor FnZ FyÞ (7)

where Eyis the tangent modulus at yield strength level and is equal to

EyZ

Eo

1 C 0:002 nEo

Fy

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The buckling stress F1given in Eq. (5) is obtained from Eq. (3) by rearranging Etand Fn

and replacing Fnby F1as follows:

F1Z EoKEt Et _F y 0:002nEo 1=nK1 !Fy (9) Let a ZEoKEt Et Z Eo Et K1 (10)

Then, Eq. (9) becomes

F1Z

aFy

0:002nE0

1_=nK1

!Fy (11)

The value of F1can be considered the proportional limit of stainless steels, which varies

with respect to the type of stainless steels.

Based on Eq. (10), the tangent modulus Etcan be expressed in terms of a

EtZ

Eo

ð1 C aÞ (12)

Substitution of Eq. (12) into Eq. (2) yields the following general expression for the flexural buckling stress: FnZ p2Et ðKL=rÞ2Z p2Eo C2_{ð1 C aÞ} (13)

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Then, for the stress level at FnZF1, the limiting slenderness ratio of C1can be obtained as follows: C1Z p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eo F1ð1 C aÞ s ðfor FnZ F1Þ (14)

The value of C1can be considered a limiting slenderness ratio of column buckling as

shown inFig. 4. When the KL/r ratio is greater than this limiting slenderness ratio, the

column buckling stress is calculated by Eq. (13); and when the KL/r ratio is smaller than this limiting ratio, the column buckling stress is determined by Eq. (5).

3.3. Approximating expressions

The parameter a plays an important role in finding the buckling stress F1in Eq. (11) and

the limiting slenderness ratio C1in Eq. (14). When the a value is known, the determination

of buckling stress Fnin Eqs. (5) and (13) becomes straight forward without iterative

calculations. This section presents an approximating technique to evaluate the parameter a used in the equations of the proposed simplified approach.

The tangent modulus in Eq. (3) can be rewritten as

EtZ EoFy FyC0:002nEoðFn=FyÞnK1 Z Eo 1 C 0:002n Eo Fy _F n Fy nK1 (15)

By comparing Eqs. (12) and (15), the parameter a can be expressed as

a Z 0:002n E_Fo y _F n Fy nK1 (16)

Substituting the value of Fnin Eq. (13), Eq. (16) can be rewritten as

a Z 0:002n _FEon y _p2_E o C2_{ð1 C aÞ} nK1 (17) From the above equation, it is noted that the parameter a can be combined to form a new polynomial function as

f ðaÞ Z að1 C aÞnK1Z 0:002n p

2 C2 nK1 Eo Fy n (18) Eq. (18) is a function of a with degree of n (nZconstant coefficient, used for determining tangent modulus as specified in ASCE Standard). Due to the complexity of the function, approximating polynomial method is used to solve this equation. Therefore, Eq. (18) can be approximately expressed by using Taylor series expansion as follows:

að1 C aÞnK1ZX N iZ0 fiðaÞ i! a i C/ (19)

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The correlation between both sides of Eq. (19) is carefully evaluated in order to obtain a better approximated solution. Higher degrees of derivatives are neglected as common engineering practice. Then, Eq. (19) can be approximately expressed as, for NZ2,

að1 C aÞnK1ya C ðnK1Þa2 (20a)

and, for NZ3,

að1 C aÞnK1ya C ðnK1Þa2C

ðnK1ÞðnK2Þ

2 a

3 _(20b)

Eqs. (20a) and (20b) are numerically evaluated for two specified n values as given in

Tables 3 and 4. These calculated values contain f(a)Za(1Ca)nK1,aC(nK1)a2, and aC

(nK1)a2C(nK1)(nK2)a3/2 for a values varies from 0 to 0.2 at intervals of

one-hundredth. As compared with the actual values of f(a), the approximating values calculated from Eq. (20b) provide better agreement than those calculated from Eq. (20a). It is an obvious outcome for this typical series expansion, i.e. the more expanded terms are used; the better approximation can be achieved. In this study, Eqs. (20a) and (20b) are separately used to determine the a value for specified column buckling mode.

3.4. Numerical approximation

It is well known that, for columns having larger slenderness ratio, the elastic buckling failure mode frequently controls the column strength. This buckling stress can be

Table 3

Comparisons of calculated values of Eqs. (20a) and (20b) for nZ4.58

a f(a)Za(1Ca)nK1 _aC(nK1)a2 aC ðnK1Þa2C ððnK1Þ! ðnK2Þ=2Þa3 0.00 0.00000 0.00000 0.00000 0.01 0.01036 0.01036 0.01036 0.02 0.02147 0.02143 0.02147 0.03 0.03335 0.03322 0.03335 0.04 0.04603 0.04573 0.04602 0.05 0.05954 0.05895 0.05953 0.06 0.07392 0.07289 0.07389 0.07 0.08919 0.08754 0.08913 0.08 0.10538 0.10291 0.10528 0.09 0.12253 0.11900 0.12236 0.10 0.14066 0.13580 0.14042 0.11 0.15983 0.15332 0.15946 0.12 0.18005 0.17155 0.17953 0.13 0.20136 0.19050 0.20065 0.14 0.22379 0.21017 0.22284 0.15 0.24739 0.23055 0.24614 0.16 0.27219 0.25165 0.27056 0.17 0.29823 0.27346 0.29615 0.18 0.32554 0.29599 0.32293 0.19 0.35417 0.31924 0.35091 0.20 0.38415 0.34320 0.38015

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determined by using Eq. (13) with a relatively small value of a. As noted inTables 3 and 4,

the approximating values of aC(nK1)a2calculated in Eq. (20a) are very close to the

function values of f(a)Za(1Ca)nK1 when a value becomes smaller. Therefore, in this

case, Eq. (20a) is used to calculate the a value for numerical approximation. Thus, it becomes a C ðnK1Þa2Z 0:002n p2 C2 nK1 Eo Fy n (21) This equation is a typical second order equation and can be solved by quadratic formula

a Z K1 C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 C 4ðnK1Þ0:002n p2 C2 nK1 Eo Fy n r 2ðnK1Þ (22)

This a is used for determining the buckling stress in Eq. (13) as shown inFig. 4.

On the other hand, a relatively large a value is found to be needed for calculating the buckling stress in the inelastic buckling mode. Because more accurate approximation is needed for this case, Eq. (20b) is used to determine the value of a. However, this equation cannot be solved with a simple formula. A reasonable estimation is needed to minimize the error of the numerical approximation. It is tentatively recommended that the maximum error between the approximating and actual values be limited to G0.3%. To meet this convergence requirement, the following limitation is required:

Table 4

Comparisons of calculated values of Eqs.(20a) and (20b) for nZ4.76

a f(a)Za(1Ca)nK1 aC(nK1)a2 aC ðnK1Þa2_{C ððnK1Þ!}

ðnK2Þ=2Þa3 0.00 0.00000 0.00000 0.00000 0.01 0.01038 0.01038 0.01038 0.02 0.02155 0.02150 0.02155 0.03 0.03353 0.03338 0.03352 0.04 0.04636 0.04602 0.04635 0.05 0.06007 0.05940 0.06005 0.06 0.07470 0.07354 0.07466 0.07 0.09028 0.08842 0.09020 0.08 0.10685 0.10406 0.10672 0.09 0.12444 0.12046 0.12424 0.10 0.14310 0.13760 0.14279 0.11 0.16286 0.15550 0.16240 0.12 0.18376 0.17414 0.18311 0.13 0.20583 0.19354 0.20494 0.14 0.22913 0.21370 0.22793 0.15 0.25370 0.23460 0.25211 0.16 0.27956 0.25626 0.27751 0.17 0.30678 0.27866 0.30416 0.18 0.33539 0.30182 0.33209 0.19 0.36544 0.32574 0.36133 0.20 0.39696 0.35040 0.39191

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ðnK1ÞðnK2Þa3₌₂

a C ðnK1Þa2 %5% (23)

Assume that the maximum value of the parameter a determined from Eq. (23) is equal to b. It yields

amaxZ b Z

1 Cqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 C₀_:05ðnK1Þ2ðnK2Þ

nK2 !0:05 (24)

in which b is a constant, which depends only on the n value for different types of stainless steels. This b value is the maximum value of a and is used to calculate the buckling stress

of F1in Eq. (11) and the limiting slenderness ratio of C1in Eq. (14).

4. Proposed design formulas

The flexural buckling stress of cold-formed stainless steel compression members can be determined based on the above-mentioned simplified approach. By using the proposed formulas, design calculation is no longer tedious because no iterative process is needed. The following design provisions are proposed herein to determine the flexural buckling

stress, Fn, for austenitic types of cold-formed stainless steel compression members.

For doubly symmetric sections, closed cross sections, and any other sections which can be shown not to be subjected to torsional or torsional–flexural buckling, the flexural

buckling stress, Fn, shall be determined as follows:

For KL/r%C1: FnZ F l_o yF l1 1 %Fy (25) For KL/rOC1: FnZ p2Eo KL r 2 ð1 C aÞ (26) where: loZ C1KKL=r C1KCo (27) l₁_{Z 1Kl}_o (28) CoZ p ffiffiffiffiffiffi Ey Fy s (29) C1Z p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eo F₁ð1 C bÞ s (30)

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EyZ Eo 1 C 0:002nEo Fy (31) F1Z Fy bFy 0:002nEo 1=nK1 (32) a Z K1 C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 C 4ðnK1Þ0:002n _ðKLp2_=rÞ2 h inK1 Eo Fy n r 2ðnK1Þ (33) b Z 0:05 Cqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:0025 C0:1ðnK2Þ_ðnK1Þ nK2 (34)

5. Comparisons and discussions

In this study, comparisons are made between the predicted flexural buckling stresses of cold-formed stainless steel columns obtained from the ASCE Standard design equations and the proposed simplified design formulas. Eqs. (2) and (3) are used for the ASCE Standard predictions. The proposed design formulas provided in Section 4 of the paper are used to determine the flexural buckling stresses. It shows that, without having iterative calculations, the proposed equations give satisfactory predictions as compared with the ASCE Standard results.

A commonly used Type 304, austenitic stainless steel column is used for the comparison. The specified material properties of the column are previously given in

Table 2. The design parameters for the same materials determined from the proposed

design equations are listed inTable 5. For columns with this type of stainless steel, the

computed buckling stresses, Fn,ASCE and Fn,prop, and the ratios of Fn,prop/Fn,ASCE for

different slenderness ratios, KL/r, in longitudinal and transverse compression are given in

Tables 6 and 7, respectively. In these tables, Fn,ASCE and Fn,prop are predicted flexural

Table 5

Design parameters used in the proposed design formulas for type 304 stainless steels

Type of stress b C0 C1 F1(MPa)

Longitudinal compression Annealed 0.1500 32.8 176.6 53.12

1/16 Hard 0.1500 32.0 137.3 87.94

1/4 Hard 0.1252 29.9 115.0 123.48

1/2 Hard 0.1429 30.2 98.4 165.91

Transverse compression Annealed 0.0526 23.2 136.1 97.72

1/16 Hard 0.0526 22.9 108.2 154.55

1/4 Hard 0.1179 27.8 80.5 259.62

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buckling stresses determined from the ASCE Standard and proposed design equations, respectively.

Based on the results of comparison, it is observed that the proposed design formulas predict slightly larger buckling stresses than those determined by the ASCE design provisions for Type 304 annealed and 1/16 Hard stainless steel columns having medium

range of KL/r. The maximum ratio of Fn,prop/Fn,ASCEfor annealed type is equal to 1.11 at

KL/rZ80, and for 1/16 Hard, the maximum ratio of Fn,prop/Fn,ASCEis equal to 1.07 at

KL/rZ60 as given inTable 6. These larger predictions of Fn,propfor annealed and 1/16

Hard types are mainly due to the relative low values of Fy and n specified in ASCE

Standard. However, for 1/4 Hard and 1/2 Hard, the proposed design formulas can provide good predictions as compared with those obtained from the ASCE design equations. Their

Table 6

Comparisons of flexural buckling stresses for type 304 stainless steel columns in longitudinal compression

KL/r Fn,ASCE (MPa) Fn,prop (MPa) Fn,prop/ Fn,ASCE Fn,ASCE (MPa) Fn,prop (Mpa) Fn,prop/ Fn,ASCE Fn,ASCE (MPa) Fn,prop (MPa) Fn,prop/ Fn,ASCE Fn,ASCE (MPa) Fn,prop (MPa) Fn,prop/ Fn,ASCE 20 193.1 193.1 1.00 282.7 282.7 1.00 344.4 344.5 1.00 448.2 448.2 1.00 40 173.4 181.0 1.04 249.1 258.6 1.04 296.3 305.4 1.03 378.5 388.4 1.03 60 137.4 151.2 1.10 193.7 207.2 1.07 232.1 239.9 1.03 283.4 290.3 1.02 80 114.2 126.4 1.11 156.7 166.0 1.06 186.0 188.4 1.01 215.6 217.0 1.01 100 96.9 105.6 1.09 128.4 133.0 1.04 148.2 148.0 1.00 162.2 161.5 1.00 120 82.9 88.3 1.07 105.0 106.5 1.01 116.0 115.7 1.00 121.3 121.3 1.00 140 71.0 73.8 1.04 85.5 85.1 1.00 90.1 90.1 1.00 91.8 91.8 1.00 160 60.7 61.7 1.02 69.4 69.4 1.00 70.6 70.6 1.00 71.1 71.1 1.00 180 51.7 51.5 1.00 56.7 56.6 1.00 56.3 56.3 1.00 56.5 56.5 1.00 200 44.0 43.9 1.00 46.7 46.7 1.00 45.8 45.8 1.00 45.8 45.8 1.00 AVG 1.05 1.02 1.01 1.01 COV 0.041 0.027 0.013 0.011 Table 7

Comparisons of flexural buckling stresses for Type 304 stainless steel columns in transverse compression

KL/r Fn,ASCE (MPa) Fn,prop (MPa) Fn,prop/ Fn,ASCE Fn,ASCE (MPa) Fn,prop (MPa) Fn,prop/ Fn,ASCE Fn,ASCE (MPa) Fn,prop (MPa) Fn,prop/ Fn,ASCE Fn,ASCE (MPa) Fn,prop (MPa) Fn,prop/ Fn,ASCE 20 206.9 206.9 1.00 310.3 310.3 1.00 613.7 613.7 1.00 827.4 827.4 1.00 40 180.2 185.1 1.03 267.4 269.8 1.01 499.8 504.4 1.01 641.0 641.6 1.00 60 160.3 162.0 1.01 234.3 229.2 0.98 368.1 364.9 0.99 434.9 431.6 0.99 80 144.8 141.9 0.98 204.9 194.6 0.95 264.4 264.0 1.00 284.1 284.0 1.00 100 130.0 124.2 0.96 171.0 165.3 0.97 184.6 184.5 1.00 188.5 188.5 1.00 120 113.6 108.8 0.96 130.5 130.5 1.00 131.2 131.2 1.00 131.9 131.9 1.00 140 93.7 93.6 1.00 97.1 97.1 1.00 97.0 97.0 1.00 97.1 97.1 1.00 160 74.0 74.0 1.00 74.4 74.4 1.00 74.4 74.4 1.00 74.4 74.4 1.00 180 58.8 58.8 1.00 58.8 58.8 1.00 58.8 58.8 1.00 58.8 58.8 1.00 200 47.6 47.6 1.00 47.7 47.7 1.00 47.6 47.6 1.00 47.6 47.6 1.00 AVG 0.99 0.99 1.00 1.00 COV 0.023 0.019 0.004 0.003

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maximum ratio of Fn,prop/Fn,ASCE is equal to 1.03 as given in Table 6. The computed

flexural buckling stresses versus slenderness ratios for Type 304 stainless steel columns, annealed, 1/16 Hard, 1/4 Hard and 1/2 Hard in longitudinal compression, are also shown in

Fig. 5.

Table 7 gives predicted flexural buckling stresses and their corresponding stress ratios for Type 304 stainless steels columns in transverse compression. It is found that the proposed design equations provide good agreements with the ASCE design formulas

for annealed, 1/16 Hard, 1/4 Hard and 1/2 Hard types. The lowest ratio of Fn,prop/

Fn,ASCEfor annealed type is equal to 0.96 at KL/rZ100 and 120, and for 1/16 Hard, the

lowest ratio of Fn,prop/Fn,ASCEis equal to 0.95 at KL/rZ80 as given inTable 7. For 1/4

Hard and 1/2 Hard, it is noted that the proposed design formulas provide good

predictions as compared with those obtained from the ASCE design equations. Fig. 6

shows the predicted flexural buckling stresses, Fn, versus the slenderness ratio, KL/r, for

Type 304 stainless steel columns, annealed, 1/16 Hard, 1/4 Hard and 1/2 Hard in transverse compression.

To demonstrate the adequacy of proposed design formulas discussed above, a design example is included in the appendix of the paper. It provides detailed calculations for determining a cold-formed stainless steel column buckling strength by using the ASCE Standard design equations and the proposed simplified formulas.

n

(MPa)

Type 304 Stainless Steel Columns (Longitudinal Compression) Proposed (Annealed) Proposed (1/16Hard) Proposed (1/4Hard) Proposed (1/2Hard) ASCE (Annealed) ASCE (1/16Hard) ASCE (1/4Hard) ASCE (1/2Hard)

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6. Conclusions

For the design of cold-formed stainless steel compression members, the flexural buckling stress is determined based on the tangent modulus theory. Because of the nonlinear stress-strain behavior of the materials, the determination of flexural buckling stress usually requires iterative process, which is often tedious and time-consuming for a typical column design. In order to simplify the iterative calculation, newly developed formulas utilizing the Taylor series expansion theory are proposed herein to determine flexural buckling stress for austenitic type of cold-formed stainless steel columns. This paper presents detailed derivations of the mathematical formulation and numerical approximation. Comparisons are made between the predicted column flexural buckling stresses determined from he ASCE design formulas and the proposed design equations. It shows that the predicted flexural buckling stresses determined by the proposed design equations are in good agreement with those calculated by the ASCE design formulas. The proposed equations can be used as an alternative to determine the flexural buckling stress for austenitic type of cold-formed stainless steel columns.

Acknowledgements

The financial support provided by the Minghsin University of Science and Technology through the Project Number MUST 93-CE-006 is gratefully acknowledged.

Flexural Buckling Stress,F

n

(MPa)

Type 304 Stainless Steel Columns (Transverse Compression) Proposed (Annealed) Proposed (1/16 Hard) Proposed (1/4Hard) Proposed (1/2Hard) ASCE (Annealed) ASCE (1/16 Hard) ASCE (1/4 Hard) ASCE (1/2 Hard)

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Appendix. Design example

The following design example is to determine the design axial strength of a cold-formed stainless steel column subjected to flexural buckling. The flexural buckling stress used to calculate the design axial strength is determined by the following two methods: (A) the design equations specified in the ASCE Standard Specification, and (B) the proposed simplified design approach presented in the paper.

Given:

1. Square tube dimensions: 101.6 mm!101.6 mm!1.65 mm as shown in Fig. 7.

2. Sectional properties: AZ652.9 mm2, IxZIyZ1,081,369 mm4, rxZryZ40.7 mm.

3. Material properties: Type 304, 1/4-Hard stainless steel, FyZ344.8 MPa, EoZ186,

200 MPa and nZ4.58 in longitudinal compression.

4. Effective length: KxLxZKyLyZ3048 mm; and KxLx/rxZKyLy/ryZ74.9.

The sectional properties of square tubular section given above are determined by using the information given in Part I of the AISI Cold-Formed Steel Design Manual

(2002) [11]. Solution: 101.6mm 101.6mm 1.65mm R=1.59mm w= 95.1mm Sectional Properties: A = 652.9 mm2 I_x = I_y = 1081369 mm4 r_x = r_y = 40.7 mm Effective Length:KxLx= KyLy = 3048 mm

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(A). The ASCE standard specification

The square tube is a doubly symmetric closed section, which is not subjected to

torsional or torsional-flexural buckling. The flexural buckling stress, Fn, can be determined

by Equation 3.4.1K1 of the ASCE Standard as given in Eq. (2) above. The tangent

modulus Etused for this example is determined by using the Modified Ramberg–Osgood

equation as given in Eq. (3). Due to the nature of the iterative process, try-and-error

calculations are necessary to determine the buckling stress, Fn.

For the first try, assume a buckling stress of FnZ230 MPa and its corresponding

tangent modulus in Eq. (3) is calculated as

EtZ ð186200 !344:8Þ=½344:8 C 0:002 !4:58 !186200 !ð230=344:8Þ

3:58

y86167 MPa

Thus, the computed buckling stress in Eq. (2) is equal to ðFnÞ1Z ðp

2_!86167Þ_=ð74:9Þ2

Z 151:6 MPa! assumed buckling stress 230 MPa ðN:G:Þ

Because the computed buckling stress is not close enough to the assumed value, further iteration is needed.

For the second try, it is assumed that FnZ191 MPa. The corresponding tangent

modulus becomes

EtZ ð186200 !344:8Þ=½344:8 C 0:002 !4:58 !186200 !ð191=344:8Þ3:58

y116599 MPa

The computed buckling stress is equal to ðFnÞ2Z ðp

2

!116599Þ=ð74:9Þ2Z 205:1 MPaO assumed buckling stress 191 MPa

ðN:G:Þ

For the third try, it is assumed that FnZ198.1 MPa. The corresponding tangent modulus is

calculated as follows:

EtZ ð186200 !344:8Þ=½344:8 C 0:002 !4:58 !186200 !ð198:1=344:8Þ3:58

y110817 MPa The computed buckling stress is

ðFnÞ3Z ðp 2

!110817Þ=ð74:9Þ2Z 195:0 MPa! assumed stress 198:1 MPaðN:G:Þ

The buckling stress is repeatedly assumed until it reaches the minimum convergence to the

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196.8 MPa is obtained, and the tangent modulus is calculated as

EtZ ð186200 !344:8Þ=½344:8 C 0:002 !4:58186200 !ð196 :8=344:8Þ3:58

y111872 MPa

Thus, the buckling stress is equal to

FnZ ðp

2

!111872Þ=ð74:9Þ2Z 196:8 MPa ðO:K:Þ

To determine the effective area of the member, the design provisions of Section 2.2 in the ASCE Standard are used to calculate the effective widths of flanges and webs of the tubular section. Based on the results of calculation, it is found that the flanges and webs of the section are not fully effective. The effective area of the tubular section is calculated in accordance with the ASCE Standard provisions as follows:

k Z 4:0; f Z FnZ 196:8 MPa l Z 1:052ffiffiffi k p w t ffiffiffiffiffiffi_f Eo s ! Z 1:052= ffiffiffiffiffiffiffi4:0 p ð95:1=1:65Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi196:8=186200 p Z 0:986O 0:673 (A1) r Z1K0_l:22=lZ ð1K0:22=0:986Þ=0:986 Z 0:788 (A2) b Z rwZ 0:788 !95:1 Z 75:2 mm (A3) AeZ AK4ðwKbÞt Z 652:9K4ð95:1K75:2Þ !1:65 Z 521:6 mm2

The design axial strength, fcPn, is determined from Eq. (1) as

fcZ 0:85; PnZ Ae FnZ 521:6 !196:8 Z 102651 N Z 102:7 kN

ðfcPnÞASCEZ 0:85 !102:7 Z 87:3 kN

(B). The proposed simplified approach

By using the proposed design formulas presented in Section 4 of this paper, the flexural buckling stress of cold-formed stainless steel column can be easily calculated without iterations.

(1) Calculate the design parameters:

Design parameters used in this example are calculated from the proposed formulas as follows:

b Z

0:05 Cqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:0025 C0:1ð4:58K2Þ_ð4_:58K1Þ

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EyZ 186200 1 C 0:002!4:58!186200₃₄₄_:8 Z 31312 MPa CoZ p ffiffiffiffiffiffiffiffiffiffiffiffiffi 31312 344:8 r Z 29:9 F1Z 344:8 0:1252!344:8 0:002!4:58!186200 1_=4:58K1 Z 123:5 MPa C1Z p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 186200 123:5ð1 C0:1252Þ s Z 115

(2) Calculate the flexural buckling stress:

Since the column slenderness ratio KL/rZ74.9!C1, the flexural buckling stress is

determined from Eq. (25) as follows:

FnZ F lo yF l1 1 %Fy where: loZ C1KKL=r C1KCo Z115K74:9 115K29:9Z 0:471; and l1Z 1 loZ 0:529 Thus, FnZ 344:8 0_:471 !123:50:529Z 200:3 MPa

(3) Determine the effective area:

The effective area of the tubular section is calculated as follows:

k Z 4:0; f Z FnZ 200:3 MPa l Z 1:052ffiffiffi k p _w t ffiffiffiffiffiffif Eo s ! Z 1:052= ffiffiffiffiffiffiffi4:0 p ð95:1=1:65Þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi200:3=186200 Z 0:994O 0:673 r Z1K0:22=l l Z ð1K0:22=0:994Þ=0:994 Z 0:783 b Z rw Z 0:783 !95:1 Z 74:5 mm AeZ AK4ðwKbÞt Z 652:9K4ð95:1K74:5Þ !1:65 Z 516:9 mm 2

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(4) Find the design axial strength:

The design axial strength, fcPn, is determined as follows:

fcZ 0:85; PnZ Ae FnZ 516:9 !200:3 Z 103535 N Z 103:5 kN

ðfPnÞprop:Z 0:85 !103:5 Z 88:0 kN

This design example provides detailed calculations to determine the design axial strength for a typical cold-formed stainless steel tubular column. Two methods are used in this example: (A) the ASCE Standard design equations and (B) the proposed design formulas presented herein. There is no iterative calculation used in the proposed design formulas. The flexural buckling stress calculated from the proposed design formulas is easier than that determined from ASCE Standard design equations. For this design example, the design axial strength determined by using the proposed design formulas is 0.8% larger than that found by using the design equations stipulated in the ASCE Standard.

References

[1] Specification for the design of cold-formed stainless steel structural members, (SEI/ASCE 8-02), American Society of Civil Engineers (ASCE); 2002.

[2] Specification for the design of cold-formed stainless steel structural members, (ANSI/ASCE 8-90), American Society of Civil Engineers (ASCE); 1991.

[3] Johnson AL, Kelsen GA. Stainless steel in structural applications, stainless steel for architecture. American Society for Testing and Materials, STP 454; Aug. 1969.

[4] Yu WW. Cold-formed steel design. New York, NY: Wiley–Interscience; 2000.

[5] Johnston BG, editor. Guide to stability design criteria for metal structures. 3rd ed. New York, NY: Wiley; 1976.

[6] Galambos TV. Structural members and frames. Englewood Cliffs, NJ: Prentice-Hall; 1968.

[7] Ramberg W, Osgood WR. Description of stress-strain curves by three parameters. NACA technical note, No. 902; July 1943.

[8] Hill HN. Determination of stress–strain relations from offset yield strength values. NACA tech. note, No. 927; Feb 1944.

[9] Rasmussen KJR, Rondal J. Column curves for stainless steel alloys. J Construct Steel Res 2000;54:89–107. [10] Rasmussen KJR, Rondal J. Explicit approach to design of stainless steel columns. J Struct Eng Am Soc Civil

Eng 1997;123(7):857–63.