Economical design procedures for built-up box sections subject to compression and bi-axial bending

    Economical design procedures for built-up box section subject to compression and bi-axial bending Osama Bedair PII: DOI: Reference:

S2352-0124(14)00005-8 doi: 10.1016/j.istruc.2014.09.001 ISTRUC 4

To appear in: Received date: Revised date: Accepted date:

29 April 2014 29 August 2014 17 September 2014

Please cite this article as: Bedair Osama, Economical design procedures for builtup box section subject to compression and bi-axial bending, (2014), doi: 10.1016/j.istruc.2014.09.001

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ECONOMICAL DESIGN PROCEDURES FOR BUILT-UP BOX SECTION SUBJECT TO COMPRESSION AND BI-AXIAL BENDING

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Osama Bedair, PhD., P.Eng

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CONSULTANT, PO BOX 45577, CHAPMAN MILLS RPO, OTTAWA, ONTARIO, K2J, 0P9, CANADA TEL: (613)-440-4741, EMAIL:[email protected],

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- ABSTRACT

The paper offers to practitioners economical procedures that can be used to optimize the design of

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built up box sections subject to compression and biaxial bending. Little emphasis appeared in the published literature that addressed this general loading condition. The analysis methodology

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and structural idealization are first presented. Diagrams are presented showing buckling behaviour of the section by accounting rotational and lateral restraints. The post-buckling response is also illustrated for various applied stress ratios. A design space concept is then

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introduced showing interaction of serviceability and strength limit states. These procedures are

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cost effective and appropriate for industrial implementation to optimize the structural design.

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KEYWORDS; Built up box sections, buckling, post-buckling, plate assemblies, steel design.

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1-INTRODUCTION Box sections are extensively used in commercial buildings, bridges, marine structures and many of the heavy industrial facilities. They provide efficient structural performance in resisting axial compression, flexural and torsional stresses. In building construction, built up thin walled box

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sections are fabricated in some cases by welding two cold-formed C-sections to; 1) increase the flexural stiffness of the member to increase the span; 2) increase the torsional stiffness by

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generating symmetric cross sections. Unfortunately, the design provisions that are available in

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practice that govern the behaviour of assembled structure are extrapolated from single members. Other type of box section fabrications is achieved by assembling plate elements. In

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this case, different flange or web thickness can be used to reduce the weight of the structure. Built-up box section may be subject to either concentric or eccentric loading. Eccentric loading may also result through load transfers by attached members. Experiments and numerical

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studies have shown that local buckling of box section is a common failure criteria in thin walled box members.

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Research into thin walled steel box sections has been a subject of interest for many years.

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Several numerical and experimental studies were conducted in the past to study the behaviour of box section. Majority of the investigations and design guidelines are limited to box sections

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subjected to either the uniform compression or pure bending. Spence and Morley [1] performed tests on box girders under different combinations of symmetrical and anti- symmetrical loads. Rasmussen and Baker [2] studied experimentally the ultimate load-carrying capacity and failure mechanisms of thin walled box section beams subject to eccentric loads. Heins and Lee [3] reported field tests for two-span curved steel single- box girder bridge. Reyes and Guzmán [4] performed experimental investigation to study the behaviour of box sections composed of two welded C-section members under uniform compression. Usami and Fukumoto [5] presented experimental results of local and overall buckling of welded box sections fabricated from high strength steel. Tests were also carried to study the influence of residual stress in hollow steel sections in references [5,6].

Ductility of the flanges was studied numerically by Zheng et al. [7] using short steel box columns. Aoki and Susantha [8] also conducted cyclic loading tests to examine the ductility of box sections by considering axial load fluctuations. Serrette [9] investigated the flexural performance built-up box sections under eccentric loading. The built-up box sections were made with two face-to-face C-shapes, with a track section cover connected to the top and bottom flanges. They presented test results showing the effect of edge loading. Jeon et al. [10] developed analytical model to study static and dynamic behaviour of composite box beams.

ACCEPTED MANUSCRIPT Page |3 Cortinez and Piovan [11] investigated the stability of composite thin- walled beams with open or closed cross-sections. Hsu and Tsao [12] studied the flexural–torsional behaviour of thin-walled steel hollow box columns subjected to a cyclic eccentric load. Bedair [13, 14] investigated the interactive buckling of stiffened box sections under uniform compression with applications to

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bridge and industrial facilities. The influence of the flange/web proportions on the behavior of section was highlighted. Industrial examples were presented showing the variation of the flange

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buckling stress for various stiffening configurations. It was shown that the behavior of the flange

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is largely affected by the restraints imposed by the webs. Other investigations by Bedair [15-22] highlighted the behaviour of W and channel steel sections under various loading combinations. Discrete models were presented to idealize realistically the restraints of the attached structural

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components. Design guidelines were proposed that can be utilized by practicing engineers and

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steel fabricators to maximize the performance of W and channel steel sections The interaction of plate assemblies was addressed by several researchers. Stamatelos et. al [23] presented a methodology to study local buckling and post-buckling behavior of plate

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assemblies. Transverse and rotational springs with varying stiffness were used to model the

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plate-stiffener interaction. Zhang, et al [24] used a triangular composite stiffened plate/shell element to analyze stiffened plates using Mindlin shear deformation theory. The rotations of ribs and the plate are determined using displacement compatibility conditions. Li and Xiaohui [25]

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presented finite element model to study the bending behavior of stiffened laminated plates. The compatibility of displacements and stresses between the plate and the stiffeners conditions were used to establish the governing equations. Nath [26] presented analytical solutions for elastic fields of a stiffened composite plate subjected to axial tension and pure bending. A potential function is expressed in terms of the displacement components that satisfy the equilibrium condition. Fourier series approximation is then used satisfying the boundary conditions. Jiang, Bao and Robert [27] presented several modeling strategies for bending and buckling of orthotropic and rectangular plates using finite element method. Numerical comparison is made using first and second order three dimensional elements with varieties of mesh intensities. It nust be noted that three dimensional solid elements require excessive computer time and consequently may not be practical for analysis of plates assemblies. Box sections are sometimes filled with concrete to enhance the member stiffness and load carrying capacity. The concrete fill acts as restraining media to the box section plates. Local buckling of box sections filled with concrete was addressed by number of researchers. Shanmugam et.al. [28] employed effective width procedure to predict the load carrying capacity of thin walled steel tubes with concrete fill subject to biaxial loadings. Uy and Bradford [29] used

ACCEPTED MANUSCRIPT Page |4 finite strip method to determine the buckling stress for various boundary conditions. A sinusoidal function is used for the longitudinal displacement and a cubic polynomial for the transverse displacement. Empirical studies were also performed for local buckling strength of steel tubes

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filled with concrete by Sakai et al. [30] and Wright [31]. Limited literature addressed buckling of box sections under combined compression and biaxial

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bending. Much of the investigations focused to develop numerical or empirical analysis procedures for box sections under compression or bending. Available local buckling

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expressions for box sections are applicable to the compression and uni-axial bending loading conditions. Furthermore, North American and European codes of practice [32-36] ignore the

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rotational and lateral restraints imposed by the attached members. The objective of the paper is to investigate the serviceability and ultimate states of box sections

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under combined axial compression and biaxial bending. The box section assemblies are treated as partially restraints against rotation and in-plane translation. The study provides useful guidelines that can be utilized by practicing engineers to maximize the section performance

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2-ANALYSIS PROCEDURE

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under this general loading.

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Consider typical built-up box section shown in Fig.(1) with unsupported length (L), subject to compressive force (P) , and biaxial moments (M1 and M2). The web width and thickness are denoted by (bw, tw) and the flange width and thickness by (bF, tF). The top and bottom flanges are denoted by (F1, F2), and the webs by (W1, W2), as illustrated in Fig.(1). The flanges and the webs are also identified by local co-ordinate systems, with origins (oW1, oF1) that are located at their centrelines. Note that the subscripts {W1, W2, F1, F2} are used to distinguish the local coordinates of each plate component. Therefore, the non-dimensional coordinates of web (W1) are ξW1=(xW1/L) and ηW1 =(yW1/bw), and for flange (F1) counterpart are; ξF1 = (xF1/L), ηF1 =(yF1/bF). In the present investigation, it is assumed that the box section is made of isotropic and elastic material. The flange (F1) out-of-plane and in-plane displacement functions (W F1, UF1, VF1 ) are assumed as follow;  M1 N1  A ijF1  i 1 j 1 F 1 W (  F1 ,  F1 )   F1   0 U (  F1 ,  F1 )     V F 1(  ,  )   F 1 F 1    0  



i

j

 B

F1 ij

0

0

0

0

0

0

0

0

0

m 1 n 1

0

M F1

C m

0

0

n

0

F1 mn

 D p

F1 pq

q

0

N F1

F

F1 rz

r

z

G s

t

F1 st

         

 iF1 ( F1 ) 1Fj (  F1 )   F1  F i ( F1 )  2 j (  F1 ) F1   x   F1   mn (  F1 , F1 )   F1    pq (  F1 , F1 )    yF  F1   (  ,  )  rz F1 F1    F1 (  , )  st F1 F1  

(1)

ACCEPTED MANUSCRIPT Page |5 Where

PF1i(ξF1)

is one dimensional function that describes the flange longitudinal displacement

profile, while the functions ωF1j (ηF1) describes the transverse displacement profiles, {AF1ij, BF1ij } are their undetermined coefficients, {ρF1mn (ξF1, ηF1), φF1pq (ξF1, ηF1), κF1rz (ξF1, ηF1), ψF1st (ξF1, ηF1)} are two dimensional functions that approximate the flange in-plane displacement profiles, and {CF1mn,

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MF1, NF1, DF1pq, FF1rz, GF1st} are their associated undetermined coefficients. Similarly, the web (W1)

i

W1 ij

j

 B i1

0

W1 ij

0

0

MW 1

0

C

W1 mn

m

0

0

0

0

n

0

 D

W1 pq

p

q

0

0

0

0

0

0

0

j

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 A

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   W 1 W ( W 1 , W 1 )   W1   U ( W 1 , W 1 )    V W 1( ,  )   W1 W1      

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out-of-plane and in-plane displacement functions (WW1, UW1, VW1) are assumed as follow;

N W1

F

W1 rz

r

z

G s

t

W1 st

          

iW 1 ( w )  Wj 1( W 1 )  W1  W1 i ( w )  j ( W 1 )   x W1   W1   mn ( W 1 ,W 1 )   W1    pq ( W 1 ,W 1 )    y W1  W1   (  ,  )  rz W 1 W 1   W1    st ( W 1 ,W 1 

(2)

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In the present formulations, it is assumed that; 1) the flanges are partially restrained against rotation and in-plan translations by the attached webs flanges; 2) the rotations of the flanges and webs are identical at their junctions, 3) the out-of-plane deflections at the webs/flanges are

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zero, and 4) the shear stresses along the longitudinal edges of the webs and the flanges vanish.

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The application of (M1, M2) causes the applied stresses to vary across the webs and the flanges. form;

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The applied stress across the flanges (ζF1xx ) and (ζF2xx ) can be expressed in the following

 xxF1

  xx

 2

 xxF 2

  xx

 2

F1

F2

(3)

 I   II  1

 I   II  1

(4)

where (ζxx) is the axial compressive stress, (ζI , ζII ) are non-dimensional stress ratios defined as;

I 

1  xx

,  II 

2  xx

(5)

Where {ζ1 , ζ2 } are the bending stresses, resulting from application of M1 and M2, respectively. The application of (M2) causes the applied stresses at the two webs (W1) and (W2) to be different, and can also be expressed in terms of the stress ratios (ζI ,ζII ) as follow;

 Wxx1   xx



 Wxx 2   xx



Where

(ζW1xx,

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 2 W 1  II  1

 I  2 W 2  II  1 ζW2xx)

(6) (7)

are the applied stress variation across the webs (W1) and (W2),

respectively. By assembling the strain energies of the box section assemblies, the local buckling stresses for the webs and the flanges can be written in the following format;

ACCEPTED MANUSCRIPT Page |6  K F1  F2 K 2  E  KW1 2 12 ( 1   )  W 2 K  

       

 F2   2  F   2   W2  W 

(8)

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F1   buc  F2   buc   W 1   buc   W  buc2     

where E = elastic modulus, v = Poisson ratio, {ζW1buc , ζW2buc , ζF1buc , ζF2buc} are the webs and the

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flange buckling stresses, αF = (tF/bF) and αW = (tW/bW). The buckling coefficients {KW1, KW1, KF1,KF2}

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are function of the assumed displacement functions {WW1ij, UW1mn, VW1rs, WW2ij, UW2mn, VW2rs ,

 b  2 F



F 1 1

,



F2

b F2  , 1

b

2 W

W 1  , 1

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 K F1   F2  2  K    L   K W1     W2   K 

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WF1ij, UF1mn, VF1rs, W F2ij, UF2mn, VF2rs } and can be expressed in the following form;

b

2 W

W 2 

1



 ijF 1   F2   ij   ijW 1   W2   ij 

(9)

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where {λW1i , λW2i , λF1i , λF2i , ψW1i , ψW2i , ψF1i , ψF2i } are given in Appendix (I).

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Local buckling stress for box section with small thickness normally occurs below the yield stress value. Therefore, it is economical to design these members in the post-buckling range. In doing so,

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the ultimate strength and post-buckling deflection become essential design parameters to evaluate. The governing equations used in the present investigation are summarized in Appendix (II). Note that due to the unequal variations of the applied stresses in the flanges and the webs the forms of the equilibrium and compatibility equations are different from the uniform compression loading. Also the form of the flanges and the webs stress functions {ΦW1, ΦW2, ΦF1, ΦF2} are different due to the unequal variation of the applied stress.

For box sections under compression and bi-axial bending, the post-buckling stress fields in the flanges along (yF1, yF2 = ±bF/2) are not equal and grow with unequal rate. Therefore, it is reasonable from design prospective to assume that the ultimate strength of the section is attained when the maximum post-buckling stress reaches the yield value, i.e. ζMAX = ζY=ζULT. The formulation can be advanced into the post yielding state and considering the strain hardening. This will be addressed in the future work.

3-DESIGN CODES COMPARISON Several design formulas are used in practice to predict the initial buckling limit state of webs under compression and uni-axial bending. These formulas are developed using the simply supported

ACCEPTED MANUSCRIPT Page |7 boundary condition (i.e. ignore the rotational/lateral restraints). No expression is available to evaluate the buckling stress for compression and the bi-axial bending. The American Iron and Steel Institute, AISI [32] and the Canadian Standards Association CAN/CSA-S16 [33] use the following formula; 3

   2 

 1 I  1  1I 

  

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 1 I K  4  2  1  1I 

(10)

The European steel Code Eurocode-3 [34] provides the following expressions;

K  7.81  6.29

1 

1I  0 1I

 1I 1I  9.78  1I 1I

  

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for

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8.2 1I 1.05  1I

2

0 

for

1I  1 1I

(11)

(12)

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K

The structural stability research council SSRC [35] presents numerical values for simply supported

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plates with several stress ratios. Gaylord and Gaylord [36] presented an approximate formula based on the German specification, DIN4114, given by;

  

1

(13)

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 1 I K  8.4  1.1  1I 

16

K

2

  1I   0.11  1  1I  

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 1I  1  1I 

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Alternative expression based on the West European standards, is provided in Reference [37]; 2

  1I    1  1I  

  

(14)

A comparison is made in Table (1) between the predictions of these equations and the analysis procedure presented in section (2) for the simply support boundary condition. It can be seen that all values are close agreements.

4- RESULTS Fig.(2) shows the variation of the top flange buckling stress (ζF1buc) with the bending stress ratio (ζ2/ζ1). The web to flange thickness ratio is fixed at (tW/tF)=1, and (ζxx/ζ1) = 1 in all cases. The solid curve represents the web/flange width ratio (bW/bF) =1, the dashed curve represents (bW/bF) =2, the dotted curve represents (bW/bF) =4 and the dashed-dotted curve represents (bW/bF) =10. All curves intersect with y axis at (ζ2/ζ1) = 0, which represents compression and uni-axial bending. By increasing the bending stress ratio (ζ2/ζ1), the buckling stress curves (ζF1buc) decrease until they approach and asymptotic values, corresponding to (ζ2/ζ1)→∞. The asymptotic value for (bW/bF) ={1, 2, 4 and 10} are {81, 90, 95, 100 MPa}, respectively. It can also be observed that as the (bW/bF) ratio increases the top flange buckling stress (ζF1buc) increases. This is due to the increase of the intensity

ACCEPTED MANUSCRIPT Page |8 of the flange rotational restraint as a result of increasing the web size. The increase is very pronounced for small variation of (bW/bF) ratio. For example, the average increase in the flange buckling stress (ζF1buc) is almost 24% by increasing the (bF/bW) ratio from 1 to 4. By increasing (bW/bF) ratio from 4 to 10 the average increase in (ζF1buc ) is only 7%. This indicates that the top

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flange buckling stress is sensitive the variations of (bW/bF) ratio. It should be emphasized that for the uni-axial bending (ζ2/ζ1)=0, the buckling stress equations provided by the North American codes are

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independent box section property. For example, for uniaxial bending, a constant value of (ζF1buc) = 114 MPa is used for all flange sizes, as identified by the dashed arrow in Fig.(3). The difference in

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the (ζF1buc) values and North American and European codes in some cases exceeds 42%.

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Fig.(3) shows the variation of the top flange buckling stress (ζF1buc) with the web/flange width ratio (bW/bF). The box section, in this case, is subject to four bending stress ratios {(ζ2/ζ1) =0, (ζ2/ζ1)=0.25,

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(ζ2/ζ1) =0.5, (ζ2/ζ1) =1}. The stress ratio (ζxx/ζ1) is fixed at 1, the web/flange thickness ratio (tW/tF)=1 and the flange aspect ratio (L/bF)=5. The variation of (bW/bF) is ranging between 0.25 and 3. It can be seen that by increasing (bW/bF) ratio, the top flange buckling stress (ζF1buc) increases. This is due to

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the increase of the rotational restraint due to the increase of the web size. The (ζF1buc) values are

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almost doubled by increasing the web/flange width ratio (bW/bF) from 0.25 to 3. Note that 86% of the increase in (ζF1buc) occurs when (bW/bF) ratio ranges between 0.25 and 1.75. Further increase in the

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web size beyond this value has little effect on the top flange buckling stress. This shows the significant influence of the web size to the buckling stress of the top flange. For the uni-axial bending (ζ2/ζ1) =0, the code prediction is (ζF1buc) =141 MPa. By accounting for the rotational restraints and using (bW/bF) = 3, the value of (ζF1buc) =225 MPa. The difference is 37%. Fig.(4) illustrates the influence of web/flange thickness ratio (tW/tF) on the top flange buckling stress (ζF1buc). The bending stress ratio, (ζ2/ζ1) = 0.5 and the web/flange width ratio (bW/bF) =3. The variation of (tW/tF) is ranging between 0.4 and 1.5. The solid curve represents the (ζxx/ζ2)=1, the dashed curve represents (ζxx/ζ2) =2, the dashed-dotted curve represents (ζxx/ζ1) =3. It can be observed that by increasing (tW/tF) ratio, the top flange buckling stress (ζF1buc) increases. In other words, by increasing web thickness (while fixing the flange thickness), the buckling stress (ζF1buc) increases. This is due to the increase of the rotational restraint as a result of increasing the web thickness. The average increase in the (ζF1buc) values is approximately 44% when (tW/tF) ranges between 0.4-1.5. It must be noted that much of the increase occurs in the early stage of the curves when (tW/tF) is between 0.4-1. Figs.(5) shows the top web post-bucking deflection of a box section with slenderness ratio (bF/tF)=150, Modulus of Elasticity (E) = 200 GPa, and Poisson ratio v =0.3. The vertical axis

ACCEPTED MANUSCRIPT Page |9 represents maximum applied stress

(ζxxF1)

and the horizontal x-axis represents maximum out-of plan

flange deflection (δF1)max. The solid curve represents stress ratios (ζxx/ζ1=1, ζ2/ζ1 =0), the dashed curve represents (ζxx/ζ1=2, ζ2/ζ1 =0.5), the dotted curve represents (ζxx/ζ1=4, ζ2/ζ1 =0.5), and the dashed-dotted curve represents (ζxx/ζ1=10, ζ2/ζ1 =0.5). It can be observed that by increasing the

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axial stress ratio, the slopes of the curves becomes shallower. A very pronounced increase in the post- buckling stiffness result by setting the minor axis bending (ζ2/ζ1 )=0, as indicated by the solid

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curve. This implies that the application of the minor axis bending (M2) weakens the web postbuckling stiffness. This type of representation is useful to use in practice to determine the member

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out of plane deflection. For example if the applied stress (ζxxF1) = 100 MPa, then the maximum flange out of plane deflection (δF1)max = 8.1 mm, for (ζxx/ζ1=1, ζ2/ζ1 =0), (δF1)max = 11.7 mm for (ζxx/ζ1=2,

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ζ2/ζ1 =0.5), (δF1)max = 13 for (ζxx/ζ1=4, ζ2/ζ1 =0.5), and (δF1)max = 14.1 for (ζxx/ζ1=10, ζ2/ζ1 =0.5).

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Fig.(6) shows the ultimate strength (ζULT) variation with the top flange slenderness ratio (bF/tF), for three stress ratios; (ζI=0.5, ζII =0), (ζI=0.5, ζII =0.25), (ζI=0.5, ζII =0.75). The flange slenderness ratio (bF/tF) ranges between 30-150. It can be seen that by decreasing (bF/tF), i.e. using thicker flanges,

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the post buckling ultimate strength increases. The ultimate strength also increases as the bending

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stress ratio (ζII) decreases. To give numerical insight, if the top flange slenderness ratio (bF/tF)= 75, the ultimate strength (ζULT)= 237 MPa for (ζI=0.5, ζII =0). By increasing the stress ratio (ζI=0.5,

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ζII=0.25), the (ζULT) is decreases to 189 MPa (i.e by 20%). The ultimate strength (ζULT) deceases further to 135 MPa (i.e., by 43%) by increasing the stress ratios to (ζI=0.5, ζII =0.75).

5- EXPERIMENTAL AND NUMERICAL VALIDATION In this section, accuracy of the analysis procedure is verified experimentally and numerically using Finite element method. The comparison is made for the axial compression and major axis bending, (i.e. M2=0). No experimental results were available to the author to compare the compression and biaxial bending load condition. Fig (7) shows the post-buckling load-deflection path computed using the procedure presented in section 2 with experimental and Finite element results. The solid line represents the elastic post-buckling load-deflection path for the perfect plate. The experimental results are shown by the solid circles and were obtained from reference [38]. The Finite Element points are shown by the solid triangles. Since no information is given regarding the initial deflection of the experimental specimen, an initial imperfection, a sinusoidal pattern was assumed with initial amplitude wo/t= 0.05. The load deflection pattern for this case is represented by the dashed curve. In the Finite Element model, the web was discreteized into 200 quadrilateral elements. As can be seen from the figure the agreement is reasonable between the three predictions. The differences in the solutions are attributed to the approximation involved in modelling of the shape and amplitude of the initial imperfection. In the current study, the initial imperfection is described by sinusoidal functions,

ACCEPTED MANUSCRIPT P a g e | 10 while in FE this pattern was approximated for each nodal point. In the experimental results, it is not clear how the initial imperfection was modeled on the test specimen.

6- OPTIMUM DESIGN OF BOX SECTIONS

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Fig.(8) shows a design space representation of a typical box section subject to compression and biaxial bending. The serviceability limit state (SLS) and the ultimate limit state (ULS) curves are plotted

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for various (L/tF) ratios. The vertical y-axis represents the applied stress (ζF1xx) and the horizontal x-

SC

axis represents the flange width to thickness ratio (bF/tF). The applied stress ratios in this case are (ζI = 0.5, ζII =0.25). The (ULS) stress curve for this stress ratio is shown by the solid legend with the

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asterisk markers. The variation of (bF/tF) ratio is ranging between 50 and 300. Each legend represents different unsupported length to thickness (L/tF) ratio. The circle-markers represent (L/tF=300), the triangle-markers represent (L/tF =500), the rectangle-markers represent (L/tF =700)

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and the parallelogram-markers represent (L/tF =900). As can be seen, the (ULS) curve intersects with each (SLS) curve at different (L/tF) value. Points of intersections are identified by the dased circles. These points represent the optimum design of the box section for the give applied stress

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ratio. The solid segments of (SLS) curves represent the governing portion, while the dotted

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segments are the inactive part (i.e., the case when the (ULS) governs the design). As illustration, when (L/tF) = 300, (SLS) curve governs the box section design when (bF/tF) ≥ 88. At this point the

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(ζSLS) =(ζULS) = 146 MPa. Therefore, as long as (bF/tF) is larger than this value, (SLS) governs the design. By decreasing (bF/tF) < 88, serviceability curve becomes inactive (as shown by the dotted segment) and (ULS) curve governs the box section design. Similarly, for (L/tF)=500, (SLS) governs the design for (bF/tF) ≥127, while (ULS) governs the design when (bF/tF) < 127. For (L/tF)={700, 900}, the optimum (bF/tF) ratios are {170, 214}, respectively. This design space representation is very useful and can be utilized in the industry for cost-effective design for box members under this type of loading. To provide further insight, assume the unsupported length of the box section is (L)=1400 mm, then by using thickness (tF)=2 mm, the optimum design is achieved when (bF/tF) =170. The flange size in this case is 340 mm and the limiting (ζxx) is 123 MPa. Fig.(9) shows alternative presentation for the (ULS) and (SLS) interaction. The variations of (ULS) and (SLS) are plotted vs. (L/tF) ratio. This scenario appears when there are constraints on the section sizes. The (ULS) is shown by the solid curves with the asterisk markers and intersects (SLS) at different stress values. The circle-markers represent (bF/tF =160), the triangle-markers represent (bF/tF =120), rectangle-markers represent (bF/tF=90), and parallelogram-markers represent (d/t=60). The active part of the (ULS) is shown by solid lines with asterisk markers. The stresses at the points of intersections between (SLS) and (ULS) for (bF/tF) = {160,120,90,60} are {124,132,145, 164MPa},respectively. The corresponding optimum (L/tF) values are {654,465,313, 193),

ACCEPTED MANUSCRIPT P a g e | 11 respectively. Therefore, if the top flange width is constrained at (bF)=240 mm, and thickness (tF)=2 mm, the optimum unsupported length (L)= 930 mm. It can also be observed that the slopes of the serviceability curves become steeper by reducing the (bF/tF) ratios. For example, the average rate of increase in the applied stress (ζF1xx), for 100 mm increments in length (L), is 17 MPa for (bF/tF)=160.

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This rate increases to 25 MPa for (bF/tF)= 90. This shows the sensitivity of stresses to the box section

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design parameters

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5- CONCLUSIONS

The paper illustrated the behaviour of built-up box sections subject to compression and bi-axial

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bending. Little emphasis appeared in the published addressing this general loading condition. Design guidelines were also presented that can be utilized by practicing engineers to maximize the section performance. Design space representations combing serviceability and ultimate states

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were also presented to assist the design engineers to determine optimum design of the box sections. The analysis methodology and structural idealization were first presented. Results

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were also presented showing the buckling loads by accounting rotational and lateral restraints. The post buckling response was illustrated or various applied stress ratios. These procedures

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6- REFERENCES

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are cost effective and appropriate for industrial implementation to optimize the structural design.

1] Spence R, Morley C. The strength of single-cell concrete box girders of deformable crosssection. Proceedings of the Institution of Civil Engineers, Part 1: Design and Construction London, 1975; 2 (59): 743–761. 2] Rasmussen L, Baker G. Large-scale experimental investigation of deformable RC box sections. Journal of Structural Engineering 1999; 125(3):227–35. 3] Heins C, Lee W. Curved box-girder bridge test: Field test. ASCE, Journal of the Structural Division 1981;107(2):317–327. 4] Reyes, W. and Guzmán, A, Evaluation of the slenderness ratio in built-up cold-formed box sections, 2011, Journal of Constructional Steel Research, 2011, 67, 929–935 5] Usami, T. and Fukumoto, Y. Local and overall buckling of welded box columns. Journal of Structural Division (ASCE), 1982;108:525–542. 6] Rasmussen K, Hancock G. Tests of high strength steel columns. Journal of Constructional Steel Research 1995;34(1):27–52. 7] Zheng Y, Usami T, Ge H. Ductility of thin-walled steel box stub-columns. ASCE, J Struct Eng, 2000;126 (11):1304–11. 8] Aoki T, Susantha KAS. Seismic performance of rectangular-shaped steel piers under cyclic loading. ASCE, J Struct Eng, 2005;131(2):240–9.

ACCEPTED MANUSCRIPT P a g e | 12 9] Serrette R. Performance of edge-loaded cold-formed steel built-up box beams. ASCE, Practice Periodical on Structural Design and Construction, 2004; 9(3): 170–4.

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10] Jeon S., Cho M., Lee I. Static and dynamic analysis of composite box beams using large deflection theory. Computers and Structures 1995; 57(4):635–42.

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11] Cortinez V, Piovan M. Stability of composite thin-walled beams with shear deformability. Computers and Structures, 2006; 84: 978–90.

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12] Hsu H, Tsao J. Flexural–torsional performance of thin-walled steel hollow box columns subjected to a cyclic eccentric load. Thin-Walled Structures 2007; 45(2):149–58.

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13] Bedair, O. “Stability Limit State Design Of Box Sections Supporting Mining and Process Facilities” International Journal of Structural Engineering and Mechanics, 2011, 39 (5), 643-659. 14] Bedair, O. ”Dynamic Analysis of Box Girders with Tee-Stiffening Using Unconstrained Optimization Techniques” J. of Structural Multidisciplinary Optimization, 2011, 42 (4), 547-558.

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15] - Bedair, O. “Analysis and Limit State Design of Stiffened Plates and Shells: A World View”, Journal of Applied Mechanics Reviews, 2009, 62 (2), 1-16.

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16] Bedair, O.” A Cost-Effective Design Procedure for Cold-Formed Lipped Channels Under Uniform Compression” Thin-Walled Structures, 2009, 47 (11), 1281-1294

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17] Bedair, O.” Stability of Web plates in W-Shape Columns Accounting For Flange/Web Interaction” Thin-Walled Structures, 2009, 47 (6-7), 768-775.

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18] Bedair, O " Analytical Effective Width Equations for Limit State Design of Thin Plates Under NonHomogeneous In-Plane Loading” Archive of Applied Mechanics Journal, 2009, 79 (12), 1173–1189 19] Bedair, O. “Stability Analysis of Plates With Partial Restraints Using Unconstrained Optimization Techniques” Journal of Structural Stability and Dynamics, 2010, 10 (3), 571-587. 20] Bedair, O. “Residual Strength Assessments of Defective Stiffened Plates Used in Marine and Aerospace Structures”, Recent Patents on Engineering, 2012, 6 (2), 96-103 21] Bedair, O. “Practical Design Considerations For Light-Weight Channels under Combined Compression, Major and Minor Axes Bending”, ASCE, Practice Periodical on Structural Design and Construction, 2011, 16 (1), 15-23. 22] Bedair, O. “Recent Developments In Modeling And Design Procedures Of Stiffened Plates And Shells ", Recent Patents on Engineering., 2013, 7 (3), 196-208 23] Stamatelos, D, Labeas, G, Tserpes, K. “Analytical calculation of local buckling and postbuckling behavior of isotropic and orthotropic stiffened panels”, Thin-Walled Structures, 2011, 49 (3), pp. 422-430. 24] Zhang, Z., Chen, H. and Ye, L. “A stiffened plate element model for advanced grid stiffened composite plates/shells” Journal of Composite Materials, 2011, 45 (2), pp. 187-202. 25] Li, L. and Xiaohui, R. “Stiffened plate bending analysis in terms of refined triangular laminated plate element” Composite Structures, 2010, 92 (12), pp. 2936-2945.

ACCEPTED MANUSCRIPT P a g e | 13 26] Nath, S.K.D, Ahmed, S.R., Kim, S. “ Analytical solution of a stiffened orthotropic plate using alternative displacement potential approach” Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 2010, 224 (1), pp. 89-99.

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27] Jiang , W. Bao, G. and Robert, J. “Finite element modeling of stiffened and unstiffened orthotropic plates” Computers & Structures, 1997, Vol.63, pp.105-117

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28] Shanmugam, N, Lakshmi, B. and Uy, B An analytical model for thin-walled steel box columns with concrete in-fill. Engineering Structures, 2002, 24 825–838

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29] Uy B, Bradford M. Elastic local buckling of steel plate in composite steel–concrete. Eng Structure, 1996; 18(3):193–200.

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30] Sakai T, Sakino K, Ishibashi H. Experimental studies on concrete-filled square steel tubular short columns subjected to cyclic shearing force and constant axial force. Trans Architectural Institute of Japan (Tokyo), 1985; 353: 81–9.

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31] Wright H Buckling of plates in contact with a rigid medium. The Structural Engineer, 1993; 71(12): 209–215.

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32] American Iron Steel Institute (AISI) “North American Specification for the Design of Cold Formed Steel Structural Members”, 2007, Washington DC, USA.

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33] Canadian Standards Association “Limit states design of steel structures.” CAN/CSA-S16-01, 2007, Mississauga, Ontario, Canada

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34] Eurocode 3: Design of steel structures Part 1.5: plated structural elements; EN 1993-1-5: 2005. 35] Ziemian, R. “Guide to Stability Design Criteria for Metal Structures”, 6th edition, 2010, John Wiley and Sons Ltd. 36] Gaylord, E. and Gaylord, C. eds., (1996)."Structural engineering handbook." McGraw-Hill, USA. 37] Beedle, L. S., ed., (1991)." Stability of metal structures, a world view.", 2 nd Ed., Structural Stability Research Council, USA 38] Walker, A. C., (1978). “Maximum loads for eccentrically loaded thin-walled channel struts.", IABSE, 28: 169-181.

ACCEPTED MANUSCRIPT P a g e | 14

- APPENDIX (I)- BUCKLING PARAMETERS Values of the buckling parameters {λW1i , λW2i , λF1i , λF2i , ψW1i , ψW2i , ψF1i , ψF2i } of Eq.(9).

SC

RI

PT

1  1   II   I 0 0 0 0 0     1     1 I II I II F 1    1   II 1  F2   0 0 0 0 0  I   II  1  I   II  1     W 1   I  1  II 0 0 0 0        1   W 2 I II I  II  1     1I 0 0 0 0 0 0  1   II   I 

    0       II 1   2   I  0

 1  2    F1   1     2 F 2     1   2 W 1     1  2    W2

(A1-1)

the box section, and are given by;



 

 3 W1      i 1 i   6 ,  W1  i    i 4 

 

 3 W1      i 1 i   6   W1  i    i 4 

 

      

 IF 1   IF 2     IW 1    IW 2

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D



 3 F2      i 1 i   6 ,  F2  i    i 4 

MA

 3   ijF 1    iF 1   F2    ij     i 1 ,  6  W 1    ij F 1   W2 i    ij     i 4

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Note that {ψijF1, ψijF2 ψijW1 ψijW2 } contains the displacement functions and the geometric parameters of

(A1-2)

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where {γW1i , γW2i , γF1i , γF2i } contain the web and the flanges geometric parameters and are given by

 6 F1  2   i   i 61    bF   F2 L   i   i 1 

4   2 0 1 2 (1   ) 2(1   ) 1      1  I II   4 1 2 (1   2 2(1   ) 1   0  1   II   I 

 6 W1  2   2 0 2 1 2 (1   ) (1   ) 1    i   b       1 I II  i 61   W   2 L  1 2 (1   2 (1   ) 1    W2  0   i   1   II   I   i 1 

 1   1   b   W   bF   1   bw     bF  bw   bw     bF  bw   bw  b b   F w  1   1   b   F   bw   1   bF     bF  bw   bF     bF  bw   bF  b  b  w  F

(A1-3)

(A1-4)

ACCEPTED MANUSCRIPT P a g e | 15

APPENDIX (II) - GOVERNING POST BUCKLING EQUATIONS The non-linear middle surface strains of the flange and the flanges (F1, F2) are given by; 2

 WoF 1     x    2  WoF 1     y    W F 1 W F 1 2 x y

0

0

0

0

0

0

0

W F 1 W F 1 x y

0

0

0

0

U F 2 x

0

0

0

V F 2 y

0

0

0

U F 2 x

0 0

       

where WF1 and WF2

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     0    0     0    0   W F 2 W F 2   x y     0

2

 WoF 2     x    2  WoF 2     y    W F 2 W F 2 2 x y

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 W F 2   x   W F 2   y  V F 2 y

0

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2

    2    

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 W F 1   x   W F 1   y  V F 1 y

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 F1  U  x   V F 1   y F1    xx  U F 1  F1     yy   x  F1   xz  F2   0  xx    F2   yy   F2  xy  0      0    

 1       0.5       0.5 1       0.5       0.5  1         

(A2-1)

are the out of plane flange deflections, WF1o and WF2o are the initial flange

imperfections, {εF1xx, εF2xx } and {εF1yy, εF2yy} are the flanges membrane strains components in the x

D

and y directions, and {εF1xy , εF2xy } are the shear strains. By similar analogy, the middle surface

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strains of the webs (W1, W2) can be determined.

   x  F1    x  F1  0    0    0    0   0    0   2  2  x F1  2  2  x F 2  2  2  xW 1  2  2  xW 2

2 y F2 1 2 y F2 2 2 yW2 1 2 yW2 2

 y F 1  y F 1

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The stress and moment equilibrium eqs. of the assembled webs and the flanges are given by; 0

0

0

0

0

0

0

0

0

0

 xF 2  xF 2

 yF 2  yF 2

0

0

0

0

0

0

0

0

0

0

0 0

  x F1 y F1  2    x F 2 y F 2  2    xW 1 yW 1  2    xW 2 yW 2 

0

0

0

0

0

0

 xW 1  xW 1

 yW 1  yW 1

0

0

0

0

0

0

2

0 0  xW 2  xW 2



M xxF1 M xxF 2 M Wxx1 M Wxx 2   F1 F2 W1 W2 M yy M yy M yy M yy    xx M F1 M F 2 M W 1 M W 2  xy xy xy  xy   

1  t 2 w F        

 0   0    0    0    0    0     yW 2     yW 2 

F1 F1  xx  xy 0 0 0 0 0 0  0   F1    F1   0 0 0 0 0 0  yx  yy 0  F2 F2  0 0  0  xx  xy 0 0 0 0      F2 F2 0  yx  yy 0 0 0 0   0 0     W1 W1 0  0 0 0  xx  xy 0 0   0   W 1 W 1  0 0 0 0  yx  yy 0 0  0   0  W2 W2 0 0 0 0 0 0    xx xy    W2 W2  0 0 0 0 0 0 0   yx yy  



 I   II  1

0



0



0

0

1 2 w  I   II  1 tF

0

0

1 2 F1  II   I  1 tW

0

0

0

0



  2W F1      x F2 1     2W F 2    0   x F2 2    2 W1    W  0   x 2    W1  1 2 F 2  II   I  1    2W W 2  tW   2   xW 2 

(A2-2)







(A2-3)

ACCEPTED MANUSCRIPT P a g e | 16 By substituting the non-linear form of the strains Eqs.(14) into the stress and moment equilibrium equations (16)& (17), yield the governing post-buckling differential equations for the flanges ;

 4  F2 2 F2 2

 4 F4 1

F1

  2W F 2         F2 F2 

F1

F2

 4 F4 2

F1

 4 WT F4 1

F2

 4 WT F4 2

 4 WT  F4 2 F2 2

2

F2

F2

 4 WT  F4 1 F2 1

  2W F 1         F1 F1 

2

  2WoF 1         F1 F1 

F1 F1  2W  2W 2 2  F 1  F 1

2

2

F2 F2  2W  2W 2 2  F 2  F 2

  2WoF 2         F2 F2 

2 F1 4 F1  W  2  2 F 1  F 1

F1 F1  2  2W  2 2  F 1  F 1

F1 F1  2  2W 2  F 1 F 1  F 1F 1

2 F2 4 F2  W  2  2  F 2  F 2



F2 F2  2  2W 2 2  F 2  F 2

2

F1 F1  2Wo  2Wo    F2 1 F2 1   F2 F2   2Wo  2Wo   F2 2 F2 2      0     0 

PT

F1

RI

 4  F2 1 F2 1

F2 F2  2  2W  F 2F 2  F 2F 2

SC

 F1   4   4 F1    4 F 2    F4 2   F1   4 WT   4 F1   4 F2 W  T    F4 2

 1   2 2    0   4    0   2  E    0   E 2     0  2   E   E  2   

(A2-4)

NU

where {ΦF1 , ΦF2} are the flange stress functions, that (WT) is the net out of plane deflection =(WWo), θF = L/bF and βF = 2 (1- ν2)/E(tF)2 . Note that the form of the stress functions {ΦF1 , ΦF2} depends upon the applied stress distribution in the flange. y similar substitution, the governing the

AC CE P

TE

D

MA

post-buckling equations of the webs (W1, W2), are determined.

ACCEPTED MANUSCRIPT P a g e | 17

- NOTATIONS L = length of the box section; bF, = width of the flange ; bW = width of the web;

PT

D = flexural rigidity; E = elastic modulus;

RI

KW1 , KW2 , KF1 , KF2= webs and flanges buckling coefficients; M1, M2= Applied bending moments.

SC

Mwxx, Mwxy , Mwxy , MF1xx, MF1xy , MF1xy, MF2xx, MF2xy , MF2xy Web and flanges bending moments; P = Applied axial force; oW1, oW2 , oF1, oF2 = webs and flanges origins

NU

ζW1buc, ζW2buc , ζF1buc , ζF2buc = webs and flanges buckling stresses; tW, tF = web and flange thicknesses;

MA

v = Poisson ratio;

xW1, yW1 , xW2, yW2 , xF1, yF1,xF2, yF2 = webs and flanges local coordinates; Pw , ωwj, TF1,θF1, QF2, γF2 = webs and flanges displacement functions; Ww, Uw,Vw , WF1, UF1 VF1 , WF2, UF2 VF2 = webs and flange displacement components;

D

ξW1, ηW1 , ξW2, ηW2 , ξF1, ηF1 , ξF2, ηF2 = non-dimensional webs and flanges coordinates;

TE

εW1xx, εW1xy , εW2xx , εW2xy , εF1xx, εF1xy , εF1xy, εF2xx, εF2xy , εF2xy =webs and flanges mid-plane strains; ζwxx, ζwxy , ζwxy , ζF1xx, ζF1xy , ζF1xy, ζF2xx, ζF2xy , ζF2xy Web and flange stresses; ζ SER , ζ ULT = Serviceability and strength limiting stresses;

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WW1, UW1,VW1 , WW2, UW2,VW2, WF1, UF1, VF1 , WF2, UF2, VF2 webs and flanges displacements; IW1- IW15, IF1 - IF5 = webs and flanges integral functions;

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ACCEPTED MANUSCRIPT

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ACCEPTED MANUSCRIPT

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ACCEPTED MANUSCRIPT

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ACCEPTED MANUSCRIPT

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ACCEPTED MANUSCRIPT P a g e | 23 AISI/CSA-S16 Eq.(10)

Eurocode 3 Eq.[11]

Ref.[36] Eq. (13)

Ref.[37] Eq. (14)

Present

6 4 3 2 1

4.6 4.9 5.3 5.9 8.0

4.7 5.0 5.3 5.9 7.8

4.6 4.9 5.3 5.9 7.6

4.7 5.0 5.3 6 7.8

4.7 5.0 5.3 6.0 8.0

RI

PT

σxx/σ1

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Table.(1)