On North American and Chinese Standards for Design of Coldformed Steel Csection Compressive Members

2014-08-08 08:42:33ZHOUXuhongYUANXi

建筑科學與工程學報 2014年1期

ZHOU+Xu+hong+YUAN+Xiao+li+XU+Lei+

建筑科學與工程學報2014年文章編號：16732049(2014)01000115

Received date：20131128

Biography：ZHOU Xuhong（1956）， male, professor, doctoral advisor, academician of CAE, PhD, Email:zhouxuhong@126.com.On North American and Chinese Standards for Design of Coldformed Steel Csection

Abstract:Nominal axial compressive strengths of coldformed steel Csections evaluated by the North American standard CSA S13607 and the Chinese standard GB 50018—2002 were investigated. The procedures of evaluating the nominal axial compressive strength associated with both standards were analyzed and compared. The study results show that discrepancies between the two standards are primarily resulted from the difference in evaluating the effective area subjected to local buckling. For the Csection compressive members, the flange effective width calculated by the Chinese standard is much smaller than that of the North American standard, whereas the web effective width evaluated by the North American standard is slightly less than that of the Chinese standard. For typical Csection wall studs, the difference on the nominal axial strength is primarily influenced by the flange and web widthtothickness ratios. When the flange widthtothickness ratio is not less than 17.8, the difference on the nominal axial compressive strength is dominated by the difference of flange effective width between the two standards and the nominal axial compressive strength evaluated by GB 50018—2002 is less than that of CSA S13607; when the flange widthtothickness ratio is less than 17.8, the difference on the nominal axial compressive strength is then primarily governed by the difference of web effective width between the two standards and the nominal axial compressive strength evaluated by GB 50018—2002 is slightly greater than that of CSA S13607.

Key words:coldformed steel; Csection member; nominal axial compressive strength; flexural buckling; lateraltorsional buckling; effective width; buckling coefficient

CLC number：TU375.4Document code：A

北美規(guī)范與中國規(guī)范關于冷彎薄壁型鋼C形截面受壓構(gòu)件設計的比較周緒紅1，苑小麗2，徐磊2，劉永健3，劉競楠2

（1. 重慶大學土木工程學院, 重慶400045; 2. 滑鐵盧大學土木與環(huán)境工程學院，安大略滑鐵盧N2L 3G1;

3.長安大學公路學院, 陜西西安710064）摘要:對比了北美規(guī)范CSA S13607和中國規(guī)范GB 50018—2002中關于冷彎薄壁型鋼C形截面軸壓構(gòu)件的名義軸壓強度。首先介紹了北美規(guī)范和中國規(guī)范計算名義軸壓強度的方法，然后針對控制構(gòu)件名義軸壓強度的2個主要參數(shù)，即屈曲應力和有效截面面積，對2本規(guī)范進行了深入對比，最后對典型C形墻架柱名義軸壓強度進行了比較。研究結(jié)果表明：2本規(guī)范具有相同的屈曲應力，但依據(jù)2本規(guī)范計算的有效截面面積卻不同；一般來說，根據(jù)GB 50018—2002計算的翼緣有效寬度遠小于根據(jù)CSA S13607計算的結(jié)果，然而依據(jù)CSA S13607計算的腹板有效寬度則略小于依據(jù)GB 50018—2002計算的結(jié)果；2本規(guī)范名義軸壓強度不同主要由C形截面翼緣和腹板有效寬厚比不同引起；當翼緣的寬厚比不小于17.8時，構(gòu)件名義軸壓強度的不同主要由翼緣有效寬厚比控制，根據(jù)GB 50018—2002計算的名義軸壓強度小于根據(jù)CSA S13607計算的結(jié)果；當翼緣的寬厚比小于17.8時，構(gòu)件名義軸壓強度的不同則主要受腹板有效寬度控制，依據(jù)GB 50018—2002計算的名義軸壓強度略大于依據(jù)CSA S13607計算的結(jié)果。

關鍵詞:冷彎薄壁型鋼；C形截面構(gòu)件；名義軸壓強度；彎曲屈曲；彎扭屈曲；有效寬度；屈曲系數(shù)

0Introduction

Csection is the most widely used section shape in coldformed steel framing construction. A typical application of Csections as compressive members is the load bearing wall studs. In North America, procedures of designing coldformed steel members are specified in CSA S13607［1］. In China, the design procedures of coldformed steel members concerning with local buckling, flexural buckling and lateraltorsional buckling strength are stipulated in GB 50018—2002［2］ while the procedure for evaluating the distortional buckling is specified in standard JGJ 227—2011［3］. Although theoretical basis for evaluating the compressive strength of coldformed steel Csection members are similar in the North American and the Chinese standards, the differences still exist. The primary objectives of this study are to identify the differences between the standards CSA S13607 and GB 50018—2002 on evaluating the nominal compressive strength of coldformed steel Csection members and to investigate how the nominal compressive strength is affected by the differences.

In the paper, the procedures associated with the two standards for evaluating the nominal compressive strength of coldformed steel Csection members are discussed. Then, two key parameters used for determining the nominal compressive strength of coldformed steel members, the buckling stress and the associated effective width, are compared, respectively. Finally, the differences in the nominal axial compressive strength between the two standards are investigated for the typical Csection load bearing wall studs.1Expression of Nominal Axial Compressive StrengthCsection members is shown in Fig.1. In Fig.1, ww is flat portion of the web; wf is flat portion of the flange; b0 is the outertoouter dimension of the flange; h0 is outertoouter depth of the Csection members; x0 is the distance from shear center to centroid along principal axis; d is flat portion of the stiffener; D is height of the stiffener; R is inside bend radius; r is centerline bend radius. Assumptions made for the comparisons on the nominal axial compressive strength of Csection members shown in Fig.1 are as follows: ① the axial compressive load is applied through the centroid of the Csection members; ② there are no holes in the Csection members; ③ distortional buckling isFig.1Profile of Csection Member

圖1C形截面構(gòu)件剖面 not considered; ④ the yield stress fy of the steel is either fy=345 MPa or fy=235 MPa. In CSA S13607, the equation to calculate the nominal axial strength Pn of a compressive member is

Pn=fnAe(1)

where fn is the nominal stress calculated based on the flexural and lateraltorsional buckling［46］; Ae is the effective area associated with the nominal stress fn.

On the other hand, in GB 50018—2002, although the nominal axial strength Pn is not explicitly expressed, the standard provides the following equation to check member stability as

NφAe≤f(2)

where N is the factored load; f is the design strength; φ is the stability coefficient; Ae is the effective area calculated at the stress φf.

To obtain the equivalent nominal axial strength Pn based on GB 50018—2002, Eq.(2) can be rewritten as

Pn=φfyAe(3)

Because the stability coefficient φ is a stress reduction factor which accounts for the flexural and lateraltorsional buckling of the compressive member, the products of φ and fy in Eq.(3) can be considered as the equivalent to the nominal stress fn in Eq. (1) as both of them are stresses calculated based on the flexural and lateraltorsional buckling.

Comparing Eq. (1) to Eq. (3), it can be seen that the two standards are similar while calculating the nominal compressive strength with an expression of the strength in terms of the product of the nominal stress fn (or φfy) and the effective area Ae. The nominal stress fn or φfy is evaluated based on the flexural and lateraltorsional buckling, and the effective area Ae is obtained with the consideration of the local buckling at the stress levels fn or φf. In order to compare the nominal compressive strengths Pn between the two standards, the procedures of evaluating the flexural and lateraltorsional buckling stress and the effective area in each standard are needed to be investigated.2Flexural and Lateraltorsional Buckling StressIn CSA S13607, the nominal stress fn is calculated as follows

fn=0.658λ2cfyλc≤1.5

0.877λ2cfyλc＞1.5(4)

where λ is slenderness factor, λ=fy/fe, fe is the least of the applicable elastic flexural buckling stress σe and flexuraltorsional buckling stress σew.

The stresses σe and σew are defined as follows

σe=π2E(KL/r)2(5)

σew=12β［σex+σt-(σex+σt)2-4βσexσt］(6)

KL/r=max{KxLx/rx,KyLy/ry}(7)

where E is the elastic modulus, E=203 GPa in CSA S13607; KL/r is the maximum of the flexural slenderness ratios about the x and y axes; Kx and Ky, Lx and Ly, and rx and ry are effective length factors, unbraced lengths and radii of gyration of fully unreduced cross section about the x and y axes, respectively; σex is the elastic flexural buckling stress about the x axis, σex=π2E/(KxLx/rx)2; β is a parameter related with the geometry of the cross section.

β is evaluated as

β=1-(x0/r0)2(8)

where r0 is the polar radius of gyration, r0=r2x+r2y+x20; σt is the elastic torsional buckling stress.

σt is calculated as

σt=1Ar20［GJ+π2ECw(KtLt)2］(9)

where A is the gross area of the section; G is the shear modulus, G=78 GPa in CSA S13607; Cw is the warping constant; J is torsional constant; Kt and Lt are the effective length factor and unbraced length for twisting, respectively.

In GB 50018—2002, tabulated values of the stability coefficient φ, which is equivalent to the ratio fn/fy in CSA S13607, are listed in Appendix A based on the maximum slenderness ratio KL/r which is defined as

KL/r=max{KxLx/rx,KyLy/ry,(KL/r)t}(10)

where (KL/r)t is equivalent lateraltorsional buckling slenderness ratio.

It is noted that the slenderness ratio KL/r in GB 50018—2002 (Eq.10) is a little different from Eq.(7) defined in CSA S13607. In CSA S13607, KL/r is the maximum of the flexural buckling slenderness ratios KxLx/rx and KyLy/ry, whereas in GB 50018—2002, it is the maximum of the flexural buckling slenderness ratios KxLx/rx, KyLy/ry and the equivalent lateraltorsional buckling slenderness ratio (KL/r)t. The equivalent lateraltorsional buckling slenderness ratio (KL/r)t is defined as

(KL/r)t=KxLxrxs2+i202s2+(s2+i202s2)2-i20-x20s2(11)

s2=(KxLx/rx)2ACwL2t+0.039G(12)

i20=(KxLxrx)2+(KyLyry)2+x20(13)

If the equivalent lateraltorsional buckling slenderness ratio (KL/r)t in Eq. (11) is substituted into Eq.(5), which is used to calculate the elastic flexural buckling stress σe, the resulting equation will be the same as Eq.(6). Since Eq.(6) is used to evaluate the elastic lateraltorsional buckling stress σew in CSA S13607, it can be concluded that the equivalent lateraltorsional buckling slenderness ratio (KL/r)t in GB 50018—2002 can also be used to calculate the elastic lateraltorsional buckling stress σew in CSA S13607. Therefore, in the following comparisons, the slenderness ratio KL/r is the maximum of the flexural slenderness ratios KxLx/rx, KyLy/ry and the equivalent lateraltorsional buckling slenderness ratio (KL/r)t. In addition, the nominal stress fn in CSA S13607 is only calculated based on the elastic flexural buckling stress σe in Eq.(5). Furthermore, the value of the stability coefficient φ is obtained from Appendix A of GB 50018—2002 based on the given value of KL/r.

The stability coefficient φ from Appendix A of GB 50018—2002 and ratio fn/fy calculated based on CSA S13607 for steel with yield stress fy =345 MPa and fy =235 MPa are presented in Fig.2. As it can be seen from Fig.2, the differences between the stability coefficient φ and ratio fn/fy for both fy=345 MPa and fy=235 MPa steel are not significant. For steel with fy =345 MPa and fy=235 MPa, the differences of stability coefficient φ in GB 50018—2002 and ratio fn/fy of CSA S13607 range from -3% to 5% and -9% to 0%, respectively. Therefore, it is concluded that the factored stress φfy in GB 50018—2002 and the nominal strength fn in CSA S13607 can be considered as equivalent.

Fig.2Comparisons of Buckling Stresses

圖2屈曲應力的比較3Procedure of Element Effective Width EvaluationSince the procedures of evaluating the effective width of the crosssectional elements in the two standards are different, it is not convenient to compare them directly. Therefore, the procedure described in GB 50018—2002 is rewritten equivalently to make the procedures of the two standards to be consistent and comparable. The comparison of the effective width calculation procedure in CSA S13607 and the rewritten effective width calculation procedure in GB 50018—2002 is shown in Tab.1. Since the different notations are used to express the dimensions of the Csection, the dimensional notations are redefined as shown in Fig.1 for the reason of clarity. According to Tab.1, both standards have the similarity of calculating the effective width based on the actual widthtothickness ratio of the flat portion w/t, the plate buckling coefficient k, and the maximum stress σmax of the considered element. However, the differences still exist. The primary differences between the two standards on evaluating the element effective width of the Csections are:

(1) In step 1 shown in Tab.1, the maximum表1Comparison of Calculation Procedure on Effective Width in CSA S13607 and GB 50018—2002

Tab.1CSA S13607與GB 50018—2002中有效寬度計算過程的比較StepsCSA S13607GB 50018—2002Step 1Calculate the maximum stress σmax and minimum stress σmin for the considered elementCalculate the maximum stress σmax and minimum stress σmin for the considered elementStep 2Determine the plate buckling coefficient kDetermine the plate buckling coefficient k and buckling coefficient related to the connected element k1Step 3Calculate slenderness factor λ

λ=wtσmax12(1-μ2)kπ2ECalculate slenderness factor λ

λ=wtσmax12(1-μ2)kk1π2EStep 4Evaluate local reduction factor ρ

ρ=1λ≤0.673

(1-0.22/λ)/λλ＞0.673Evaluate local reduction factor ρ

ρ=1 λ≤0.60α

0.72αλ-0.100.60α＜λ＜1.26α

0.83αλ λ≥1.26α

α=1.15-0.15Ψ≤1.15, Ψ=σminσmaxStep 5Calculate effective width be

be=ρwCalculate effective width be

be=ρbcStep 6Distribute effective widthDistribute effective widthNote: As the two standards use different notations, the notations are redefined as follows (Fig.1 and Fig.4):bc is compressed flat portion

of the element; t is thickness; μ is Poisson ratio, μ=0.3 for coldformed steel; positive value for compressive stress; E=203 GPa

in CSA S13607, E=206 GPa in GB 50018—2002.stresses to calculate the local buckling σmax defined in the two standards are different. In CSA S13607, the maximum stress for compressive members σmax=fn, whereas in GB 50018—2002, σmax=φf. According to the requirement of GB 50018—2002, f=300 MPa for fy=345 MPa and f=205 MPa for fy=235 MPa. As discussed in section 2, since φ is basically the same as the ratio fn/fy, it is concluded that the maximum stress σmax defined in GB 50018—2002 is approximately 87% of that specified in CSA S13607.

It is noted that in both standards, both the maximum stress σmax and minimum stress σmin for each element of the Csection, such as the web, flange and stiffener, are calculated based on the flexural and lateraltorsional buckling stresses of the member, not related to the actual applied load.

Since the maximum stresses specified in the two standards are different, to avoid confusion, the maximum stress σmax in the following discussion refers to the one defined in CSA S13607 unless otherwise indicated.

(2) In step 2 and step 3, the influence of the support condition on the effective width of each element in CSA S13607 is only represented by the plate buckling coefficient k. However, in GB 50018—2002, this influence is represented by the product of the plate buckling coefficient k and a modification coefficient k1. The modification coefficient k1 is introduced in GB 50018—2002 to explicitly account for the restraining effect of the connected element on the element under the consideration and it is calculated as

k1=0.11+0.93(ζ-0.05)2≤k1-limζ＞1.1

1ζ≤k1-limζ≤1(14)

ζ=cbkkc(15)

where c is the flat portion of the connected element; b is the flat portion of the element under the consideration; k and kc are the buckling coefficients of the element under the consideration and the connected element; k1-lim is the upper limit of k1, k1-lim=1.7 for stiffened elements, k1-lim=2.4 for partially stiffened elements, and 3.0 for unstiffened elements.

Comparisons between the plate buckling coefficient k in CSA S13607 and the product kk1 in GB 50018—2002 for stiffener, flange, and web element of the compressive Csection members are discussed in section 4.

(3) In step 4, the local reduction factor ρ in CSA S13607 is only associated with the slenderness factor λ. However, in GB 50018—2002, ρ is also related to the parameter α, a function of the stress distribution parameter Ψ as shown Tab.1.As shown in Fig.3(a), α ranges from 1 to 1.15: ① α=1 when the element is subjected to uniform compressive stress (Ψ=1); ② α=1.15 when the element is experienced to gradient compressive and tensile stress (Ψ≤0); ③ whereas the element is subjected to gradient compressive stress, α can be linearly interpolated by Ψ (0<Ψ<1).

Fig.3Comparisons of Local Reduction Factor ρ

圖3局部折減系數(shù)ρ的比較The influence of α on the local reduction factor ρ in GB 50018—2002 is shown in Fig.3(b). The local reduction factor ρ increases slightly with the increase of α. Although the two standards use different equations to calculate the local reduction factor ρ in step 4, the magnitude of the difference in the local reduction factor ρ is not significant: ① when α=1, the local reduction factor ρ in GB 50018—2002 is slightly smaller than that in CSA S13607; ② when α=1.15, the local reduction factor ρ in GB 50018—2002 is slightly larger than that in CSA S13607. Therefore, results obtained in step 4 in the two procedures can be considered as almost the same.

(4) In step 5, the effective width be is calculated based on the entire flat portion of the element w in CSA S13607, whereas in GB 50018—2002, the effective width be is calculated based on the compressed flat portion of the element bc (Fig.4). However, for compressive members, since there is only compressive stress in the considered element, bc and w are identical, as shown in Fig.4(a). Therefore, step 5 in both standards for compressive member is considered the same.

Fig.4Maximum and Minimum Stresses of

Csection Member

圖4C形截面構(gòu)件應力的最大值和最小值Fig.5Comparison of Effective Width Distribution

圖5有效寬度分布的比較(5) In step 6, the flange effective width distribution in CSA S13607 for a compressive member is different from that in GB 50018—2002. As shown in Fig.5, for the flange of Csection, be1 is effective width adjacent to the web whereas be2 is the effective width adjacent to the stiffener. In CSA S13607, be1=be-be2 and be2=0.5beRI, whereas in GB 50018—2002, be1=0.4be and be2=0.6be, as shown in Fig.5. The parameter RI is discussed in section 4.1. As RI≤1, be1≥be2 according to CSA S13607, whereas based on GB 50018—2002, be1

From the foregoing analysis, it can be concluded that the differences between the effective widths calculated by using CSA S13607 and GB 50018—2002 are primarily resulted from: ① the difference in the determination of the maximum stress σmax. The maximum stress σmax in GB 50018—2002 is approximately 87% of that defined in CSA S13607; ② the difference between the plate buckling coefficient k specified in CSA S13607 and the product kk1 defined in GB 50018—2002. The resulted differences between the effective widths evaluated based on CSA S13607 and GB 50018—2002 for the stiffener, flange and web of Csections in compressive member are discussed in section 4.4Element Effective Width of Compressive Member4.1Effective Width of Stiffener

The buckling coefficients of stiffener between the two standards are essentially identical being 0.425 and 0.43 for GB 50018—2002 and CSA S13607, respectively. However, there is a considerable difference on how to evaluate the effective width of the stiffener between the two standards. The difference is primarily associated with the difference between coefficient k1L defined in GB 50018—2002 and coefficient RI specified in CSA S13607 (Fig.6). Forwf/tS≤0.328, be=wf, be1=be2=wf/2, RI=1, de=d′e; forwf/tS＞0.328, RI=IsIa≤1, de=d′eRI, Ia=399t4(wf/tS-0.328)3≤t4(115wf/tS+5), Is=d3t(sin(θ))212, d=D-(R+t); S=128E/σmax, de is reduced effective width, d′e is effective width of the stiffener calculated based on the unstiffened element. Buckling coefficient of flange kf: for D/wf≤0.25, 3.57RnI+0.43≤4; for 0.25＜D/wf≤0.8, (4.82-5Dwf)RnI+0.43≤4; n=0.582-wf/t4S≥13. The stiffener effective width de of CSA S13607 is calculated as the products of the coefficient RI and d′e, with d′e being evaluated on the buckling coefficient kL to consider the strength reduction effect caused by local buckling. The coefficient RI is introduced to consider the effective width reduction caused the distortional buckling. If RI is less than 1.0, distortional buckling may occur.

Fig.6Flange Buckling Coefficient kf and

Coefficient RI in CSA S13607

圖6CSA S13607中的翼緣屈曲系數(shù)kf與系數(shù)RIFig.7Plate Restraint Coefficient k1L of Stiffener in

GB 50018—2002 and Coefficient RI in

CSA S13607 when d/t=9.5

圖7d/t=9.5時 GB 50018—2002中的卷邊板組

約束系數(shù)k1L與CSA S13607中的系數(shù)RIAs shown in Fig.7(a), the coefficient k1L specified in GB 50018—2002 is associated with the ratio d/wf between the flat portion of stiffener depth and the flat portion of flange width. The coefficient k1L gradually increases from 0.18 to 0.95 as the ratio d/wf increases from 0.18 to 0.6. On the other hand, RI defined in CSA S13607 is not only associated with the ratio d/wf but also related to the maximum stress σmax as illustrated in Fig.7. Generally, it is observed that: ① RI increases with the increase of ratio d/wf since distortional buckling normally occurs when the stiffener size is small ［Fig.7(a)］; ② RI decreases with the increase of the maximum stress σmax as lower magnitude of σmax indicates the strength of the member is likely controlled by flexural or lateraltorsional buckling not the distortional buckling ［Fig.7(b)］.

A demonstration of the stiffener effective width calculated based on the two standards is presented in Fig.8 for a Csection with stiffener widthtothickness ratio d/t=9.5. As illustrated in Fig.8(a), ratio d/wf has significant influence on Fig.8Comparisons of Stiffener Effective Widths

when d/t=9.5

圖8d/t=9.5時卷邊有效寬度的比較the effective widths calculated based on the both two standards. For a stiffener with ratio d/t=9.5 and the maximum stress σmax=fy, the followings are observed:

(1) When ratio d/wf is 0.18, although the coefficient k1L of GB 50018—2002 is always less than the coefficient RI of CSA S13607 for both fy=345 MPa and fy =235 MPa steel as shown in Fig.7(a), the stiffener effective widthtothickness ratio de/t obtained from CSA S13607 is less than that from GB 50018—2002. This is because coefficient RI applies directly to effective width of the stiffener calculated based on the unstiffened element as discussed previously.

(2) With the increase of d/wf, effective widthtothickness de/t in CSA S13607 increases rapidly due to the speedy increase of RI shown in Fig.7(a), whereas the increasing of de/t based on GB 50018—2002 is much less than that for both cases of fy=345 MPa and fy=235 MPa. Therefore, when ratio d/wf ≥0.32 for fy=345 MPa and d/wf≥0.27 for fy=235 MPa in this case, the stiffener effective widthtothickness ratio obtained from CSA S13607 becomes greater than that from GB 50018—2002, as shown in Fig.8(a).

(3) Eventually, with the further increase of d/wf, the stiffener is fully effective based on both standards.In addition to the ratio d/wf, the maximum stress σmax also has considerable influence on the stiffener effective width. According to Fig.8(b), for the cases of d/t=9.5 and d/wf=0.22, the followings are perceived:

(1) When the magnitude of the maximum stress σmax is small (σmax≤78 MPa in this case), the stiffener is fully effective in accordance with both CSA S13607 and GB 50018—2002.

(2) With the increase of σmax, the stiffener is still fully effective based on CSA S13607 as the value of RI remains as 1.0 and ratio de/t associated with CSA S13607 is greater than that of GB 50018—2002. However, the stiffener is not fully effective when σmax≥78 MPa based on GB 50018—2002 due to the smaller value k1L.

(3) With the further increase of σmax, RI begins to decrease rapidly as shown Fig.7(b) whereas coefficient k1L remains as a constant. Consequently, the stiffener effective widthtothickness ratio de/t associated with CSA S13607 becomes less than that of GB 50018—2002 when maximum stress σmax≥154 MPa.

4.2Effective Width of Flange

The plate buckling coefficients kf of flange adopted in the two standards are quite different. In GB 50018—2002, the flange buckling coefficient is 0.98. However, in CSA S13607, the flange buckling coefficient kf is associated with the ratio d/wf between flat portion of stiffener depth and flat portion of flange width, the ratio D/wf of stiffener depth and the flat portion of flange width (assume the inside corner radius R=2t［7］ which leads to D=d+3t), the maximum stress σmax and the flange widthtothickness ratio wf/t as discussed in section 4.1 (Fig.6). The influence of d/wf and σmax on the flange buckling coefficient kf for the case when wf/t=45 is demonstrated in Fig.9(a). From Fig.9(a), it can be seen that: ① when d/wf is small (d/wf =0.21 in this case), RI is usually less than 1.0 (RI =0.41 and RI=0.49 for fy=345 MPa and fy=235 MPa in this case, respectively) as distortional buckling Fig.9Flange Buckling Coefficient kf in CSA S13607 and

k1f of Flange Plate Restraint Coefficient in

GB 50018—2002

圖9CSA S13607中的翼緣屈曲系數(shù)kf與GB 50018—

2002中的翼緣板組約束系數(shù)k1fmay occur; therefore, the resulted value of the flange buckling coefficient kf is small (kf=2.98 and kf=3.15 for fy=345 MPa and fy=235 MPa, respectively); ② with the increase of d/wf, the value k increases as the increase of value of RI; ③ with further increase of ratio d/wf, RI reaches to 1.0 and distortional buckling will not occur. However, the stiffener becomes prone to the local buckling, and the flange buckling coefficient kf decreases linearly with the increase of ratio d/wf. Therefore, in order to stiffen the flange, the size of stiffener needs to be designed appropriately. Very large or small stiffeners may not be effective.

The maximum stress σmax also has certain influence on the flange buckling coefficient kf according to CSA S13607. In the left side range of the peak amplitude of the coefficient kf shown in Fig.9(a), the increase in the flange buckling coefficient kf is contributed by the increase of RI shown in Fig.7(a). Since RI is associated with the maximum stress σmax as shown in Fig.7(b), therefore, the flange buckling coefficient kf in that range is also associated with maximum stress σmax. With the increase of σmax, the distortional buckling may occur; thus, the flange buckling coefficient kf decreases as shown in Fig.9(a).

Although the flange buckling coefficient kf in CSA S13607 is greatly influenced by d/wf, for common Csections with sizes of stiffener and flange satisfied requirements in both standards shown in Tab.2, the influence of the different flange buckling coefficients on the flange effective width is small as illustrated in Fig.10(a). The lines denoted as the maximum and minimum values shown in Fig.10(a) respectively represent the maximum and minimum values of the flange local reduction factors ρ for various d/wf corresponding to the values d/t shown in Tab.2. It can be seen from Fig.10(a), for given values of the widthtothickness ratio wf/t and the maximum stress σmax, the variation of the flange local reduction factor is relatively small. This is because for Csections satisfied requirements listed in Tab.2, the stiffener sizes can effectively stiffen the flange to ensure that the local buckling of the flange occurs prior to the buckling of the stiffeners.

Fig.10Comparisons of Flange Effective Widths

圖10翼緣有效寬度的比較A relationship of the flange effective width evaluated based on the two standards is illustrated

Tab.2Maximum and Minimum Stiffener Sizes for Flange

表2翼緣加勁肋的最大尺寸和最小尺寸wft-115202530354045505560dt-1Maximum9121212121212121212Minimum5.46.37.28.08.59.09.510.010.511.0in Fig.10(b) for a case of wf/t=45 and σmax=fy. It is observed from Fig.10(b) that:

(1) As the effective widthtothickness ratio be/t of the flange in CSA S13607 is not only associated with wf/t and σmax, but also the value of d/wf, the two lines are used to describe the effective ratio be/t for each specified value of fy. The upper and lower lines represent the maximum and minimum effective ratios be/t for various d/wf, respectively.

Based on Tab.2, for wf/t=45, the associated maximum and minimum values of ratio d/t are 12 and 9.5, respectively. The resulted maximum and minimum values of d/wf=0.27 and d/wf=0.21 correspondingly, which are close to each other in this case. Therefore, the effective widthtothickness ratio be/t of the flange in CSA S13607 is not much influenced by d/wf. be/t ranges from 31.2 to 33.8 for fy=345 MPa and 37.2 to 39.2 for fy=235 MPa.

(2) The effective width of the flange evaluated based on GB 50018—2002 is considerably less than that of CSA S13607 due to the smaller value of product kfk1f in GB 50018—2002. For the common Csections, when wf/t=45 and σmax=fy, the flange buckling coefficient kf in CSA S13607 ranges from 3.0 to 3.4 for fy=345 MPa and 3.1 to 3.6 for fy=235 MPa. However, the flange buckling coefficient kf is only 0.98 in GB 50018—2002. Moreover, the flange buckling coefficient k1f which is associated with the connected element in GB 50018—2002 is also much smaller than 1.0 as shown in Fig.9(b). Therefore, the flange buckling coefficient kf evaluated by CSA S13607 is much larger than the coefficient kfk1f specified in GB 50018—2002.

(3) Since coefficient k1f is associated with the ratio ww/wf, from Fig.9(b), k1f of flange in GB 50018—2002 decreases from 0.56 to 0.15 when ww/wf increases from 3 to 10. Therefore, the differences in the flange effective widths between the two standards gradually become larger as the increase of ww/wf, as shown in Fig.10(b).When ratio ww/wf increases from 3 to 10, the differences in the flange effective width between the two standards increase from 54.6% to 76.7% and 52.7% to 75.6% for fy =345 MPa and fy =235 MPa steel, respectively.

For Csections satisfied with the requirements in both standards shown in Tab.2, the flange buckling coefficient kf calculated as per CSA S13607 is in the range of 1.25 to 4.0. Therefore, it is concluded that for all Csections satisfied with the requirements listed in Tab.2, the flange buckling coefficient kf evaluated based on CSA S13607 is much larger than the coefficient kfk1f specified in GB 50018—2002, which results the effective width of the flange associated with GB 50018—2002 is much smaller than that of CSA S13607.

4.3Effective Width of Web

There is no difference on plate buckling coefficient kw of the web of a Csection between the two standards: both of them specify 4.0 as the web buckling coefficient. The primary difference in the effective width comes from plate buckling coefficient k1w related to the connected element. As shown in Fig.11, coefficient k1w for the web defined in GB 50018—2002 is associated with the ratio ww/wf. For Csections with ratio ww/wf ranging from 3.0 to 10.0, coefficient k1w increases from 1.22 to 1.7 when ww/wf increases from 3.0 to 6.0, and after that it remains as a constant of 1.7. For the web of Csections, due to the large value of k1w, the products of kw and k1w (kwk1w) in GB 50018—2002 is larger than the value kw evaluated based on CSA S13607. In addition, as the maximum stress σmax in GB 50018—2002 is only 87% of the maximum stress σmax specified in CSA S13607, the effective width of the web in GB 50018—2002 is usually slightly greater than that in CSA S13607 as shown Fig.12. For a case of ww/t=120 and σmax=fy: the effective widthtothickness ratio of web is 42.2 and 50.2 for fy=345 MPa and fy =235 MPa steel in CSA S13607, reFig.11Restraining Coefficient k1w of Web in

GB 50018—2002

圖11GB 50018—2002中的腹板板組約束系數(shù)k1wFig.12Comparisons of Web Effective Widths

when ww/t=120, σmax=fy

圖12ww/t=120, σmax=fy時腹板有效寬度的比較spectively, whereas in GB 50018—2002, the effective widthtothickness ratio of web gradually increases from 45.6 to 53.9 for fy=345 MPa and 55.2 to 65.2 for fy=235 MPa when ww/wf increases from 3 to 6. When ww/wf>6, the effective width of web in GB 50018—2002 remains as a constant due to the upper limit k1w≤1.7 for the web. For this case, the web effective width of GB 50018—2002 is 8.1% to 27.6% and 10.0% to 30.0% greater than that of CSA S13607 for fy=345 MPa and fy=235 MPa steel, respectively.5Comparison of Nominal Axial Compressive StrengthTwo Csections, with section thicknesses t=2.58 mm and t=0.879 mm as shown in Tab.3 are selected from the Handbook of Steel Construction［8］ for the nominal axial strength comparison in this study. The thicknesses t=2.58 mm and t=0.879 mm are likely the maximum and minimum thicknesses in load bearing wall stud application. The length of each member is 3.0 m and the weak Tab.3Sizes of Members

表3構(gòu)件尺寸MemberSection Dimension/mmh0b0DtRrdwfwwArea/

mm2Length/

mBracingA15241.312.72.5803.875.166.2528.40139.10621.323.02B15241.312.70.8791.942.389.8835.66146.36221.863.02Tab.4Comparison of Nominal Axial Compressive Strength

表4名義軸壓強度的比較MemberStandardfy=345 MPafy=235 MPaStress fn or

φfy/MPaEffective Area

Ae/mm2Nominal Axial

Strength Pn/kNStress fn or

φfy/MPaEffective Area

Ae/mm2Nominal Axial

Strength Pn/kNACSA S13607235.96549.19129.58181.42575.93104.49GB 50018—2002238.32573.06136.57179.07610.24109.27Difference/%1.04.35.4-1.36.04.6BCSA S13607241.41 125.12 30.21 184.27137.0825.26GB 50018—2002244.03 92.39 22.55 181.68104.0218.90Difference/%1.1-26.2-25.4-1.4-24.1-25.2Tab.5Comparison of Effective Width when fy=345 MPa

表5fy=345 MPa時有效寬度的比較MemberElementwt-1bet-1CSA S13607GB 50018—2002CSA S13607GB 50018—2002Difference/%kRIkk1AWeb53.9143.0848.6212.94.00N/A4.001.56Flange11.0111.0110.03-8.9N/AN/A0.980.27Stiffener2.422.422.420.00.43 N/A0.430.16BWeb166.5151.1258.6714.84.00 N/A4.001.43Flange40.5736.7214.33-61.03.37N/A0.980.35Stiffener11.2410.187.62-25.10.43 0.890.430.24Note:N/A means not applicable.

Tab.6Comparison of Effective Width when fy=235 MPa

表6fy=235 MPa時有效寬度的比較MemberElementwt-1bet-1CSA S13607GB 50018—2002CSA S13607GB 50018—2002Difference/%kRIkk1AWeb53.9147.1052.5811.64.00N/A4.001.56Flange11.0111.0110.85-1.5N/AN/A0.980.27Stiffener2.422.422.420.00.43N/A0.430.16BWeb166.5157.8567.8917.44.00N/A4.001.43Flange40.5740.0416.58-58.63.47N/A0.980.35Stiffener11.2411.248.28-26.30.431.000.430.24axis of the member is braced at the 1/3 point and 2/3 point.

The comparisons on the buckling stress, effective crosssectional area, nominal axial strength and the effective width for each member calculated in accordance with the two standards are presented in Tab.4 to Tab.6. It can be seen that the difference of the buckling stresses between the two standards is negligible as discussed in section 2. The difference in the nominal axial strength of the member is primarily resulted from the difference of the effective crosssectional area.

For member A, since the actual widthtothickness ratios w/t of the flange and the stiffener are small, flange and stiffener are both fully effective according to CSA S13607, whereas in GB 50018—2002, although the flange is not fully effective, the effective widths are only 8.9 % and 1.5% less than those of CSA S13607 for fy=345 MPa and fy=235 MPa steel. On the other hand, the web effective widths calculated based on GB 50018—2002 are 12.9% and 11.6% greater than those of CSA S13607 for fy=345 MPa and fy=235 MPa steel as shown in Tab.5 and Tab.6. Therefore, there is no significant difference on the nominal axial strengths obtained from the two standards for this member. It can be seen from Tab.4, the nominal axial strengths calculated based on GB 50018—2002 are about 5.4% and 4.6% greater than those evaluated based on CSA S13607 for fy=345 MPa and fy=235 MPa steel, respectively.

However, for member B, the nominal axial strengths associated with GB 50018—2002 are 25.4% and 25.2% less than those of CSA S13607 for fy=345 MPa and fy=235 MPa steel, respectively, as shown in Tab.4. The smaller values of the nominal axial strength associated with GB 50018—2002 are primarily resulted from the smaller values of flange and stiffener effective widths. The flange effective widths calculated in accordance with GB 50018—2002 are 61.0% and 58.6% less than those of CSA S13607 for fy=345 MPa and for fy=235 MPa steel, respectively, whereas the stiffener effective widths associated with GB 50018—2002 are 25.1% and 26.3% less than those of CSA S13607 for fy=345 MPa and for fy=235 MPa steel, respectively, as shown in Tab.5 and Tab.6. Although the web effective widths calculated based on GB 50018—2002 are 14.8% and 17.4% respectively greater than those of CSA S13607 for fy=345 MPa and for fy=235 MPa steel, it does not contribute significantly to the difference in the effective crosssectional areas.

From the foregoing analysis, it can be seen the yield stress doesnt have significant influence on the difference of the nominal axial strength evaluated based on the two standards. The differences of the nominal axial strength between the two standards is greatly influenced by the crosssectional dimensions of Csections. Specifically, by comparing member A to member B, it is found that the difference on the nominal axial strength is primarily influenced by the flange widthtothickness ratio wf/t. For member A, as the flange widthtothickness ratio is small (wf/t=11.01), the flange is almost fully effective in the both standards. The difference on the nominal axial strength is mainly controlled by the difference of the web effective width. Since the web effective width associated with GB 50018—2002 is usually greater than that of CSA S13607 as discussed in section 4.3, the nominal axial strengths calculated based on GB 50018—2002 are about 5.4% and 4.6% greater than those evaluated based on CSA S13607 for fy=345 MPa and fy=235 MPa steel, respectively, as shown in Tab.4. However, for member B, the flange widthtothickness ratio wf/t is relatively large (wf/t=40.57). The difference on the nominal axial strength is primarily dominated by the difference of the flange effective width, whereas the difference of the web effective width doesnt have a significant contribution. Since the flange effective width associated with GB 50018—2002 is much less than that of CSA S13607 as discussed in section 4.2, the nominal axial strengths associated with GB 50018—2002 are 25.4% and 25.2% less than those of CSA S13607 for fy=345 MPa and fy=235 MPa steel, respectively, as shown in Tab.4.

In order to further investigate effects of the flange widthtothickness ratio wf/t on the difference of the nominal axial strength between the two standards, comparisons of nominal axial strength between the two standards are carried out for typical Csection loadbearing wall studs with the section depth ranging from 92.1 mm to 203 mm. The first nineteen typical Csections listed in Tab.7 are selected from the Handbook of Steel Construction［8］ and the rest are from the Lightweight Steel Framing Metric Section Properties［9］. The length of the stud is still assumed to be 3.0 m and the weak axis of the member is braced at the 1/3 point and 2/3 point. In addition, as the yield stressTab.7Comparison of Nominal Axial Compressive Strength for Typical Csection Wall Studs when fy=345 MPa

表7fy=345 MPa時典型C形截面墻架柱名義軸壓強度的比較MemberSection Dimension/mmh0b0DtRwft-1www-1fPn/kNCSA S13607GB 50018—2002Difference/%1203.063.419.102.5803.8719.573.76200.07190.86-4.62203.063.419.101.8102.7230.023.57120.4599.96-17.03203.063.419.101.4402.1639.033.4885.4864.56-24.54203.063.419.101.1501.8149.983.4361.0841.62-31.95203.050.819.102.5803.8714.695.02174.32179.082.76203.050.819.101.8102.7223.064.65109.7797.84-10.97203.050.819.101.4402.1630.284.4981.8763.36-22.68203.050.819.101.1501.8139.034.3956.6241.31-27.09203.050.819.100.8791.9451.384.3735.8325.03-30.110203.041.312.702.5803.8711.016.69130.03141.608.911203.041.312.701.8102.7217.816.0281.2984.534.012203.041.312.701.4402.1623.685.7460.1355.40-7.913203.041.312.701.1501.8130.775.5744.9036.22-19.314203.041.312.700.8791.9440.575.5330.1822.33-26.015152.041.312.702.5803.8711.014.90129.58136.575.416152.041.312.701.8102.7217.814.4382.2182.690.617152.041.312.701.4402.1623.684.2560.5456.28-7.018152.041.312.701.1501.8130.774.1345.0836.68-18.619152.041.312.700.8791.9440.574.1030.2122.55-25.420152.050.815.902.5803.8714.693.67154.59155.160.421152.050.815.901.8102.7223.063.4297.4691.58-6.022152.050.815.901.4402.1630.283.3272.4560.21-16.923152.050.815.901.1501.8139.033.2551.7539.31-24.024152.050.815.900.8791.9451.383.2433.7923.89-29.325101.641.312.702.5803.8711.013.1392.8791.44-1.526101.641.312.701.8102.7217.802.8763.0861.62-2.327101.641.312.701.4402.1623.672.7747.1744.14-6.428101.641.312.701.1501.8130.752.7135.4031.30-11.629101.641.312.700.8791.9440.552.6925.0619.90-20.630101.631.84.761.4402.1617.053.8534.0433.58-1.431101.631.84.761.1501.8122.463.7024.3223.68-2.632101.631.84.760.8791.9429.713.6816.6015.84-4.63392.141.312.702.5803.8711.012.7979.4477.61-2.33492.141.312.701.8102.7217.802.5856.3555.43-1.63592.141.312.701.4402.1623.672.4942.4840.00-5.83692.141.312.701.1501.8130.752.4432.0128.49-11.03792.141.312.700.8791.9440.552.4322.8718.97-17.03892.131.84.761.4402.1617.053.4631.1130.58-1.73992.131.84.761.1501.8122.463.3422.3321.58-3.44092.131.84.760.8791.9429.713.3115.3514.45-5.8Fig.13Comparison of Nominal Axial Compressive

Strength for Typical Csection Wall Studs when

fy=345 MPa

圖13fy=345 MPa時典型C形截面墻架柱名義

軸壓強度的比較doesnt have significant influence on the difference of the nominal axial strength, comparisons are only carried out for steel with fy=345 MPa.

The differences of the nominal axial strength between the two standards are listed in Tab.7 and are illustrated in Fig.13. From Tab.7, it is found that for given values of flange widthtothickness ratio wf/t, the ratio ww/wf has certain influence on the difference of the nominal axial strength, especially when the flange widthtothickness ratio wf/t is small. For example, for wf/t=11.1, as ratio ww/wfincreases from 2.79 to 6.69, the difference of the nominal axial strength between the two standards increases from -2.3% to 8.9%. This is because the difference of the web effective width between the two standards increases when ww/wf increases, as discussed in section 4.3.

However, it can be seen from Fig.13 compared to the influence of ww/wf, the influence of the flange widthtothickness ratio wf/t is far more significant. The difference of the nominal axial strength between the two standards generally decreases with the increase of flange widthtothickness ratio wf/t. Approximately, if the flange widthtothickness ratio wf/t is not less than 17.8, the difference on the nominal axial strength is dominated by the difference of the flange effective width and the nominal axial strength evaluated by GB 50018—2002 is less than that of CSA S13607, with the maximum difference being 31.9%. However, for the case that wf/t is approximately less than 17.8, the difference on the nominal axial strength is dominated by the difference of the web effective width and the nominal axial strength evaluated by GB 50018—2002 is generally slightly greater than that of CSA S13607, with the maximum magnitude being 8.9%.6Conclusions

The differences on evaluating the nominal axial compressive strength of coldformed steel Csection based on the North American standard CSA S13607 and the Chinese standard GB 50018—2002 are investigated. The investigation unveils that the differences are primarily resulted from the difference in computing the effective crosssectional area at the specified buckling stresses. More specifically, it is contributed mainly from using different flange buckling coefficients and the maximum stress while evaluating the effective widths of the crosssection. The maximum stress σmax used in GB 50018—2002 is approximately 87% of that used in CSA S13607. However, it is also found in this study that the difference between the two standards in evaluating the flexural and lateraltorsional buckling stresses is negligible. The conclusions are obtained as follows:

(1) The flange effective width calculated based on the North American standard CSA S13607 is considerably greater than that evaluated in Chinese standard GB 50018—2002. This is because GB 50018—2002 adopts smaller values of the flange buckling coefficients kf and k1f. In GB 50018—2002, the flange buckling coefficient kf=0.98 and k1f is usually less than 1.0. As a result, the products of kf and k1f (kfk1f) in GB 50018—2002 is practically less than 1.0. On the contrast, the flange buckling coefficient kf in CSA S13607 normally ranges from 1.25 to 4.0.

(2) The web effective width calculated based on the Chinese standard is slightly greater than that computed from the North American standard. This is because the web buckling coefficient associated with the connected element k1w is usually larger than 1.0 and the maximum stress σmax in GB 50018—2002 is approximately 87% of that in CSA S13607.

(3) The difference of the stiffener effective width between the two standards is a little bit complicated than that of the flange or web effective width. Generally, when the stiffener is small, the stiffener effective width calculated in CSA S13607 may be less than that of GB 50018—2002. However, if the size of stiffener is large, the effective width

associated with CSA S13607 will be greater than that of GB 50018—2002.

(4) The difference of the nominal axial strength between the two standards is primarily influenced by the flange widthtothickness ratio. For typical Csection wall studs investigated herein, the difference on the nominal axial strength is primarily influenced by the flange widthtothickness ratio wf/t. If the flange widthtothickness ratio wf/t is not less than 17.8, the difference on the nominal axial strength is dominated by the difference of flange effective width, and the nominal axial strength evaluated by GB 50018—2002 is less than that of CSA S13607, with the maximum difference being 31.9%. However, when wf/t is approximately less than 17.8, then the difference on the nominal axial strength is primarily governed by the difference of the web effective width and the nominal axial strength evaluated by GB 50018—2002 is slightly greater than that of CSA S13607, with the maximum difference being 8.9%.References:［1］CSA S13607,North American Specification for the Design of Coldformed Steel Structural Members［S］.

［2］GB 50018—2002,Technical Code of Coldformed Thinwall Steel Structures［S］.

［3］JGJ 227—2011,Technical Specification for Lowrise Coldformed Thinwall Steel Buildings［S］.

［4］CHEN J.Stability of Steel Structures:Theory and Design［M］.Beijing：Science Press,2008.

［5］YU W W,LABOUBE R A.Coldformed Steel Design［M］.New York:John Wiley & Sons,2010.

［6］ZHOU X H,WANG S J.Stability Theory and Its Applications of Thinwalled Members［M］.Beijing：Science Press,2009.

［7］XU L.Advanced Structural Steel Design［R］.Waterloo:University of Waterloo,2012.

［8］Canadian Institute of Steel Construction.Handbook of Steel Construction［M］.10th ed.Markham:Canadian Institute of Steel Construction,2010.

［9］Canadian Sheet Steel Building Institute (CSSBI).Lightweight Steel Frame Metric Section Properties［R］.Cambridge:Canadian Sheet Steel Building Institute,2011.