anewsimplemethodforthestrengthofhigh-strengthsteel thin...

Research ArticleA New Simple Method for the Strength of High-Strength SteelThin-Walled Box Columns Subjected to Axial Force and BiaxialEnd Moments

Hong-Xia Shen

School of Civil Engineering Xirsquoan University of Architecture and Technology Xirsquo an Shaanxi 710055 China

Correspondence should be addressed to Hong-Xia Shen shenhongxia888163com

Received 9 April 2019 Accepted 1 July 2019 Published 18 July 2019

Academic Editor Eric Lui

Copyright copy 2019 Hong-Xia Shen +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

+e strength of Q460 steel welded thin-walled box columns under biaxial bending is investigated by an FE model +e numericalresults together with available experimental data are compared with the American Specification ANSIAISC 360-10 It shows thatANSIAISC 360-10 provides a good estimate over a wide range of column slenderness ratios but it severely underestimates thestrengths of beam-columns with large slenderness ratios and slightly underestimates those of beam-columns made fromHSS withfy 741MPa and the calculation is more complicated because of using an effective width concept+erefore a simple and effectivecalculation method is proposed Also a comparison is made between the proposed formulas and the available experimentalresults It shows that the proposed formulas can precisely evaluate the local-overall interactive buckling strength of HSS beam-columns and inmost cases are also suitable for beam-columns fabricated frommild steels+e proposed formulas are simple andmeanwhile achieve the same level of accuracy as ANSIAISC 360-10

1 Introduction

Structural steel with a nominal yield stress (denoted by fy)greater than or equal to 420MPa is often defined as high-strength steel (HSS) HSSs such as Q460 and Q690 thenominal yield stress of which is 460MPa and 690MParespectively have been successfully applied in large-spanstructures high-rise buildings and transmission towers inChina owing to having high plasticity good toughnessand excellent weldability Economical design of HSSstructures leads to thin-walled members +ese thin-walled members will very likely fail in a mode of in-teraction between local plate buckling and overall memberbuckling also called local-overall interactive buckling inwhich local buckling of plate elements occurs prior tooverall buckling of members and the local bucklingweakens the overall stiffness and reduces the overallbuckling strength

A large amount of research work has been done on thelocal-overall interaction buckling strength of welded I- and

box-section columns [1ndash14] and beam-columns [2 6 1112 15ndash23] Nonetheless at present there are a few researchstudies about HSS beam-columns [11 12 20ndash23] In theearly 1980s an experimental study was performed by Usamiand Fukumoto [11 12] on the local-overall interactionbuckling strength of HSS welded box columns and anempirical formula for the buckling strength was suggested Atotal of 23 specimens with fy 460MPa and 27 specimenswith fy 690MPa were tested to failure under concentricload and uniaxial eccentric load material properties re-sidual stresses and column initial deflections were alsomeasured +e column slenderness ratios ranged from 30 to65 and the plate width-to-thickness ratios from 22 to 58Until recent years a series of numerical simulations of in-plane and out-of-plane ultimate carrying capacities of Q460steel welded I- and box-section columns loaded with aneccentricity in a plane of symmetry were conducted by Shenand Yang [20] Shen and Liu [21] Shen [22] and Shen andZhao [23] respectively +e beam-columns were selected tocover a wide range of plate width-to-thickness ratios ranging

HindawiAdvances in Civil EngineeringVolume 2019 Article ID 7495890 16 pageshttpsdoiorg10115520197495890

from 40 to 70 and a wide range of member slenderness ratiosfrom 20 to 120 +e effects of plate width-to-thickness ratiocolumn slenderness ratio and eccentricity on the ultimatecarrying capacities were investigated and the calculationformulas for the ultimate strengths were proposed How-ever all these studies focused on the HSS beam-columnsunder uniaxial bending and more attention should be paidto those under biaxial bending

In addition no strength formulas are provided speciallyfor HSS welded beam-columns in some existing standardsGB 50017-2017 [24] ANSIAISC 360-10 [25] EN 1993-1-1[26] and EN 1993-1-5 [27] +ese standards are all based onthe research results of mild steels and are also used for HSSsbut it is unknown whether they are applicable for HSSmembers Furthermore when the strength of thin-walledbox beam-columns under biaxial bending is calculated inaccordance with the above standards a cumbersome cal-culation has to be made because of the effective widthmethod which is used in these standards Hence a simplemethod needs to be provided

In this study the local-overall interaction buckling be-havior of Q460 steel welded thin-walled box beam-columnsunder biaxial bending is analyzed by the commercial soft-ware ANSYS 80 [28 29] +e numerical results as well asavailable experimental results of different steel grades arecompared with ANSIAISC 360-10 to check whether ANSIAISC 360-10 is suitable for such types of beam-columnsSimple formulas are presented for predicting the local-overall interactive buckling strength of Q460 steel beam-columns subjected to axial compression and biaxial bendingmoments +e proposed formulas are verified by comparingthem with the available experimental results

2 Finite Element Analysis

21 An FE Model and Its Verification A pin-ended beam-column as shown in Figure 1 with a welded thin-walled boxcross section fabricated from Q460 subjected to combinedaxial compression force P and biaxial bending momentsMx and My is investigated in this paper Here the bendingmoments around the x- and y-axesMx andMy are taken asMx Pey and My Pex where ex and ey are the eccentricitiesin the x and y directions respectively Also given inFigure 1(b) are the dimensions of the welded box section inwhich b and h are the widths of plate elements and t is thethickness of the plate elements

A geometrical and material nonlinear FE model wasestablished by Shen [9 22] for HSS box columns loaded withaxial compression and uniaxial eccentric compression re-spectively by using the ANSYS program In this study theFE model is modified for HSS box columns loaded withbiaxial eccentric compression

As described by Shen [9 22] the effects of initial geo-metrical imperfections and residual stresses are taken intoaccount +e initial geometrical imperfections include bothoverall member imperfections and local plate imperfections+e member imperfections are assumed to bend in a halfsine wave about the major and minor axes respectively asfollows

]0 l

1000sin

πz

l (1a)

μ0 l

1000sin

πz

l (1b)

where l is the length of the memberIn equations (1a) and (1b) the maximum value of out-of-

straightness of a member is taken equal to l1000 as specifiedby GB 50017-2003 [30] and ANSIAISC 360-10 [25]

+e plate imperfections are assumed as

ω ω1 sinmπz

lcos

πx

by plusmn

h

2+

t

21113888 11138891113888 1113889 (2a)

ω ω2 sinmπz

lcos

πy

hx plusmn

b

2+

t

21113888 11138891113888 1113889 (2b)

where ω1 b1000 ω2 h1000 and m is the buckling half-wave number along the longitudinal axis of the member

+e magnitudes of plate imperfections were taken asb100 and b1000 (b is the plate width of a square box section)[8] and it showed that the magnitude of plate imperfectionshad a significant effect on the compressive strength of weldedbox columns No uniform value was specified for the max-imum of plate imperfections and it was taken as 001 [4] andb1000 [3] but it was demonstrated that the value of b1000seemed to be reasonable [3] when residual stresses weretaken into consideration Hence for a square box sectionω1 ω2 b1000 But for a rectangular one according to thedeformation compatibility condition the angle between thetwo adjacent plates remained unchanged before and after theplate buckling so ω1 b1000 and ω2 h1000 herein

Residual stress measurements of welded square andrectangular box sections were made by Usami and Fuku-moto [11] and Pavlovcic et al [6] respectively According totheir measurement results a simplified pattern of residualstress distribution as shown in Figure 2 is adopted wheretensile residual stresses are positive values and compressivestresses are negative values An SM58 steel plate was used[11] its nominal yield strength was 460MPa and itsmeasured yield strength was 568MPa +e maximum valueof tensile stresses was about 80 of the measured yieldstrength ie 08times 568 454MPa which is very close to thenominal yield strength of 460MPa+e value of compressiveresidual stresses decreased along with the increase of thewidth-to-thickness ratio of the plate [6 11] +e distributionwidth of tensile residual stresses was about 3t [9] As aconsequence the maximum value of tensile residual stressesσrt is taken as the nominal yield stress of steel and those ofcompressive residual stresses σrcf and σrcw along the flangeand web respectively may be obtained according to equi-librium conditions

+e Shell181 element is employed to take account ofplate imperfections Shell181 is suitable for analyzing thin tomoderately thick shell structures It is a 4-node element withsix degrees of freedom at each node translations in the x yand z directions and rotations about the x- y- and z-axes Itis well suited for linear large-rotation andor large-strainnonlinear applications Also the effects of plasticity of steel

2 Advances in Civil Engineering

member imperfections and residual stresses can be con-sidered by using Shell181 As a result the Shell181 elementsatises the requirements of this study

e experiment results [31] showed that stress-straincurves for Q460 steel as shown in Figure 3 were similar tothose for mild steel and also had a yield plateauematerialmodels with and without strain hardening had almost noinuence on the compressive strength of welded box col-umns [8]e similar nding was also noted by Pircher et al[7] in the simulation of the buckling of thin-walled steel boxtubes So an ideal elastic-plastic material model is assumedto represent the stress-strain relationship for Q460 steel e

nominal material properties yield stress fy 460Nmm2Youngrsquos modulus E 206000Nmm2 Poissonrsquos ratio v 03and tangent modulus Et 0 are used

e eect of geometrical imperfections is taken intoaccount by direct modeling according to the following stepsFirstly the number of nodes required in the FE model andthe order in which the nodes should be generated are de-termined Secondly all nodes are generated according to thex y and z coordinates considering geometrical imperfec-tions irdly the element attributes including the elementtype material model and real constant have been set andshell elements are automatically generated within each areadened by four nodes For any plate element in a boxcolumn element sizes are b8 or h8 along its width and h4

x

x

y

y

z

Mx

Mx

My

MyP

(a)

h

t

y

x

tt

b

t

(b)

Figure 1 A beam-column subjected to biaxial bending (a) a force diagram (b) cross section and its dimensions

+

ndash

ndash

σrcf

σrt

σrt

x

b

tt

t

t

h

y

+

+ +

σrcw

σrt σrt

Figure 2 Residual stress distribution pattern

0

100

200

300

400

500

600

700

0 005 01 015 02 025

Stre

ss (M

Pa)

Strain

10mm thickness12mm thickness14mm thickness

Figure 3 Stress-strain curves for Q460 steel

Advances in Civil Engineering 3

along its length and thus a finite element grid is b8times h4 orh8times h4 In this paper the minimum grid is approximately20mmtimes 40mm and the maximum one 50mmtimes 100mmwhich satisfies the accuracy requirement of the numericalsimulation Figure 4 shows the initial geometrical imper-fections generated by direct modeling in which the initialdeflections are magnified 5000 times only to be observedclearly

Residual stresses are treated as initial stresses that areloads and must be applied at the integration points ofShell181 elements For a Shell181 element there are fourintegration points in its plane together with 5 integrationpoints in the thickness direction to take account of nonlinearbending properties and thus the total number of integrationpoints is 20 In order to input initial stresses in the FEmodeltwo steps must be performed Firstly an initial stress fileneeds to be written based on the integration point locationsSecondly the initial stress file must be read by using theISFILE command only at the first substep of the first loadstep as specified in the ANSYS program In this study thesame residual stress distributions are assumed along thethickness direction because of thin plate elements

+e loading and boundary conditions of the beam-columns studied in this paper are different greatly fromthose of the beam-columns [22] Figure 5 displays the loadsand boundary conditions in the FE model As shown inFigure 5 a 30mm thick end plate is attached to both ends ofa beam-column [32] and an eccentric load at every end ofthe member is equivalent to combined axial compressiveforce and biaxial bending moments +ese forces as well asthe boundary conditions are applied at the center points ofboth end plates [32] At one end z 0 besides the bendingmomentsMx andMy the three translationsUxUy andUz inthe x y and z directions and the rotation Rotz about the z-axis are prevented ie Ux Uy Uz Rotz 0 At the otherend z l besides P Mx and My the boundary conditionssatisfy Ux Uy Rotz 0 In addition both ends of themember are restrained against the rotation about the lon-gitudinal axis by applying Ux Uy 0 at the four cornerpoints of the end plate

+e available experimental data are used to verify the FEmodel 35 experimental specimens fabricated from two typesof HSSs having fy 460MPa and 690MPa of which 21 werecentrally loaded and 14 eccentrically loaded in a symmetryplane by Usami and Fukumoto [11 12] were simulated byShen [22] and the ratios of the numerical and experimentalultimate strength change from 0901 to 1084 with an averagevalue of 1012 and a standard deviation of 453 Here theFE model [22] was modified for biaxial bending but nopublished experimental information could be found on thebeam-columns made from HSSs and subjected to biaxialbending Hence in this section 28 experimental specimenstested by Richard Liew et al [16] fabricated from mild steelswith the nominal yield strength of 353MPa 268MPa and293MPa were simulated to verify the effectiveness of themodified FEmodel Among these specimens 10 were loadedwith uniaxial eccentricity and labeled with a single letter X orY as shown in Table 1 and 18 were loaded with biaxialeccentricity and marked with two letters XY +e second

field S of the specimen number indicates a square boxsection while R indicates a rectangular box section +ethird field refers to the web depth-to-thickness ratio and thelast field denotes the column slenderness ratio about theminor axis Also listed in Table 1 are the main parameters ofthe specimens including the width-to-thickness ratio btdepth-to-thickness ratio ht maximum initial deflectionsδ0xL and δ0yL about x- and y-axes respectively eccen-tricities ex and ey in x and y directions respectively andmeasured yield stress fy All these parameters as well as thematerialrsquos Youngrsquos modulus Poissonrsquos ratio and residualstress patternsmeasured by Richard Liew et al [16] are used

A comparison between the experimental and numericalresults is also presented in Table 1 where Pexp and Pfem arethe ultimate strengths obtained from the experimental resultand numerical simulation respectively +e ratios of theexperimental and numerical ultimate strength PexpPfemvary within a range of 086 to 112 with an average value of097 and a standard deviation of 68 It can be seen that thenumerical results are in good agreement with the experi-mental ones and the FE model established in this study canprecisely predict the local-overall interactive bucklingstrength of welded box beam-columns under both uniaxialbending and biaxial bending

22 Buckling Behavior of Q460 SteelWelded9in-Walled BoxBeam-Columns By using the FE model mentioned above agreat amount of numerical simulation work with differentparameters has been performed on the local-overall in-teractive buckling of Q460 steel thin-walled box columnsunder the biaxial eccentric loading condition +e param-eters include the cross-sectional shape plate width-to-thickness ratio column slenderness ratio and applied ec-centricities ex and ey

In the calculation the cross sections include square andrectangular box sections and the depth-to-width ratio of therectangular boxes is 125+e thickness of the plate elementst is taken as 4mm and kept constant

Plate width-to-thickness ratios together with columnslenderness ratios are selected elaborately to ensure that thelocal-overall interaction buckling occurs In accordance withthe Chinese Code GB 50017-2003 [30] the limiting width-to-thickness ratio is 40

235fy

1113969 40 times

235460

radic 286 for

compression elements of members subjected to axialcompression and bending+is limiting value is 14

EFy

1113969

14 times206000460

radic 296 in ANSIAISC 360-10 [25] Both

of them are basically the same When the width-to-thicknessratio is greater than the limiting value the local-overallinteraction buckling will occur +erefore the width-to-thickness ratio bt is taken as 40 50 60 70 and 80 re-spectively whether a square or a rectangular box crosssection If it is a rectangular box the corresponding depth-to-thickness ratio ht will be 50 625 75 875 and 100respectively

To obtain the predicated failure mode of local-overallinteraction buckling the beam-columns with mediumslenderness ratios should be selected+erefore the memberslenderness ratio about the major axis λx (λx lix where l is


(a) (b)

Figure 4 Initial geometrical imperfections plate imperfections and member imperfections about the (a) x-axis and (b) y-axis

z = 0

z = l

(a) (b) (c)

Figure 5 Loading and boundary conditions (a) overall model (b) loads and boundary conditions at z 0 (c) loads and boundaryconditions at z l

Table 1 Comparison between the experimental and numerical results [16]

No Specimen bt ht δ0xL (times10minus4) δ0yL (times10minus4) ex (mm) ey (mm) fy (Nmm2) Pexp (kN) Pfem (kN) PexpPfem

1 X-S-30-85 295 298 mdash 06 15 0 353 20999 20462 1032 XY-S-30-85a 295 298 175 minus367 5 10 353 22462 20784 1083 XY-S-30-85b 292 292 063 075 5 5 353 25437 22788 1124 X-S-45-57 439 446 mdash 08 15 0 353 32530 33481 0975 XY-S-45-57a 442 442 377 214 5 15 353 31224 33087 0946 XY-S-45-57b 442 442 minus189 minus233 15 15 353 28674 30655 0947 X-S-52-48 524 524 mdash 088 20 0 353 32473 31724 1028 XY-S-52-48a 524 524 189 251 15 10 353 33872 31420 1089 XY-S-52-48b 521 518 151 176 10 10 353 34863 33561 10410 Y-S-64-64 642 642 300 mdash 0 10 268 11439 11742 09711 XY-S-64-64a 642 637 233 641 10 5 268 10818 11563 09412 XY-S-64-64b 648 642 453 503 15 15 268 8325 8830 09413 X-S-75-55 759 759 mdash 277 15 0 268 11811 12463 09514 XY-S-75-55a 759 753 189 378 15 10 268 10924 11773 09315 XY-S-75-55b 755 753 06 189 15 15 268 10263 11360 09016 Y-S-85-48 86 856 minus276 mdash 0 20 268 11766 12450 09517 XY-S-85-48a 85 861 minus466 466 20 10 268 11530 12150 09518 XY-S-85-48b 86 856 minus189 minus441 15 15 268 11159 12000 09319 X-R-53-64 403 537 mdash minus378 19 0 293 26176 25157 10420 XY-R-53-64a 396 54 378 428 20 10 293 26037 23700 11021 XY-R-53-64b 396 54 minus441 441 20 20 293 23670 23700 10022 XY-R-53-64c 399 537 minus554 minus315 10 20 293 26897 26400 10223 Y-R-53-64 399 54 minus441 minus125 0 20 293 31195 31478 09924 X-R-86-62 70 861 mdash minus251 20 0 268 10521 11406 09225 XY-R-86-62a 70 861 396 minus466 20 10 268 9714 11085 08826 XY-R-86-62b 70 861 minus293 409 20 20 268 9372 10921 08627 XY-R-86-62c 70 861 252 378 10 20 268 10272 11550 08928 Y-R-86-62 70 861 minus466 267 0 20 268 11452 12300 093Average 097Standard deviation() 68


the effective length of a member and ix is the gyration radiusabout the major axis ie x-axis) is taken equal to 40 60 80and 100 respectively

Eccentricities ex and ey are taken as two of 10mm15mm 20mm 25mm 30mm 35mm 40mm and 50mmrespectively satisfying ex ey or eyminus ex 5mm in order tosave materials

+e beam-columns are marked with numbers such asS-40-40 (10 10) in which the first S indicates a square boxsection the second is the width-to-thickness ratio the thirdis the column slenderness ratio about the x-axis and thefigures in brackets are the eccentricities ex and ey re-spectively If it is a rectangular box section S is replaced by Rand another figure following the second part is addedrepresenting the depth-to-thickness ratio

221 Load-Displacement or Load-Rotation Curves Figure 6shows the flexural deformation of a beam-column whereμmax and vmax are the maximum deflections in the x and ydirections respectively and the section 1-1 is the crosssection of the member at midlength Points A and B are themidpoints of the upper flange and left web of the crosssection 1-1 respectively Owing to the interaction betweenlocal buckling and biaxial bending the maximum deflectionsin the x and y directions of the points on the cross-section 1-1are different from each other In this study μmax is taken asthe deflection of the point B and vmax as that of the point A

+ere is a large difference in values of the displacementbetween a long and a short column so their load-dis-placement and load-rotation curves are plotted separatelyFigures 7 and 8 show the typical ones for the member S-40-40 (40 40) that represents a short column and the ones forthe member S-50-100 (30 35) that represents a long columnrespectively In Figures 7 and 8 the curves of the axialcompressive force P versus the axial compressive dis-placement Δ the maximum deflection in the x directionμmax the maximum deflection in the y direction vmax andthe maximum rotation about the z-axis θmax which is takenas the rotation of the point A as above are shown re-spectively It can be seen from Figures 7 and 8 that whenreaching the ultimate strengths (44036 kN and 33845 kN)for the member S-40-40 (40 40) Δ 721mmμmax 1224mm vmax 1286mm and θmax 0005 radwhile for the member S-50-100 (30 35) Δ 1271mmμmax 6492mm vmax 8493mm and θmax minus0001 radFrom these values a conclusion can be drawn that thebending deflections about the x- and y-axes and axialcompressive displacement are large and the rotation aboutthe z-axis is very small for both the short and long box-section beam-columns subjected to biaxial bending Such asmall twisting effect is due to the fact that a box section haslarge torsional rigidity

222 Von Mises Stress Distribution of the Cross Section atMidlength A biaxial bending beam-column of this typemight be in the elastic region when it reaches the maximumstrength It can be determined by viewing von Mises stressdistribution A typical member R-50-625-60 with bt 50

ht 625 and λx 60 is selected to exhibit the von Misesstress distribution of the cross-section 1-1 as shown inFigure 6 where the maximum bending moments occur

To make it easier to describe the von Mises stress dis-tribution the coordinate system as shown in Figure 6 isdefined again +e top-left corner point of the cross-section1-1 is taken as the coordinate origin the x-axis is horizontalwith values increasing from left to right and the y-axis isvertical with values increasing from top to bottom +e vonMises stress distributions at the ultimate loads are plotted inFigure 9 under different eccentricities In Figure 9 all ec-centricities ex and ey are in mm A little difference in vonMises stress distributions exists in the bottom flange andright web under eight different eccentricities but obviousdifference is observed in the top flange and left web +emaximum von Mises stress of 45277MPa occurs at thecorner point between the top flange and the left web whenex 50mm and ey 50mm All these stresses are smallerthan the steel yield stress of 460MPa and the beam-columnis still in the elastic range and does not enter the elastic-plastic range under any eccentricity condition

223 Interaction between Bending Moment and Axial ForceIt is very important for a beam-column subjected to biaxialbending to find the relationship equation between the axialload and end bending moments +e scatter diagrams be-tween PuPy and MuxMxy (or MuyMyy) of Q460 steel thin-walled box-section beam-columns with five different width-to-thickness ratios are essentially the same so Figures 10 and11 only show the ones for beam-columns with square boxsections and bt 40 60 and 80 and for those with rectan-gular box sections and bt 60 and 80 respectively whereMux Puey Muy Puex Py Afy MxyWx fy andMyyWyfy (A is the gross area and Wx and Wy are the grosssectionmoduli about x- and y-axes respectively) In Figures 10and 11 the scatter plots are the finite element results and alsothe straight lines that predict the changing trends of all thecoordinate points are added It is seen in Figures 10 and 11 thatthe straight lines can fit very well the overall trend of the scatterplots between PuPy and MuxMxy (or MuyMyy)

3 Evaluation of the AmericanNational Standard

31 Comparison between EN 1993-1-1 andANSIAISC 360-10In the Eurocode EN 1993-1-1 [26] for a member undercombined axial compression and biaxial bending the fol-lowing conditions should be satisfied

N

χxNR

+ kxx

Mx + ΔMx

χLTMxR

+ kxy

My + ΔMy

MyR

le 1 (3a)

N

χyNR

+ kyx

Mx + ΔMx

χLTMxR

+ kyy

My + ΔMy

MyR

le 1 (3b)

where N Mx and My are the axial compression force andthe maximum moments about the x- and y-axes re-spectively ΔMx and ΔMy are the moments due to the shift


of the centroidal axis for class 4 sections NR is the re-sistance to axial compression force MxR and MyR are theresistances to bending moments about the x- and y-axesrespectively χx and χy are the reduction factors due toexural buckling χLT is the reduction factor due to lateraltorsional buckling and kxx kxy kyx and kyy are the in-teraction factors

is type of cross section studied in this paper is calledclass 4 section in EN 1993-1-1 [26] For class 4 cross sectionsthese conditions are expressed as

N

χxAefffy+ kxx

Mx +NeNxχLTWeff xfy

+ kxyMy +NeNyWeff yfy

le 1 (4a)

N

χyAefffy+ kyx


+ kyyMy +NeNyWeff yfy

le 1 (4b)

where Aeff Weff x andWeff y are the eective cross-sectionalarea and eective section moduli about the x- and y-axesrespectively and eNx and eNy are the eccentricities withrespect to the neutral axes

According to the American National Standard ANSIAISC 360-10 [25] for members subjected to axial force andbiaxial end moments the following conditions should bemet

PrPn+89

Mrx

Mnx+Mry

Mny( )le 10 when

PrPnge 02 (5a)

Pr2Pn

+Mrx

Mnx+Mry

Mny( )le 10 when

PrPnlt 02 (5b)

where Pr is the required compressive strength Pn is thenominal compressive strengthMrx andMry are the required

νmax

y

z

Mx Mx

l

P P1-1

(a)

l

x

1-1P

Pz

My My

μmax

(b)

A

1-1

B

(c)

Figure 6 Flexural deformation about the (a) x-axis (b) y-axis and (c) cross section 1-1 at midlength

0

100

200

300

400

500

0 2 4 6 8 10

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

500

0 5 10 15 20P

(kN

)

μmax (mm)

(b)

0

100

200

300

400

500

0 5 10 15 20

P (k

N)

νmax (mm)

(c)

0

100

200

300

400

500

0 001 002 003

P (k

N)

θmax (rad)

(d)

Figure 7 Load-displacement and load-rotation curves for the member S-40-40 (40 40) (a) P-Δ (b) P-μmax (c) P-vmax (d) P-θmax


exural strengths about the x- and y-axes respectively andMnx andMny are the nominal exural strengths about the x-and y-axes respectively

In ANSIAISC 360-10 [25] the cross section studiedherein is called the slender section For slender sections PnMnx and Mny are determined on the basis of the eectivewidth concept (see Section 32)

Comparing the two standards the following can beenfound (1) Linear interaction equations between the com-pression force and biaxial bendingmoments are used in boththe standards but two formulas in EN 1993-1-1 and onlyone formula in ANSIAISC 360-10 (2) e second-ordereect of P-δ is taken into account It is considered in theinteraction factors kxx kxy kyx and kyy in EN 1993-1-1 whileit is included in the required exural strengths Mrx and Mryin ANSIAISC 360-10 but the calculations of kxx kxy kyxand kyy are very complicated while those ofMrx andMry aresimple (3) For a beam-column with a thin-walled sectionboth of the specications adopt the eective width conceptIn short compared to EN 1993-1-1 the calculationaccording to ANSIAISC 360-10 becomes much simplerHence a comparison between ANSIAISC 360-10 and theavailable experimental results and that between ANSIAISC360-10 and the numerical results will be made in the fol-lowing sections to evaluate the applicability of ANSIAISC360-10 to HSS thin-walled beam-columns

32 Comparison between ANSIAISC 360-10 and the Avail-able Experimental Results When making a calculation byemploying equations (5a) and (5b) the values of Pn MnxMny Mrx and Mry must be determined rst

For slender sections the nominal compressive strengthPn should be taken as

Pn FcrA (6)

where Fcr is the exural buckling stress and determined by

Fcr Q 0658QFyFe[ ]Fy forKL

rle 471

E

QFy

radic(7a)

Fcr 0877Fe forKL

rgt 471

E

QFy

radic(7b)

where K is the eective length factor L is the laterallyunbraced length of the member r is the governing radius ofgyration Fy is the steel yield stress Fe is the elastic criticalbuckling stress and Q is the reduction factor for slenderstiened elements QQa

Qa is dened as

Qa Ae

A (8)

where Ae is the eective cross-sectional area based on ef-fective width be

0

100

200

300

400

0 5 10 15 20

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

0 25 50 75 100 125

P (k

N)

μmax (mm)

(b)

νmax (mm)

0

100

200

300

400

0 50 100 150

P (k

N)

(c)

0

100

200

300

400

ndash001 0 001 002

P (k

N)

θmax (rad)

(d)



e eective width be is expressed as follows

be 192t

E

f1

radic

1minus038(bt)

E

f1

radic le b (9)

wheref1

PnAe (10)

It requires iteration to calculate the strength of a columnaccording to equations (6)ndash(10) In order to simplify thecalculation f1 is taken equal to Fy as specied in ANSIAISC

360-10 [25] is will result in a slightly conservative esti-mate of available column strengths

For slender sections the nominal exural strengths Mnxand Mny shall be determined as follows

Mnx FySex (11a)

Mny FySey (11b)

where Sex and Sey are the eective section moduli about thex- and y-axes and determined with the eective width berespectively

140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)

ex = 10 ey = 10ex = 10 ey = 15ex = 20 ey = 20ex = 20 ey = 25ex = 30 ey = 30

ex = 30 ey = 35ex = 40 ey = 40ex = 50 ey = 50fy = 460MPa

(a)

60

110

160

210

260

310

360

410

460

0 76 152 228 304y (mm)

Von

Mise

s str

ess (

MPa

)



(b)

140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(c)

60

110

160

210

260

310

360

410

460

0 76 152 228 304



y (mm)

Von

Mise

s str

ess (

MPa

)

(d)

Figure 9 Von Mises stress distributions (a) top ange (b) left web (c) bottom ange (d) right web


In order to consider the second-order eect of P-δ Mrxand Mry are calculated using the following equations

Mrx CmxPrey

1minus PrPEx( ) (12a)

Mry CmyPrex

1minus PrPEy( ) (12b)

where Cmx and Cmy are the equivalent moment factors andPEx and PEy are Eulerrsquos critical loads

Equations (5a) and (5b) are veried by the availableexperimental results (for slender box sections) as shown inTable 2 ese experimental results are taken from dierentliteratures with the measured steel yield stress of268sim741MPa By substituting these experimental values intothe left-hand side of equations (5a) and (5b) the calculationresults marked with ldquo(1)rdquo in Table 2 are obtained and are

closer to 1 except for the three specimens with λx 85 [16]and the two specimens with fy 741MPa [12] e averagevalues of ldquo(1)rdquo in Table 2 are about 158 and 126 for thespecimens with λx 85 (fy 353MPa) and those withfy 741MPa respectively Comparisons show that thestandard ANSIAISC 360-10 agrees in most instances verywell with the experimental results e ANSIAISC 360-10standard provides a better evaluation of the strengths ofbeam-columns with medium slenderness ratios andfy 248MPasim373MPa It underestimates the strengths ofbeam-columns with fy 741MPa but overestimates thecapacities of members with fy 568MPa

33ComparisonbetweenANSIAISC360-10andtheNumericalResults e standard ANSIAISC 360-10 is also comparedwith the numerical results as shown in Figure 12 S-40represents a series of square box beam-columns with bt 40

018

027

036

045

054

004 011 018 025 032

λx = 40

λx = 60λx = 80λx = 100

P uP

y

MuxMxy

(a)

018

027

036

045

054

004 011 018 025 032

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

017

024

031

038

045

002 006 01 014 018

P uP

y

MuxMxy

λx = 40

λx = 60λx = 80λx = 100

(c)

017

024

031

038

045

002 006 01 014 018P u

Py

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(d)

017

024

031

038

003 006 009 012

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(e)

017

022

027

032

037

042

002 007 012

P uP

y

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(f)

Figure 10 Interaction curves between PuPy andMuxMxy (orMuyMyy) for Q460 steel thin-walled square box beam-columns subjected toaxial force and biaxial end moments PuPy versusMuxMxy for (a) bt 40 (c) bt 60 (e) bt 80 PuPy versus MuyMyy for (b) bt 40(d) bt 60 (f ) bt 80


and R-40-50 represents another series of rectangular oneswith bt 40 and ht 50 e meaning of other symbols (S-50 S-60 ) is the same as that of the two members emaximum value of the left-hand side of equation (5) is 179the minimum one is 083 the average one is 109 and thestandard deviation is 148 For Q460 steel the same phe-nomenon as observed in the comparison between ANSIAISC 360-10 and the available experimental results is foundin Figure 12 and ANSIAISC 360-10 severely underestimatesthe strengths of beam-columns with large slenderness ratiosand small width-to-thickness ratios such as S-40 with λx 80and 100 S-60 with λx 100 R-40-50 with λx 80 and 100 andR-50-625 with λx 100 especially for rectangular box sec-tions Except for these members ANSIAISC 360-10 givesgood estimation of the strength of Q460 steel thin-walled boxbeam-columns subjected to axial force and biaxial end mo-ments It seems that ANSIAISC 360-10 is not suitable for therectangular box sections with a large depth-to-width ratioowing to the symmetry of the calculation formulas for the x-and y-axes

4 Proposal Strength Formulas for Q460 SteelThin-Walled Box Beam-ColumnsSubjected to Axial Force and BiaxialEnd Moments

41 Proposed Strength Formulas e ANSIAISC 360-10standard is also suitable for Q460 steel thin-walled box

beam-columns under biaxial bending over a wide range ofcolumn slenderness ratios However a complex calculationmust be done to acquire the eective section properties to (1)calculate the eective width (2) determine the centroidal axisof the eective section and (3) calculate the eective sectionproperties Moreover the available experimental results to-gether with the nite element results show that ANSIAISC360-10 severely underestimates the strengths of beam-col-umns with large slenderness ratios (fy 353MPa and460MPa) and slightly underestimates those of beam-columnswith a yield stress of 741MPa As a result a simple methodwhich uses the gross section properties is proposed

For a beam-column the interaction formula between theaxial compression load and the bending moment gives atheoretical basis to develop the strength formulas Asmentioned before a linear relationship between PuPy andMuxMxy or between PuPy and MuyMyy is found in mostcases for Q460 steel box cross-sectional biaxial bendingbeam-columns with slender plates In addition Salem et al[17] investigated the ultimate strength of biaxially loadedslender I-section beam-columns made from mild steel andconcluded that the linear interaction relationship couldpredict satisfactorily the strength of beam-columns withmedium slenderness ratios erefore the following linearinteraction equation that is used in EN 1993-1-1 [26] andNAS AISI 2007 [33] is adopted

P

Pc+αxMx

Mxc+αyMy

Mycle 1 (13)

016

025

034

043

001 006 011 016

λx = 40

λx = 60

λx = 80λx = 100

P uP

y

MuxMxy

(a)

016

025

034

043

002 007 012 017

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

015

02

025

03

035

0016 0044 0072 01

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(c)

015

02

025

03

035

0018 0041 0064 0087 011P u

Py

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(d)

Figure 11 Interaction curves between PuPy and MuxMxy (or MuyMyy) for Q460 steel thin-walled rectangular box beam-columnssubjected to axial force and biaxial end moments PuPy versus MuxMxy for (a) bt 60 ht 75 (c) bt 80 ht 100 PuPy versusMuyMyy for (b) bt 60 ht 75 (d) bt 80 ht 100


where P is the axial force Pc is the allowable pure axial forceMxc andMyc are the allowable pure bending moments aboutthe x- and y-axes respectively and αx and αy are the am-plification factors for bending moments

When equation (13) is used for class 4 cross sections inEN 1993-1-5 [27] it is changed into equations (4a) and (4b)

When equation (13) is used for cold-formed sections inNAS AISI 2007 [33] it is modified as follows

Table 2 Comparisons between the available experimental results and those obtained according to ANSIAISC 360-10 and the proposedformulas respectively

Literature bt ht t (mm) λx λy ex (mm) ey (mm) fy (Nmm2) Pexp (kN) ANSIAISC360-10 (1)

Proposedformulas (2)

Richard Liew et al [16]

295 298 305 853 854 15 0 353 20999 135 109295 298 305 853 854 5 10 353 22462 157 119292 292 305 871 833 5 5 353 25437 181 120439 446 305 572 576 15 0 353 32530 115 098442 442 305 575 573 5 15 353 31224 113 102442 442 305 575 573 15 15 353 28674 121 109524 524 305 488 487 20 0 353 32473 106 091524 524 305 488 487 15 10 353 33872 114 103521 518 305 491 488 10 10 353 34863 111 100642 642 187 644 642 0 10 268 11439 095 096642 637 187 657 634 10 5 268 10818 106 099648 642 187 644 642 15 15 268 8325 098 097759 759 186 548 547 15 0 268 11811 105 110759 753 186 557 547 15 10 268 10924 108 119755 753 186 557 547 15 15 268 10263 106 11986 856 185 487 486 0 20 268 11766 092 13385 861 185 485 488 20 10 268 11530 112 14486 856 185 487 486 15 15 268 11159 090 143403 537 298 51 624 19 0 293 26176 102 096396 54 298 501 633 20 10 293 26037 114 111396 54 298 501 633 20 20 293 23670 113 111399 537 298 502 628 10 20 293 26897 113 110399 54 298 50 628 0 20 293 31195 121 10870 861 187 496 617 20 0 268 10521 103 10370 861 187 496 617 20 10 268 9714 102 10770 861 187 496 617 20 20 268 9372 105 11470 861 187 496 617 10 20 268 10272 098 10970 861 187 496 617 0 20 268 11452 090 103

Usami and Fukumoto [11]

293 220 446 396 293 0 104 568 742 098 095439 329 447 40 296 0 157 568 906 098 091582 437 448 40 304 0 209 568 932 090 086444 333 445 40 295 0 314 568 743 097 093295 221 442 644 456 0 104 568 524 095 083439 329 45 649 479 0 157 568 740 103 090582 437 448 646 488 0 208 568 743 087 082439 329 45 649 479 0 314 568 593 097 085582 437 448 646 488 0 416 568 639 089 083442 332 447 396 368 0 204 568 914 098 081582 437 448 40 378 0 269 568 986 095 081

Usami and Fukumoto [12]221 166 601 488 390 0 107 741 1299 114 101271 203 601 49 393 0 131 741 1681 133 115331 248 601 492 395 0 160 741 1877 130 111

Chiew et al [2]40 40 2 337 337 0 82 2612 13634 112 125571 571 14 337 337 0 82 2536 7112 103 11780 80 1 337 337 0 82 2486 3688 096 138

Pavlovcic et al [6]44 39 4 614 562 0 20 3734 56676 118 10944 39 4 614 562 0 60 3734 38485 114 11044 39 4 614 562 0 200 3734 20609 118 121

Average 108 106Standard deviation () 172 157

(1) (PexpPn) + (89)[(Pexpey(Mnx(1minusPexpPEx))) + (Pexpex(Mny(1minusPexpPEy)))](2) (PexpNm) + (Pexpey(αWxfy(1minus (φxPexpNEx

prime )))) + (Pexpex(αWyfy(1minus (φyPexpNEyprime )))) fy is the measured yield stress


N

φminAefy+

βmxMx

Wexfy 1minus NNEx( )( )+

βmyMy

Weyfy 1minus NNEy( )( )le 1

(14)

where βmx and βmy are the equivalent moment factors aboutthe x- and y-axes respectively φmin is the smaller one of φxand φyWex andWey are the eective section moduli aboutthe x- and y-axes respectively andNEx andNEy are Eulerrsquoscritical loads

Also the eective section properties are adopted inequation (14) thus leading to a complex calculation Hencea simple method is proposed in the following section

Being in the elastic range is found when the Q460 steelbeam-columns reach the ultimate carrying capacities fromthe von Mises stress distributions in Figure 9 e samephenomenon was also discovered by Shen [22] A simpleformula was proposed by Shen [22] to evaluate the strengthof HSS rectangular cross-sectional beam-columns withslender webs under uniaxial bending e formula isexpressed as follows

N

φxA+

βmxMx

Wx 1minus φxNNExprime( )( )le αfy (15)

where φx is the strength reduction factor of a centrallyloaded member about the x-axisNExprime is a parameterNExprime π2EA(11λ2x) (where 11 is a partial factor for resistance) αis the modication factor for the steel yield strength and fyis the steel yield strength

In addition for box-section beam-columns under biaxialbending as mentioned in Section 221 the bending de-ections about the x- and y-axes and axial compressive dis-placement are two major deformations and the rotation aboutthe z-axis is very small For this reason using equation (14) as

the reference equation (15) can be modied for the ultimatestrength of Q460 steel thin-walled box-section columnssubjected to axial compressive force and major- and minor-axis bending momentsemodied equation is expressed asfollows

N

φminA+

βmxMx

Wx 1minus φxNNExprime( )( )+

βmyMy

Wy 1minus φyNNEyprime( )( )le αfy

(16)

where φy is the strength reduction factor of a centrallyloaded member about the y-axis and NEyprime is a parameterNEyprime π2EA(11λ2y)

e ultimate carrying capacity of columns with nons-lender sections and made from Q345 Q390 Q420 and SM58steels (the nominal yield stresses are 345MPa 390MPa420MPa and 460MPa respectively) was simulated to de-termine which column curve should be used when calculatingthe values of φx and φy by Shen [9] e results show that thecurve a in GB 50017-2003 [30] should be adopted Degee et al[3] also suggested that the curve a in EN 1993-1-1 [26] shouldbe employed when the local and global interaction buckling ofwelded box-section compression members fabricated fromS355 S460 and S690 steels (the nominal yield stresses are355MPa 460MPa and 690MPa respectively) was in-vestigated ere is a little dierence in the value of theequivalent imperfection between GB 50017-2003 and EN1993-1-1 when determining the stability reduction factor butthe values of three design column curves a b and c in GB50017-2003 are very close to those of three column curves a band c in EN 1993-1-1 As a result in equation (16) φx and φyare obtained according to λx

fy235radic

and λyfy235radic

re-spectively and the column curve a recommended in theChinese Code GB 50017-2003 [30]

Besides in equation (16) βmx βmy 10 for a beam-column with equal end exural moments N is taken as PuMx asMux ie Puey andMy asMuy ie Puex and then thenumerical result Pu is substituted into equation (16) Afternishing all these the modied factor α can be obtained bydividing the left-hand side of equation (16) by the steel yieldstrength fy

e ultimate strength of Q460 steel welded thin-walledbox cross-sectional beam-columns is related to the slen-derness ratios λx and λy width-to-thickness ratio bt anddepth-to-thickness ratio ht For rectangular box sections inthis paper the depth-to-width ratio is a constant of 125hence ht is related to bt and similarly λy to λx ereforethe modication factor α may be treated as a function ofonly two variables bt and λx e study shows that in mostcases the nondimensional ultimate carrying capacityPuAfy (Pu is the ultimate carrying capacity) is nearly linearto each of λx and bt respectively By using a curve-ttingtechnique the following equation can be obtained

α 12 + 0003λx minus0011bt

(17)

Substituting the numerical results and equation (17) intoequation (16) and introducing the symbolsNm αφminAfy

02

04

06

08

1

0 02 04 06 08 1

Nu

P n

(89)(MrxMnx + MryMny)

S-40S-50S-60S-70

S-80R-40-50R-50-625R-60-75

R-70-875R-80-100ANSIAISC 360-10

Figure 12 Comparison of formula (5) and numerical results


Mmx αWx(1minus (φxNNExprime ))fy and Mmy αWy(1minus(φyNNEyprime ))fy the ratios of the left side to the right side ofequation (16) are obtained and plotted in Figure 13 InFigure 13 the ratio values change over a range of 0818 to1239 with an average value of 1015 and a standard de-viation of 792 Hence equation (16) coincides quite wellwith the numerical results

42 Verication of the Proposed Formulas e availableexperimental results as listed in Table 2 are also used toverify the proposed formulas In order to compare themeasily equation (16) is changed into the following equation

N

φminAαfy+

βmxMx

Wxαfy 1minus φxNNExprime( )( )

+βmyMy

Wyαfy 1minus φyNNEyprime( )( )le 1

(18)

Similarly substituting the experimental results andequation (17) into the left-hand side of equation (18) thecalculation results labeled with ldquo(2)rdquo in Table 2 are obtainedese values are very close to 1 except for four beam-col-umns with larger width-to-thickness ratios (bt 80 and 85)and fabricated from the mild steels with fy 2486MPa andfy 268MPa For the four members the average value ofldquo(2)rdquo in Table 2 is about 140 and equation (18) gives a lowestimate of their strengths In general the simple formulasproposed herein give a reasonably accurate estimate of thestrength of the beam-columns over a wide range of the steelyield stress and for the most part they are applicable to thethin-walled box beam-columns made from both mild andhigh-strength steels Meanwhile a standard deviation of157 listed in Table 2 compared to 172 shows that thesimple formulas achieve the same level of accuracy as theAmerican Standard ANSIAISC 360-10 For the HSS

specimens with fy 741MPa the simple formulas based onthe numerical results provide a good estimate of theirstrengths for those with fy 568MPa they give a slightoverestimate however all these specimens were loaded withuniaxial eccentricity so more experiments on the HSSbeam-columns under biaxial bending are needed to beconducted to verify the proposed formulas

5 Conclusions

e behavior and strength of Q460 steel welded thin-walledbox-section beam-columns under combined compressionforce and biaxial bending are analyzed by an established FEmodel e American Specication ANSIAISC 360-10 isveried with the numerical results and the available ex-perimental results Simple and eective formulas are pro-posed for predicting the maximum strength of the beam-columns Also the proposed formulas are compared withthe available experimental results Some conclusions can bedrawn from this paper

(1) e established FE model can simulate the local-overall interaction buckling behaviors of the beam-columns subjected to combined axial compressionforce and biaxial bending

(2) It is observed that in most cases the von Misesstresses are much smaller than the steel yield strengthwhen the beam-columns reach the ultimate loads

(3) e correlation curves between the axial force PuPyand the exural moment MuxMxy (or MuyMyy) ofQ460 steel thin-walled box-section beam-columnsunder compression and biaxial bending momentsare nearly linear

(4) e code ANSIAISC 360-10 agrees very well withboth the numerical and experimental results exceptfor two types of beam-columns one with largeslenderness ratios and the other made from the HSSwith a yield stress of 741MPa For the former ANSIAISC 360-10 seriously underestimates the strengthsand for the latter it slightly underestimates

(5) After having been modied the formula proposedfor the local-overall interactive buckling strength ofHSS box beam-columns under uniaxial bending canbe used for Q460 steel box beam-columns underbiaxial bendinge proposed modied formulas arein good agreement with the numerical and experi-mental results over a wide range of the steel yieldstress e maximum of nominal yield stress in theavailable tests is 690MPa and hence fyle 690MPahas been adopted as a limit of application of theproposed formulas hb 10 or 125 in this studyand hb or bh 10sim133 in the available experi-ments [2 6 11 12 16] In practical projects foreconomic consideration hb or bh of box sectionsshould be close to or slightly greater than 1 to avoidthat the carrying capacity in one direction is muchhigher than that in the other directionerefore themodication factor α which should be a function of

0

02

04

06

08

1

12

0 02 04 06 08 1

Nu

Nm

MuxMmx + MuyMmy

NuNm + MuxMmx + MuyMmy = 1Numerical results



bt ht λx and λy can be simplified as a function ofbt and λx as given in equation (17) But the availableexperimental data on HSSs are very limited andmore experiments are needed to be conducted toverify these proposed formulas

Data Availability

+e data used to support the findings of this study areavailable from the author upon request

Conflicts of Interest

+e author declares that there are no conflicts of interest

Acknowledgments

+e research reported in this paper was supported by Na-tional Science Basic Research Plan in Shaanxi ProvinceChina (Grant no 2018JM5079) and the Steel StructureInnovation Team of Xirsquoan University of Architecture andTechnology China +eir financial support is highlyappreciated

References

[1] L Bai and M A Wadee ldquoSlenderness effects in thin-walledI-section struts susceptible to localndashglobal mode interactionrdquoEngineering Structures vol 124 pp 128ndash141 2016

[2] S P Chiew S L Lee and N E Shanmugam ldquoExperimentalstudy of thin-walled steel box columnsrdquo Journal of StructuralEngineering vol 113 no 10 pp 2208ndash2220 1987

[3] H Degee A Detzel and U Kuhlmann ldquoInteraction of globaland local buckling in welded RHS compression membersrdquoJournal of Constructional Steel Research vol 64 no 7-8pp 755ndash765 2008

[4] Y B Kwon N G Kim and G J Hancock ldquoCompression testsof welded section columns undergoing buckling interactionrdquoJournal of Constructional Steel Research vol 63 no 12pp 1590ndash1602 2007

[5] E L Liu and M A Wadee ldquoGeometric factors affectingI-section struts experiencing local and strong-axis globalbuckling mode interactionrdquo 9in-Walled Structures vol 109pp 319ndash331 2016

[6] L Pavlovcic B Froschmeier U Kuhlmann and D BegldquoFinite element simulation of slender thin-walled box col-umns by implementing real initial conditionsrdquo Advances inEngineering Software vol 44 no 1 pp 63ndash74 2012

[7] M Pircher M D OrsquoShea and R Q Bridge ldquo+e influence ofthe fabrication process on the buckling of thin-walled steelbox sectionsrdquo 9in-Walled Structures vol 40 no 2pp 109ndash123 2002

[8] H X Shen ldquoUltimate capacity of welded box section columnswith slender plate elementsrdquo Steel amp Composite Structuresvol 13 no 1 pp 15ndash33 2012

[9] H X Shen ldquoOn the direct strength and effective yield strengthmethod design of medium and high strength steel weldedsquare section columns with slender plate elementsrdquo Steel andComposite Structures vol 17 no 4 pp 497ndash516 2014

[10] J J Shen M A Wadee and A J Sadowski ldquoInteractivebuckling in long thin-walled rectangular hollow sectionstrutsrdquo International Journal of Non-Linear Mechanicsvol 89 pp 43ndash58 2017

[11] T Usami and Y Fukumoto ldquoWelded box compressionmembersrdquo Journal of Structural Engineering vol 110 no 10pp 2457ndash2470 1984

[12] T Usami and Y Fukumoto ldquoLocal and overall buckling ofwelded box columnsrdquo Journal of Structural Division vol 108no 3 pp 525ndash542 1982

[13] J J Shen and M A Wadee ldquoLength effects on interactivebuckling in thin-walled rectangular hollow section strutsrdquo9in-Walled Structures vol 128 pp 152ndash170 2018

[14] J J Shen and M A Wadee ldquoImperfection sensitivity of thin-walled rectangular hollow section struts susceptible to in-teractive bucklingrdquo International Journal of Non-LinearMechanics vol 99 pp 112ndash130 2018

[15] S L Lee N E Shanmugam and S P Chiew ldquo+in-walledbox columns under arbitrary end loadsrdquo Journal of StructuralEngineering vol 114 no 6 pp 1390ndash1402 1988

[16] J Y Richard Liew N E Shanmugam and S L Lee ldquoBehaviorof thin-walled steel box columns under biaxial loadingrdquoJournal of Structural Engineering vol 115 no 12 pp 3076ndash3094 1989

[17] A H Salem M El Aghoury F F El Dib and M T HannaldquoStrength of biaxially loaded slender I-section beam-col-umnsrdquo Canadian Journal of Civil Engineering vol 34 no 2pp 219ndash227 2007

[18] N E Shanmugam S P Chiew and S L Lee ldquoStrength ofthin-walled square steel box columnsrdquo Journal of StructuralEngineering vol 113 no 4 pp 818ndash831 1987

[19] N E Shanmugam J Y Richard Liew and S L Lee ldquo+in-walled steel box columns under biaxial loadingrdquo Journal ofStructural Engineering vol 115 no 11 pp 2706ndash2726 1989

[20] H X Shen and C H Yang ldquoAnalysis of local-overall in-teraction buckling of high-strength steel welded I-sectioncompression-bending membersrdquo Building Structure vol 43no 22 pp 33ndash38 2013 in Chinese

[21] H X Shen and X Liu ldquoAnalysis on the ultimate carryingcapacities of local-overall interaction of high-strength steelwelded square-box section columns loaded with eccentricityrdquoBuilding Structure vol 44 no 4 pp 35ndash38 2014 in Chinese

[22] H X Shen ldquoBehavior of high-strength steel welded rectan-gular section beamndashcolumns with slender websrdquo9in-WalledStructures vol 88 pp 16ndash27 2015

[23] H X Shen and K X Zhao ldquoFinite element analysis of localand overall flexural-torsional interactive buckling of Q460high strength steel welded I-section beam-columnsrdquo Progressin Steel Building Structures vol 17 no 4 pp 1ndash9 2015 inChinese

[24] GB 50017-2017 Standard for Design of Steel StructuresGuobiao Standards Beijing China 2017

[25] ANSIAISC 360-10 Specification for Structural Steel Build-ings AISC Chicago IL USA 2010

[26] EN 1993-1-1 Eurocode 3 Design of Steel Structures-Part 1-1General Rules and Rules for Buildings European Committeefor Standardization Brussels Belgium 2005

[27] EN 1993-1-5 Eurocode 3-Design of Steel Structures-Part 1-5Plated Structural Elements European Committee for Stan-dardization Brussels Belgium 2007

[28] ANSYS Inc ANSYS 9eory Reference Electronic Release 80SAS IP Inc Cary NC USA 1998

[29] ANSYS Inc ANSYS Element Reference Electronic Release 80SAS IP Inc Cary NC USA 1998

[30] GB 50017-2003 Code for Design of Steel Structures GuobiaoStandards Beijing China 2003

[31] H Y Ban G Shi Y J Shi and Y Q Wang ldquoOverall bucklingbehavior of 460MPa high strength steel columns


experimental investigation and design methodrdquo Journal ofConstructional Steel Research vol 74 pp 140ndash150 2012

[32] Y Liu and L B Hui ldquoFinite element study of steel single anglebeamndashcolumnsrdquo Engineering Structures vol 32 no 8pp 2087ndash2095 2010

[33] NAS AISI North American Specifications for the Design ofCold-Formed Steel Structural Members AISI WashingtonDC USA 2007


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from 40 to 70 and a wide range of member slenderness ratiosfrom 20 to 120 +e effects of plate width-to-thickness ratiocolumn slenderness ratio and eccentricity on the ultimatecarrying capacities were investigated and the calculationformulas for the ultimate strengths were proposed How-ever all these studies focused on the HSS beam-columnsunder uniaxial bending and more attention should be paidto those under biaxial bending

In addition no strength formulas are provided speciallyfor HSS welded beam-columns in some existing standardsGB 50017-2017 [24] ANSIAISC 360-10 [25] EN 1993-1-1[26] and EN 1993-1-5 [27] +ese standards are all based onthe research results of mild steels and are also used for HSSsbut it is unknown whether they are applicable for HSSmembers Furthermore when the strength of thin-walledbox beam-columns under biaxial bending is calculated inaccordance with the above standards a cumbersome cal-culation has to be made because of the effective widthmethod which is used in these standards Hence a simplemethod needs to be provided

In this study the local-overall interaction buckling be-havior of Q460 steel welded thin-walled box beam-columnsunder biaxial bending is analyzed by the commercial soft-ware ANSYS 80 [28 29] +e numerical results as well asavailable experimental results of different steel grades arecompared with ANSIAISC 360-10 to check whether ANSIAISC 360-10 is suitable for such types of beam-columnsSimple formulas are presented for predicting the local-overall interactive buckling strength of Q460 steel beam-columns subjected to axial compression and biaxial bendingmoments +e proposed formulas are verified by comparingthem with the available experimental results

2 Finite Element Analysis

21 An FE Model and Its Verification A pin-ended beam-column as shown in Figure 1 with a welded thin-walled boxcross section fabricated from Q460 subjected to combinedaxial compression force P and biaxial bending momentsMx and My is investigated in this paper Here the bendingmoments around the x- and y-axesMx andMy are taken asMx Pey and My Pex where ex and ey are the eccentricitiesin the x and y directions respectively Also given inFigure 1(b) are the dimensions of the welded box section inwhich b and h are the widths of plate elements and t is thethickness of the plate elements

A geometrical and material nonlinear FE model wasestablished by Shen [9 22] for HSS box columns loaded withaxial compression and uniaxial eccentric compression re-spectively by using the ANSYS program In this study theFE model is modified for HSS box columns loaded withbiaxial eccentric compression

As described by Shen [9 22] the effects of initial geo-metrical imperfections and residual stresses are taken intoaccount +e initial geometrical imperfections include bothoverall member imperfections and local plate imperfections+e member imperfections are assumed to bend in a halfsine wave about the major and minor axes respectively asfollows

]0 l

1000sin

πz

l (1a)

μ0 l

1000sin

πz

l (1b)

where l is the length of the memberIn equations (1a) and (1b) the maximum value of out-of-

straightness of a member is taken equal to l1000 as specifiedby GB 50017-2003 [30] and ANSIAISC 360-10 [25]

+e plate imperfections are assumed as

ω ω1 sinmπz

lcos

πx

by plusmn

h

2+

t

21113888 11138891113888 1113889 (2a)

ω ω2 sinmπz

lcos

πy

hx plusmn

b

2+

t

21113888 11138891113888 1113889 (2b)

where ω1 b1000 ω2 h1000 and m is the buckling half-wave number along the longitudinal axis of the member

+e magnitudes of plate imperfections were taken asb100 and b1000 (b is the plate width of a square box section)[8] and it showed that the magnitude of plate imperfectionshad a significant effect on the compressive strength of weldedbox columns No uniform value was specified for the max-imum of plate imperfections and it was taken as 001 [4] andb1000 [3] but it was demonstrated that the value of b1000seemed to be reasonable [3] when residual stresses weretaken into consideration Hence for a square box sectionω1 ω2 b1000 But for a rectangular one according to thedeformation compatibility condition the angle between thetwo adjacent plates remained unchanged before and after theplate buckling so ω1 b1000 and ω2 h1000 herein

Residual stress measurements of welded square andrectangular box sections were made by Usami and Fuku-moto [11] and Pavlovcic et al [6] respectively According totheir measurement results a simplified pattern of residualstress distribution as shown in Figure 2 is adopted wheretensile residual stresses are positive values and compressivestresses are negative values An SM58 steel plate was used[11] its nominal yield strength was 460MPa and itsmeasured yield strength was 568MPa +e maximum valueof tensile stresses was about 80 of the measured yieldstrength ie 08times 568 454MPa which is very close to thenominal yield strength of 460MPa+e value of compressiveresidual stresses decreased along with the increase of thewidth-to-thickness ratio of the plate [6 11] +e distributionwidth of tensile residual stresses was about 3t [9] As aconsequence the maximum value of tensile residual stressesσrt is taken as the nominal yield stress of steel and those ofcompressive residual stresses σrcf and σrcw along the flangeand web respectively may be obtained according to equi-librium conditions

+e Shell181 element is employed to take account ofplate imperfections Shell181 is suitable for analyzing thin tomoderately thick shell structures It is a 4-node element withsix degrees of freedom at each node translations in the x yand z directions and rotations about the x- y- and z-axes Itis well suited for linear large-rotation andor large-strainnonlinear applications Also the effects of plasticity of steel






x

x

y

y

z

Mx

Mx

My

MyP

(a)

h

t

y

x

tt

b

t

(b)


+

ndash

ndash

σrcf

σrt

σrt

x

b

tt

t

t

h

y

+

+ +

σrcw

σrt σrt


0

100

200

300

400

500

600

700

0 005 01 015 02 025

Stre

ss (M

Pa)

Strain













235fy

1113969 40 times

235460

radic 286 for


EFy

1113969

14 times206000460





(a) (b)


z = 0

z = l

(a) (b) (c)

















N

χxNR

+ kxx

Mx + ΔMx

χLTMxR

+ kxy

My + ΔMy

MyR

le 1 (3a)

N

χyNR

+ kyx

Mx + ΔMx

χLTMxR

+ kyy

My + ΔMy

MyR

le 1 (3b)





N

χxAefffy+ kxx



le 1 (4a)

N

χyAefffy+ kyx



le 1 (4b)



PrPn+89

Mrx

Mnx+Mry

Mny( )le 10 when

PrPnge 02 (5a)

Pr2Pn

+Mrx

Mnx+Mry

Mny( )le 10 when

PrPnlt 02 (5b)


νmax

y

z

Mx Mx

l

P P1-1

(a)

l

x

1-1P

Pz

My My

μmax

(b)

A

1-1

B

(c)


0

100

200

300

400

500

0 2 4 6 8 10

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

500

0 5 10 15 20P

(kN

)

μmax (mm)

(b)

0

100

200

300

400

500

0 5 10 15 20

P (k

N)

νmax (mm)

(c)

0

100

200

300

400

500

0 001 002 003

P (k

N)

θmax (rad)

(d)








Pn FcrA (6)



rle 471

E

QFy

radic(7a)

Fcr 0877Fe forKL

rgt 471

E

QFy

radic(7b)


Qa is dened as

Qa Ae

A (8)


0

100

200

300

400

0 5 10 15 20

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

0 25 50 75 100 125

P (k

N)

μmax (mm)

(b)

νmax (mm)

0

100

200

300

400

0 50 100 150

P (k

N)

(c)

0

100

200

300

400

ndash001 0 001 002

P (k

N)

θmax (rad)

(d)




be 192t

E

f1

radic

1minus038(bt)

E

f1

radic le b (9)

wheref1

PnAe (10)




Mnx FySex (11a)

Mny FySey (11b)


140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(a)

60

110

160

210

260

310

360

410

460

0 76 152 228 304y (mm)

Von

Mise

s str

ess (

MPa

)



(b)

140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(c)

60

110

160

210

260

310

360

410

460

0 76 152 228 304



y (mm)

Von

Mise

s str

ess (

MPa

)

(d)




Mrx CmxPrey


Mry CmyPrex






018

027

036

045

054

004 011 018 025 032

λx = 40

λx = 60λx = 80λx = 100

P uP

y

MuxMxy

(a)

018

027

036

045

054

004 011 018 025 032

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

017

024

031

038

045

002 006 01 014 018

P uP

y

MuxMxy

λx = 40

λx = 60λx = 80λx = 100

(c)

017

024

031

038

045

002 006 01 014 018P u

Py

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(d)

017

024

031

038

003 006 009 012

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(e)

017

022

027

032

037

042

002 007 012

P uP

y

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(f)








P

Pc+αxMx

Mxc+αyMy

Mycle 1 (13)

016

025

034

043

001 006 011 016

λx = 40

λx = 60

λx = 80λx = 100

P uP

y

MuxMxy

(a)

016

025

034

043

002 007 012 017

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

015

02

025

03

035

0016 0044 0072 01

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(c)

015

02

025

03

035

0018 0041 0064 0087 011P u

Py

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(d)










295 298 305 853 854 15 0 353 20999 135 109295 298 305 853 854 5 10 353 22462 157 119292 292 305 871 833 5 5 353 25437 181 120439 446 305 572 576 15 0 353 32530 115 098442 442 305 575 573 5 15 353 31224 113 102442 442 305 575 573 15 15 353 28674 121 109524 524 305 488 487 20 0 353 32473 106 091524 524 305 488 487 15 10 353 33872 114 103521 518 305 491 488 10 10 353 34863 111 100642 642 187 644 642 0 10 268 11439 095 096642 637 187 657 634 10 5 268 10818 106 099648 642 187 644 642 15 15 268 8325 098 097759 759 186 548 547 15 0 268 11811 105 110759 753 186 557 547 15 10 268 10924 108 119755 753 186 557 547 15 15 268 10263 106 11986 856 185 487 486 0 20 268 11766 092 13385 861 185 485 488 20 10 268 11530 112 14486 856 185 487 486 15 15 268 11159 090 143403 537 298 51 624 19 0 293 26176 102 096396 54 298 501 633 20 10 293 26037 114 111396 54 298 501 633 20 20 293 23670 113 111399 537 298 502 628 10 20 293 26897 113 110399 54 298 50 628 0 20 293 31195 121 10870 861 187 496 617 20 0 268 10521 103 10370 861 187 496 617 20 10 268 9714 102 10770 861 187 496 617 20 20 268 9372 105 11470 861 187 496 617 10 20 268 10272 098 10970 861 187 496 617 0 20 268 11452 090 103


293 220 446 396 293 0 104 568 742 098 095439 329 447 40 296 0 157 568 906 098 091582 437 448 40 304 0 209 568 932 090 086444 333 445 40 295 0 314 568 743 097 093295 221 442 644 456 0 104 568 524 095 083439 329 45 649 479 0 157 568 740 103 090582 437 448 646 488 0 208 568 743 087 082439 329 45 649 479 0 314 568 593 097 085582 437 448 646 488 0 416 568 639 089 083442 332 447 396 368 0 204 568 914 098 081582 437 448 40 378 0 269 568 986 095 081


Chiew et al [2]40 40 2 337 337 0 82 2612 13634 112 125571 571 14 337 337 0 82 2536 7112 103 11780 80 1 337 337 0 82 2486 3688 096 138

Pavlovcic et al [6]44 39 4 614 562 0 20 3734 56676 118 10944 39 4 614 562 0 60 3734 38485 114 11044 39 4 614 562 0 200 3734 20609 118 121





N

φminAefy+

βmxMx


βmyMy


(14)




N

φxA+

βmxMx





N

φminA+

βmxMx


βmyMy


(16)



fy235radic

and λyfy235radic





(17)


02

04

06

08

1

0 02 04 06 08 1

Nu

P n


S-40S-50S-60S-70

S-80R-40-50R-50-625R-60-75

R-70-875R-80-100ANSIAISC 360-10





N

φminAαfy+

βmxMx


+βmyMy


(18)



5 Conclusions







0

02

04

06

08

1

12

0 02 04 06 08 1

Nu

Nm

MuxMmx + MuyMmy





Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






x

x

y

y

z

Mx

Mx

My

MyP

(a)

h

t

y

x

tt

b

t

(b)


+

ndash

ndash

σrcf

σrt

σrt

x

b

tt

t

t

h

y

+

+ +

σrcw

σrt σrt


0

100

200

300

400

500

600

700

0 005 01 015 02 025

Stre

ss (M

Pa)

Strain













235fy

1113969 40 times

235460

radic 286 for


EFy

1113969

14 times206000460





(a) (b)


z = 0

z = l

(a) (b) (c)

















N

χxNR

+ kxx

Mx + ΔMx

χLTMxR

+ kxy

My + ΔMy

MyR

le 1 (3a)

N

χyNR

+ kyx

Mx + ΔMx

χLTMxR

+ kyy

My + ΔMy

MyR

le 1 (3b)





N

χxAefffy+ kxx



le 1 (4a)

N

χyAefffy+ kyx



le 1 (4b)



PrPn+89

Mrx

Mnx+Mry

Mny( )le 10 when

PrPnge 02 (5a)

Pr2Pn

+Mrx

Mnx+Mry

Mny( )le 10 when

PrPnlt 02 (5b)


νmax

y

z

Mx Mx

l

P P1-1

(a)

l

x

1-1P

Pz

My My

μmax

(b)

A

1-1

B

(c)


0

100

200

300

400

500

0 2 4 6 8 10

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

500

0 5 10 15 20P

(kN

)

μmax (mm)

(b)

0

100

200

300

400

500

0 5 10 15 20

P (k

N)

νmax (mm)

(c)

0

100

200

300

400

500

0 001 002 003

P (k

N)

θmax (rad)

(d)








Pn FcrA (6)



rle 471

E

QFy

radic(7a)

Fcr 0877Fe forKL

rgt 471

E

QFy

radic(7b)


Qa is dened as

Qa Ae

A (8)


0

100

200

300

400

0 5 10 15 20

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

0 25 50 75 100 125

P (k

N)

μmax (mm)

(b)

νmax (mm)

0

100

200

300

400

0 50 100 150

P (k

N)

(c)

0

100

200

300

400

ndash001 0 001 002

P (k

N)

θmax (rad)

(d)




be 192t

E

f1

radic

1minus038(bt)

E

f1

radic le b (9)

wheref1

PnAe (10)




Mnx FySex (11a)

Mny FySey (11b)


140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(a)

60

110

160

210

260

310

360

410

460

0 76 152 228 304y (mm)

Von

Mise

s str

ess (

MPa

)



(b)

140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(c)

60

110

160

210

260

310

360

410

460

0 76 152 228 304



y (mm)

Von

Mise

s str

ess (

MPa

)

(d)




Mrx CmxPrey


Mry CmyPrex






018

027

036

045

054

004 011 018 025 032

λx = 40

λx = 60λx = 80λx = 100

P uP

y

MuxMxy

(a)

018

027

036

045

054

004 011 018 025 032

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

017

024

031

038

045

002 006 01 014 018

P uP

y

MuxMxy

λx = 40

λx = 60λx = 80λx = 100

(c)

017

024

031

038

045

002 006 01 014 018P u

Py

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(d)

017

024

031

038

003 006 009 012

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(e)

017

022

027

032

037

042

002 007 012

P uP

y

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(f)








P

Pc+αxMx

Mxc+αyMy

Mycle 1 (13)

016

025

034

043

001 006 011 016

λx = 40

λx = 60

λx = 80λx = 100

P uP

y

MuxMxy

(a)

016

025

034

043

002 007 012 017

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

015

02

025

03

035

0016 0044 0072 01

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(c)

015

02

025

03

035

0018 0041 0064 0087 011P u

Py

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(d)










295 298 305 853 854 15 0 353 20999 135 109295 298 305 853 854 5 10 353 22462 157 119292 292 305 871 833 5 5 353 25437 181 120439 446 305 572 576 15 0 353 32530 115 098442 442 305 575 573 5 15 353 31224 113 102442 442 305 575 573 15 15 353 28674 121 109524 524 305 488 487 20 0 353 32473 106 091524 524 305 488 487 15 10 353 33872 114 103521 518 305 491 488 10 10 353 34863 111 100642 642 187 644 642 0 10 268 11439 095 096642 637 187 657 634 10 5 268 10818 106 099648 642 187 644 642 15 15 268 8325 098 097759 759 186 548 547 15 0 268 11811 105 110759 753 186 557 547 15 10 268 10924 108 119755 753 186 557 547 15 15 268 10263 106 11986 856 185 487 486 0 20 268 11766 092 13385 861 185 485 488 20 10 268 11530 112 14486 856 185 487 486 15 15 268 11159 090 143403 537 298 51 624 19 0 293 26176 102 096396 54 298 501 633 20 10 293 26037 114 111396 54 298 501 633 20 20 293 23670 113 111399 537 298 502 628 10 20 293 26897 113 110399 54 298 50 628 0 20 293 31195 121 10870 861 187 496 617 20 0 268 10521 103 10370 861 187 496 617 20 10 268 9714 102 10770 861 187 496 617 20 20 268 9372 105 11470 861 187 496 617 10 20 268 10272 098 10970 861 187 496 617 0 20 268 11452 090 103


293 220 446 396 293 0 104 568 742 098 095439 329 447 40 296 0 157 568 906 098 091582 437 448 40 304 0 209 568 932 090 086444 333 445 40 295 0 314 568 743 097 093295 221 442 644 456 0 104 568 524 095 083439 329 45 649 479 0 157 568 740 103 090582 437 448 646 488 0 208 568 743 087 082439 329 45 649 479 0 314 568 593 097 085582 437 448 646 488 0 416 568 639 089 083442 332 447 396 368 0 204 568 914 098 081582 437 448 40 378 0 269 568 986 095 081


Chiew et al [2]40 40 2 337 337 0 82 2612 13634 112 125571 571 14 337 337 0 82 2536 7112 103 11780 80 1 337 337 0 82 2486 3688 096 138

Pavlovcic et al [6]44 39 4 614 562 0 20 3734 56676 118 10944 39 4 614 562 0 60 3734 38485 114 11044 39 4 614 562 0 200 3734 20609 118 121





N

φminAefy+

βmxMx


βmyMy


(14)




N

φxA+

βmxMx





N

φminA+

βmxMx


βmyMy


(16)



fy235radic

and λyfy235radic





(17)


02

04

06

08

1

0 02 04 06 08 1

Nu

P n


S-40S-50S-60S-70

S-80R-40-50R-50-625R-60-75

R-70-875R-80-100ANSIAISC 360-10





N

φminAαfy+

βmxMx


+βmyMy


(18)



5 Conclusions







0

02

04

06

08

1

12

0 02 04 06 08 1

Nu

Nm

MuxMmx + MuyMmy





Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











235fy

1113969 40 times

235460

radic 286 for


EFy

1113969

14 times206000460





(a) (b)


z = 0

z = l

(a) (b) (c)

















N

χxNR

+ kxx

Mx + ΔMx

χLTMxR

+ kxy

My + ΔMy

MyR

le 1 (3a)

N

χyNR

+ kyx

Mx + ΔMx

χLTMxR

+ kyy

My + ΔMy

MyR

le 1 (3b)





N

χxAefffy+ kxx



le 1 (4a)

N

χyAefffy+ kyx



le 1 (4b)



PrPn+89

Mrx

Mnx+Mry

Mny( )le 10 when

PrPnge 02 (5a)

Pr2Pn

+Mrx

Mnx+Mry

Mny( )le 10 when

PrPnlt 02 (5b)


νmax

y

z

Mx Mx

l

P P1-1

(a)

l

x

1-1P

Pz

My My

μmax

(b)

A

1-1

B

(c)


0

100

200

300

400

500

0 2 4 6 8 10

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

500

0 5 10 15 20P

(kN

)

μmax (mm)

(b)

0

100

200

300

400

500

0 5 10 15 20

P (k

N)

νmax (mm)

(c)

0

100

200

300

400

500

0 001 002 003

P (k

N)

θmax (rad)

(d)








Pn FcrA (6)



rle 471

E

QFy

radic(7a)

Fcr 0877Fe forKL

rgt 471

E

QFy

radic(7b)


Qa is dened as

Qa Ae

A (8)


0

100

200

300

400

0 5 10 15 20

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

0 25 50 75 100 125

P (k

N)

μmax (mm)

(b)

νmax (mm)

0

100

200

300

400

0 50 100 150

P (k

N)

(c)

0

100

200

300

400

ndash001 0 001 002

P (k

N)

θmax (rad)

(d)




be 192t

E

f1

radic

1minus038(bt)

E

f1

radic le b (9)

wheref1

PnAe (10)




Mnx FySex (11a)

Mny FySey (11b)


140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(a)

60

110

160

210

260

310

360

410

460

0 76 152 228 304y (mm)

Von

Mise

s str

ess (

MPa

)



(b)

140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(c)

60

110

160

210

260

310

360

410

460

0 76 152 228 304



y (mm)

Von

Mise

s str

ess (

MPa

)

(d)




Mrx CmxPrey


Mry CmyPrex






018

027

036

045

054

004 011 018 025 032

λx = 40

λx = 60λx = 80λx = 100

P uP

y

MuxMxy

(a)

018

027

036

045

054

004 011 018 025 032

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

017

024

031

038

045

002 006 01 014 018

P uP

y

MuxMxy

λx = 40

λx = 60λx = 80λx = 100

(c)

017

024

031

038

045

002 006 01 014 018P u

Py

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(d)

017

024

031

038

003 006 009 012

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(e)

017

022

027

032

037

042

002 007 012

P uP

y

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(f)








P

Pc+αxMx

Mxc+αyMy

Mycle 1 (13)

016

025

034

043

001 006 011 016

λx = 40

λx = 60

λx = 80λx = 100

P uP

y

MuxMxy

(a)

016

025

034

043

002 007 012 017

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

015

02

025

03

035

0016 0044 0072 01

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(c)

015

02

025

03

035

0018 0041 0064 0087 011P u

Py

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(d)










295 298 305 853 854 15 0 353 20999 135 109295 298 305 853 854 5 10 353 22462 157 119292 292 305 871 833 5 5 353 25437 181 120439 446 305 572 576 15 0 353 32530 115 098442 442 305 575 573 5 15 353 31224 113 102442 442 305 575 573 15 15 353 28674 121 109524 524 305 488 487 20 0 353 32473 106 091524 524 305 488 487 15 10 353 33872 114 103521 518 305 491 488 10 10 353 34863 111 100642 642 187 644 642 0 10 268 11439 095 096642 637 187 657 634 10 5 268 10818 106 099648 642 187 644 642 15 15 268 8325 098 097759 759 186 548 547 15 0 268 11811 105 110759 753 186 557 547 15 10 268 10924 108 119755 753 186 557 547 15 15 268 10263 106 11986 856 185 487 486 0 20 268 11766 092 13385 861 185 485 488 20 10 268 11530 112 14486 856 185 487 486 15 15 268 11159 090 143403 537 298 51 624 19 0 293 26176 102 096396 54 298 501 633 20 10 293 26037 114 111396 54 298 501 633 20 20 293 23670 113 111399 537 298 502 628 10 20 293 26897 113 110399 54 298 50 628 0 20 293 31195 121 10870 861 187 496 617 20 0 268 10521 103 10370 861 187 496 617 20 10 268 9714 102 10770 861 187 496 617 20 20 268 9372 105 11470 861 187 496 617 10 20 268 10272 098 10970 861 187 496 617 0 20 268 11452 090 103


293 220 446 396 293 0 104 568 742 098 095439 329 447 40 296 0 157 568 906 098 091582 437 448 40 304 0 209 568 932 090 086444 333 445 40 295 0 314 568 743 097 093295 221 442 644 456 0 104 568 524 095 083439 329 45 649 479 0 157 568 740 103 090582 437 448 646 488 0 208 568 743 087 082439 329 45 649 479 0 314 568 593 097 085582 437 448 646 488 0 416 568 639 089 083442 332 447 396 368 0 204 568 914 098 081582 437 448 40 378 0 269 568 986 095 081


Chiew et al [2]40 40 2 337 337 0 82 2612 13634 112 125571 571 14 337 337 0 82 2536 7112 103 11780 80 1 337 337 0 82 2486 3688 096 138

Pavlovcic et al [6]44 39 4 614 562 0 20 3734 56676 118 10944 39 4 614 562 0 60 3734 38485 114 11044 39 4 614 562 0 200 3734 20609 118 121





N

φminAefy+

βmxMx


βmyMy


(14)




N

φxA+

βmxMx





N

φminA+

βmxMx


βmyMy


(16)



fy235radic

and λyfy235radic





(17)


02

04

06

08

1

0 02 04 06 08 1

Nu

P n


S-40S-50S-60S-70

S-80R-40-50R-50-625R-60-75

R-70-875R-80-100ANSIAISC 360-10





N

φminAαfy+

βmxMx


+βmyMy


(18)



5 Conclusions







0

02

04

06

08

1

12

0 02 04 06 08 1

Nu

Nm

MuxMmx + MuyMmy





Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


(a) (b)


z = 0

z = l

(a) (b) (c)

















N

χxNR

+ kxx

Mx + ΔMx

χLTMxR

+ kxy

My + ΔMy

MyR

le 1 (3a)

N

χyNR

+ kyx

Mx + ΔMx

χLTMxR

+ kyy

My + ΔMy

MyR

le 1 (3b)





N

χxAefffy+ kxx



le 1 (4a)

N

χyAefffy+ kyx



le 1 (4b)



PrPn+89

Mrx

Mnx+Mry

Mny( )le 10 when

PrPnge 02 (5a)

Pr2Pn

+Mrx

Mnx+Mry

Mny( )le 10 when

PrPnlt 02 (5b)


νmax

y

z

Mx Mx

l

P P1-1

(a)

l

x

1-1P

Pz

My My

μmax

(b)

A

1-1

B

(c)


0

100

200

300

400

500

0 2 4 6 8 10

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

500

0 5 10 15 20P

(kN

)

μmax (mm)

(b)

0

100

200

300

400

500

0 5 10 15 20

P (k

N)

νmax (mm)

(c)

0

100

200

300

400

500

0 001 002 003

P (k

N)

θmax (rad)

(d)








Pn FcrA (6)



rle 471

E

QFy

radic(7a)

Fcr 0877Fe forKL

rgt 471

E

QFy

radic(7b)


Qa is dened as

Qa Ae

A (8)


0

100

200

300

400

0 5 10 15 20

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

0 25 50 75 100 125

P (k

N)

μmax (mm)

(b)

νmax (mm)

0

100

200

300

400

0 50 100 150

P (k

N)

(c)

0

100

200

300

400

ndash001 0 001 002

P (k

N)

θmax (rad)

(d)




be 192t

E

f1

radic

1minus038(bt)

E

f1

radic le b (9)

wheref1

PnAe (10)




Mnx FySex (11a)

Mny FySey (11b)


140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(a)

60

110

160

210

260

310

360

410

460

0 76 152 228 304y (mm)

Von

Mise

s str

ess (

MPa

)



(b)

140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(c)

60

110

160

210

260

310

360

410

460

0 76 152 228 304



y (mm)

Von

Mise

s str

ess (

MPa

)

(d)




Mrx CmxPrey


Mry CmyPrex






018

027

036

045

054

004 011 018 025 032

λx = 40

λx = 60λx = 80λx = 100

P uP

y

MuxMxy

(a)

018

027

036

045

054

004 011 018 025 032

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

017

024

031

038

045

002 006 01 014 018

P uP

y

MuxMxy

λx = 40

λx = 60λx = 80λx = 100

(c)

017

024

031

038

045

002 006 01 014 018P u

Py

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(d)

017

024

031

038

003 006 009 012

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(e)

017

022

027

032

037

042

002 007 012

P uP

y

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(f)








P

Pc+αxMx

Mxc+αyMy

Mycle 1 (13)

016

025

034

043

001 006 011 016

λx = 40

λx = 60

λx = 80λx = 100

P uP

y

MuxMxy

(a)

016

025

034

043

002 007 012 017

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

015

02

025

03

035

0016 0044 0072 01

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(c)

015

02

025

03

035

0018 0041 0064 0087 011P u

Py

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(d)










295 298 305 853 854 15 0 353 20999 135 109295 298 305 853 854 5 10 353 22462 157 119292 292 305 871 833 5 5 353 25437 181 120439 446 305 572 576 15 0 353 32530 115 098442 442 305 575 573 5 15 353 31224 113 102442 442 305 575 573 15 15 353 28674 121 109524 524 305 488 487 20 0 353 32473 106 091524 524 305 488 487 15 10 353 33872 114 103521 518 305 491 488 10 10 353 34863 111 100642 642 187 644 642 0 10 268 11439 095 096642 637 187 657 634 10 5 268 10818 106 099648 642 187 644 642 15 15 268 8325 098 097759 759 186 548 547 15 0 268 11811 105 110759 753 186 557 547 15 10 268 10924 108 119755 753 186 557 547 15 15 268 10263 106 11986 856 185 487 486 0 20 268 11766 092 13385 861 185 485 488 20 10 268 11530 112 14486 856 185 487 486 15 15 268 11159 090 143403 537 298 51 624 19 0 293 26176 102 096396 54 298 501 633 20 10 293 26037 114 111396 54 298 501 633 20 20 293 23670 113 111399 537 298 502 628 10 20 293 26897 113 110399 54 298 50 628 0 20 293 31195 121 10870 861 187 496 617 20 0 268 10521 103 10370 861 187 496 617 20 10 268 9714 102 10770 861 187 496 617 20 20 268 9372 105 11470 861 187 496 617 10 20 268 10272 098 10970 861 187 496 617 0 20 268 11452 090 103


293 220 446 396 293 0 104 568 742 098 095439 329 447 40 296 0 157 568 906 098 091582 437 448 40 304 0 209 568 932 090 086444 333 445 40 295 0 314 568 743 097 093295 221 442 644 456 0 104 568 524 095 083439 329 45 649 479 0 157 568 740 103 090582 437 448 646 488 0 208 568 743 087 082439 329 45 649 479 0 314 568 593 097 085582 437 448 646 488 0 416 568 639 089 083442 332 447 396 368 0 204 568 914 098 081582 437 448 40 378 0 269 568 986 095 081


Chiew et al [2]40 40 2 337 337 0 82 2612 13634 112 125571 571 14 337 337 0 82 2536 7112 103 11780 80 1 337 337 0 82 2486 3688 096 138

Pavlovcic et al [6]44 39 4 614 562 0 20 3734 56676 118 10944 39 4 614 562 0 60 3734 38485 114 11044 39 4 614 562 0 200 3734 20609 118 121





N

φminAefy+

βmxMx


βmyMy


(14)




N

φxA+

βmxMx





N

φminA+

βmxMx


βmyMy


(16)



fy235radic

and λyfy235radic





(17)


02

04

06

08

1

0 02 04 06 08 1

Nu

P n


S-40S-50S-60S-70

S-80R-40-50R-50-625R-60-75

R-70-875R-80-100ANSIAISC 360-10





N

φminAαfy+

βmxMx


+βmyMy


(18)



5 Conclusions







0

02

04

06

08

1

12

0 02 04 06 08 1

Nu

Nm

MuxMmx + MuyMmy





Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia













N

χxNR

+ kxx

Mx + ΔMx

χLTMxR

+ kxy

My + ΔMy

MyR

le 1 (3a)

N

χyNR

+ kyx

Mx + ΔMx

χLTMxR

+ kyy

My + ΔMy

MyR

le 1 (3b)





N

χxAefffy+ kxx



le 1 (4a)

N

χyAefffy+ kyx



le 1 (4b)



PrPn+89

Mrx

Mnx+Mry

Mny( )le 10 when

PrPnge 02 (5a)

Pr2Pn

+Mrx

Mnx+Mry

Mny( )le 10 when

PrPnlt 02 (5b)


νmax

y

z

Mx Mx

l

P P1-1

(a)

l

x

1-1P

Pz

My My

μmax

(b)

A

1-1

B

(c)


0

100

200

300

400

500

0 2 4 6 8 10

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

500

0 5 10 15 20P

(kN

)

μmax (mm)

(b)

0

100

200

300

400

500

0 5 10 15 20

P (k

N)

νmax (mm)

(c)

0

100

200

300

400

500

0 001 002 003

P (k

N)

θmax (rad)

(d)








Pn FcrA (6)



rle 471

E

QFy

radic(7a)

Fcr 0877Fe forKL

rgt 471

E

QFy

radic(7b)


Qa is dened as

Qa Ae

A (8)


0

100

200

300

400

0 5 10 15 20

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

0 25 50 75 100 125

P (k

N)

μmax (mm)

(b)

νmax (mm)

0

100

200

300

400

0 50 100 150

P (k

N)

(c)

0

100

200

300

400

ndash001 0 001 002

P (k

N)

θmax (rad)

(d)




be 192t

E

f1

radic

1minus038(bt)

E

f1

radic le b (9)

wheref1

PnAe (10)




Mnx FySex (11a)

Mny FySey (11b)


140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(a)

60

110

160

210

260

310

360

410

460

0 76 152 228 304y (mm)

Von

Mise

s str

ess (

MPa

)



(b)

140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(c)

60

110

160

210

260

310

360

410

460

0 76 152 228 304



y (mm)

Von

Mise

s str

ess (

MPa

)

(d)




Mrx CmxPrey


Mry CmyPrex






018

027

036

045

054

004 011 018 025 032

λx = 40

λx = 60λx = 80λx = 100

P uP

y

MuxMxy

(a)

018

027

036

045

054

004 011 018 025 032

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

017

024

031

038

045

002 006 01 014 018

P uP

y

MuxMxy

λx = 40

λx = 60λx = 80λx = 100

(c)

017

024

031

038

045

002 006 01 014 018P u

Py

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(d)

017

024

031

038

003 006 009 012

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(e)

017

022

027

032

037

042

002 007 012

P uP

y

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(f)








P

Pc+αxMx

Mxc+αyMy

Mycle 1 (13)

016

025

034

043

001 006 011 016

λx = 40

λx = 60

λx = 80λx = 100

P uP

y

MuxMxy

(a)

016

025

034

043

002 007 012 017

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

015

02

025

03

035

0016 0044 0072 01

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(c)

015

02

025

03

035

0018 0041 0064 0087 011P u

Py

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(d)










295 298 305 853 854 15 0 353 20999 135 109295 298 305 853 854 5 10 353 22462 157 119292 292 305 871 833 5 5 353 25437 181 120439 446 305 572 576 15 0 353 32530 115 098442 442 305 575 573 5 15 353 31224 113 102442 442 305 575 573 15 15 353 28674 121 109524 524 305 488 487 20 0 353 32473 106 091524 524 305 488 487 15 10 353 33872 114 103521 518 305 491 488 10 10 353 34863 111 100642 642 187 644 642 0 10 268 11439 095 096642 637 187 657 634 10 5 268 10818 106 099648 642 187 644 642 15 15 268 8325 098 097759 759 186 548 547 15 0 268 11811 105 110759 753 186 557 547 15 10 268 10924 108 119755 753 186 557 547 15 15 268 10263 106 11986 856 185 487 486 0 20 268 11766 092 13385 861 185 485 488 20 10 268 11530 112 14486 856 185 487 486 15 15 268 11159 090 143403 537 298 51 624 19 0 293 26176 102 096396 54 298 501 633 20 10 293 26037 114 111396 54 298 501 633 20 20 293 23670 113 111399 537 298 502 628 10 20 293 26897 113 110399 54 298 50 628 0 20 293 31195 121 10870 861 187 496 617 20 0 268 10521 103 10370 861 187 496 617 20 10 268 9714 102 10770 861 187 496 617 20 20 268 9372 105 11470 861 187 496 617 10 20 268 10272 098 10970 861 187 496 617 0 20 268 11452 090 103


293 220 446 396 293 0 104 568 742 098 095439 329 447 40 296 0 157 568 906 098 091582 437 448 40 304 0 209 568 932 090 086444 333 445 40 295 0 314 568 743 097 093295 221 442 644 456 0 104 568 524 095 083439 329 45 649 479 0 157 568 740 103 090582 437 448 646 488 0 208 568 743 087 082439 329 45 649 479 0 314 568 593 097 085582 437 448 646 488 0 416 568 639 089 083442 332 447 396 368 0 204 568 914 098 081582 437 448 40 378 0 269 568 986 095 081


Chiew et al [2]40 40 2 337 337 0 82 2612 13634 112 125571 571 14 337 337 0 82 2536 7112 103 11780 80 1 337 337 0 82 2486 3688 096 138

Pavlovcic et al [6]44 39 4 614 562 0 20 3734 56676 118 10944 39 4 614 562 0 60 3734 38485 114 11044 39 4 614 562 0 200 3734 20609 118 121





N

φminAefy+

βmxMx


βmyMy


(14)




N

φxA+

βmxMx





N

φminA+

βmxMx


βmyMy


(16)



fy235radic

and λyfy235radic





(17)


02

04

06

08

1

0 02 04 06 08 1

Nu

P n


S-40S-50S-60S-70

S-80R-40-50R-50-625R-60-75

R-70-875R-80-100ANSIAISC 360-10





N

φminAαfy+

βmxMx


+βmyMy


(18)



5 Conclusions







0

02

04

06

08

1

12

0 02 04 06 08 1

Nu

Nm

MuxMmx + MuyMmy





Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




N

χxAefffy+ kxx



le 1 (4a)

N

χyAefffy+ kyx



le 1 (4b)



PrPn+89

Mrx

Mnx+Mry

Mny( )le 10 when

PrPnge 02 (5a)

Pr2Pn

+Mrx

Mnx+Mry

Mny( )le 10 when

PrPnlt 02 (5b)


νmax

y

z

Mx Mx

l

P P1-1

(a)

l

x

1-1P

Pz

My My

μmax

(b)

A

1-1

B

(c)


0

100

200

300

400

500

0 2 4 6 8 10

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

500

0 5 10 15 20P

(kN

)

μmax (mm)

(b)

0

100

200

300

400

500

0 5 10 15 20

P (k

N)

νmax (mm)

(c)

0

100

200

300

400

500

0 001 002 003

P (k

N)

θmax (rad)

(d)








Pn FcrA (6)



rle 471

E

QFy

radic(7a)

Fcr 0877Fe forKL

rgt 471

E

QFy

radic(7b)


Qa is dened as

Qa Ae

A (8)


0

100

200

300

400

0 5 10 15 20

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

0 25 50 75 100 125

P (k

N)

μmax (mm)

(b)

νmax (mm)

0

100

200

300

400

0 50 100 150

P (k

N)

(c)

0

100

200

300

400

ndash001 0 001 002

P (k

N)

θmax (rad)

(d)




be 192t

E

f1

radic

1minus038(bt)

E

f1

radic le b (9)

wheref1

PnAe (10)




Mnx FySex (11a)

Mny FySey (11b)


140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(a)

60

110

160

210

260

310

360

410

460

0 76 152 228 304y (mm)

Von

Mise

s str

ess (

MPa

)



(b)

140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(c)

60

110

160

210

260

310

360

410

460

0 76 152 228 304



y (mm)

Von

Mise

s str

ess (

MPa

)

(d)




Mrx CmxPrey


Mry CmyPrex






018

027

036

045

054

004 011 018 025 032

λx = 40

λx = 60λx = 80λx = 100

P uP

y

MuxMxy

(a)

018

027

036

045

054

004 011 018 025 032

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

017

024

031

038

045

002 006 01 014 018

P uP

y

MuxMxy

λx = 40

λx = 60λx = 80λx = 100

(c)

017

024

031

038

045

002 006 01 014 018P u

Py

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(d)

017

024

031

038

003 006 009 012

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(e)

017

022

027

032

037

042

002 007 012

P uP

y

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(f)








P

Pc+αxMx

Mxc+αyMy

Mycle 1 (13)

016

025

034

043

001 006 011 016

λx = 40

λx = 60

λx = 80λx = 100

P uP

y

MuxMxy

(a)

016

025

034

043

002 007 012 017

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

015

02

025

03

035

0016 0044 0072 01

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(c)

015

02

025

03

035

0018 0041 0064 0087 011P u

Py

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(d)










295 298 305 853 854 15 0 353 20999 135 109295 298 305 853 854 5 10 353 22462 157 119292 292 305 871 833 5 5 353 25437 181 120439 446 305 572 576 15 0 353 32530 115 098442 442 305 575 573 5 15 353 31224 113 102442 442 305 575 573 15 15 353 28674 121 109524 524 305 488 487 20 0 353 32473 106 091524 524 305 488 487 15 10 353 33872 114 103521 518 305 491 488 10 10 353 34863 111 100642 642 187 644 642 0 10 268 11439 095 096642 637 187 657 634 10 5 268 10818 106 099648 642 187 644 642 15 15 268 8325 098 097759 759 186 548 547 15 0 268 11811 105 110759 753 186 557 547 15 10 268 10924 108 119755 753 186 557 547 15 15 268 10263 106 11986 856 185 487 486 0 20 268 11766 092 13385 861 185 485 488 20 10 268 11530 112 14486 856 185 487 486 15 15 268 11159 090 143403 537 298 51 624 19 0 293 26176 102 096396 54 298 501 633 20 10 293 26037 114 111396 54 298 501 633 20 20 293 23670 113 111399 537 298 502 628 10 20 293 26897 113 110399 54 298 50 628 0 20 293 31195 121 10870 861 187 496 617 20 0 268 10521 103 10370 861 187 496 617 20 10 268 9714 102 10770 861 187 496 617 20 20 268 9372 105 11470 861 187 496 617 10 20 268 10272 098 10970 861 187 496 617 0 20 268 11452 090 103


293 220 446 396 293 0 104 568 742 098 095439 329 447 40 296 0 157 568 906 098 091582 437 448 40 304 0 209 568 932 090 086444 333 445 40 295 0 314 568 743 097 093295 221 442 644 456 0 104 568 524 095 083439 329 45 649 479 0 157 568 740 103 090582 437 448 646 488 0 208 568 743 087 082439 329 45 649 479 0 314 568 593 097 085582 437 448 646 488 0 416 568 639 089 083442 332 447 396 368 0 204 568 914 098 081582 437 448 40 378 0 269 568 986 095 081


Chiew et al [2]40 40 2 337 337 0 82 2612 13634 112 125571 571 14 337 337 0 82 2536 7112 103 11780 80 1 337 337 0 82 2486 3688 096 138

Pavlovcic et al [6]44 39 4 614 562 0 20 3734 56676 118 10944 39 4 614 562 0 60 3734 38485 114 11044 39 4 614 562 0 200 3734 20609 118 121





N

φminAefy+

βmxMx


βmyMy


(14)




N

φxA+

βmxMx





N

φminA+

βmxMx


βmyMy


(16)



fy235radic

and λyfy235radic





(17)


02

04

06

08

1

0 02 04 06 08 1

Nu

P n


S-40S-50S-60S-70

S-80R-40-50R-50-625R-60-75

R-70-875R-80-100ANSIAISC 360-10





N

φminAαfy+

βmxMx


+βmyMy


(18)



5 Conclusions







0

02

04

06

08

1

12

0 02 04 06 08 1

Nu

Nm

MuxMmx + MuyMmy





Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







Pn FcrA (6)



rle 471

E

QFy

radic(7a)

Fcr 0877Fe forKL

rgt 471

E

QFy

radic(7b)


Qa is dened as

Qa Ae

A (8)


0

100

200

300

400

0 5 10 15 20

P (k

N)

Δ (mm)

(a)

0

100

200

300

400

0 25 50 75 100 125

P (k

N)

μmax (mm)

(b)

νmax (mm)

0

100

200

300

400

0 50 100 150

P (k

N)

(c)

0

100

200

300

400

ndash001 0 001 002

P (k

N)

θmax (rad)

(d)




be 192t

E

f1

radic

1minus038(bt)

E

f1

radic le b (9)

wheref1

PnAe (10)




Mnx FySex (11a)

Mny FySey (11b)


140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(a)

60

110

160

210

260

310

360

410

460

0 76 152 228 304y (mm)

Von

Mise

s str

ess (

MPa

)



(b)

140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(c)

60

110

160

210

260

310

360

410

460

0 76 152 228 304



y (mm)

Von

Mise

s str

ess (

MPa

)

(d)




Mrx CmxPrey


Mry CmyPrex






018

027

036

045

054

004 011 018 025 032

λx = 40

λx = 60λx = 80λx = 100

P uP

y

MuxMxy

(a)

018

027

036

045

054

004 011 018 025 032

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

017

024

031

038

045

002 006 01 014 018

P uP

y

MuxMxy

λx = 40

λx = 60λx = 80λx = 100

(c)

017

024

031

038

045

002 006 01 014 018P u

Py

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(d)

017

024

031

038

003 006 009 012

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(e)

017

022

027

032

037

042

002 007 012

P uP

y

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(f)








P

Pc+αxMx

Mxc+αyMy

Mycle 1 (13)

016

025

034

043

001 006 011 016

λx = 40

λx = 60

λx = 80λx = 100

P uP

y

MuxMxy

(a)

016

025

034

043

002 007 012 017

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

015

02

025

03

035

0016 0044 0072 01

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(c)

015

02

025

03

035

0018 0041 0064 0087 011P u

Py

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(d)










295 298 305 853 854 15 0 353 20999 135 109295 298 305 853 854 5 10 353 22462 157 119292 292 305 871 833 5 5 353 25437 181 120439 446 305 572 576 15 0 353 32530 115 098442 442 305 575 573 5 15 353 31224 113 102442 442 305 575 573 15 15 353 28674 121 109524 524 305 488 487 20 0 353 32473 106 091524 524 305 488 487 15 10 353 33872 114 103521 518 305 491 488 10 10 353 34863 111 100642 642 187 644 642 0 10 268 11439 095 096642 637 187 657 634 10 5 268 10818 106 099648 642 187 644 642 15 15 268 8325 098 097759 759 186 548 547 15 0 268 11811 105 110759 753 186 557 547 15 10 268 10924 108 119755 753 186 557 547 15 15 268 10263 106 11986 856 185 487 486 0 20 268 11766 092 13385 861 185 485 488 20 10 268 11530 112 14486 856 185 487 486 15 15 268 11159 090 143403 537 298 51 624 19 0 293 26176 102 096396 54 298 501 633 20 10 293 26037 114 111396 54 298 501 633 20 20 293 23670 113 111399 537 298 502 628 10 20 293 26897 113 110399 54 298 50 628 0 20 293 31195 121 10870 861 187 496 617 20 0 268 10521 103 10370 861 187 496 617 20 10 268 9714 102 10770 861 187 496 617 20 20 268 9372 105 11470 861 187 496 617 10 20 268 10272 098 10970 861 187 496 617 0 20 268 11452 090 103


293 220 446 396 293 0 104 568 742 098 095439 329 447 40 296 0 157 568 906 098 091582 437 448 40 304 0 209 568 932 090 086444 333 445 40 295 0 314 568 743 097 093295 221 442 644 456 0 104 568 524 095 083439 329 45 649 479 0 157 568 740 103 090582 437 448 646 488 0 208 568 743 087 082439 329 45 649 479 0 314 568 593 097 085582 437 448 646 488 0 416 568 639 089 083442 332 447 396 368 0 204 568 914 098 081582 437 448 40 378 0 269 568 986 095 081


Chiew et al [2]40 40 2 337 337 0 82 2612 13634 112 125571 571 14 337 337 0 82 2536 7112 103 11780 80 1 337 337 0 82 2486 3688 096 138

Pavlovcic et al [6]44 39 4 614 562 0 20 3734 56676 118 10944 39 4 614 562 0 60 3734 38485 114 11044 39 4 614 562 0 200 3734 20609 118 121





N

φminAefy+

βmxMx


βmyMy


(14)




N

φxA+

βmxMx





N

φminA+

βmxMx


βmyMy


(16)



fy235radic

and λyfy235radic





(17)


02

04

06

08

1

0 02 04 06 08 1

Nu

P n


S-40S-50S-60S-70

S-80R-40-50R-50-625R-60-75

R-70-875R-80-100ANSIAISC 360-10





N

φminAαfy+

βmxMx


+βmyMy


(18)



5 Conclusions







0

02

04

06

08

1

12

0 02 04 06 08 1

Nu

Nm

MuxMmx + MuyMmy





Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



be 192t

E

f1

radic

1minus038(bt)

E

f1

radic le b (9)

wheref1

PnAe (10)




Mnx FySex (11a)

Mny FySey (11b)


140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(a)

60

110

160

210

260

310

360

410

460

0 76 152 228 304y (mm)

Von

Mise

s str

ess (

MPa

)



(b)

140

180

220

260

300

340

380

420

460

0 61 122 183 244x (mm)

Von

Mise

s str

ess (

MPa

)



(c)

60

110

160

210

260

310

360

410

460

0 76 152 228 304



y (mm)

Von

Mise

s str

ess (

MPa

)

(d)




Mrx CmxPrey


Mry CmyPrex






018

027

036

045

054

004 011 018 025 032

λx = 40

λx = 60λx = 80λx = 100

P uP

y

MuxMxy

(a)

018

027

036

045

054

004 011 018 025 032

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

017

024

031

038

045

002 006 01 014 018

P uP

y

MuxMxy

λx = 40

λx = 60λx = 80λx = 100

(c)

017

024

031

038

045

002 006 01 014 018P u

Py

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(d)

017

024

031

038

003 006 009 012

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(e)

017

022

027

032

037

042

002 007 012

P uP

y

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(f)








P

Pc+αxMx

Mxc+αyMy

Mycle 1 (13)

016

025

034

043

001 006 011 016

λx = 40

λx = 60

λx = 80λx = 100

P uP

y

MuxMxy

(a)

016

025

034

043

002 007 012 017

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

015

02

025

03

035

0016 0044 0072 01

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(c)

015

02

025

03

035

0018 0041 0064 0087 011P u

Py

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(d)










295 298 305 853 854 15 0 353 20999 135 109295 298 305 853 854 5 10 353 22462 157 119292 292 305 871 833 5 5 353 25437 181 120439 446 305 572 576 15 0 353 32530 115 098442 442 305 575 573 5 15 353 31224 113 102442 442 305 575 573 15 15 353 28674 121 109524 524 305 488 487 20 0 353 32473 106 091524 524 305 488 487 15 10 353 33872 114 103521 518 305 491 488 10 10 353 34863 111 100642 642 187 644 642 0 10 268 11439 095 096642 637 187 657 634 10 5 268 10818 106 099648 642 187 644 642 15 15 268 8325 098 097759 759 186 548 547 15 0 268 11811 105 110759 753 186 557 547 15 10 268 10924 108 119755 753 186 557 547 15 15 268 10263 106 11986 856 185 487 486 0 20 268 11766 092 13385 861 185 485 488 20 10 268 11530 112 14486 856 185 487 486 15 15 268 11159 090 143403 537 298 51 624 19 0 293 26176 102 096396 54 298 501 633 20 10 293 26037 114 111396 54 298 501 633 20 20 293 23670 113 111399 537 298 502 628 10 20 293 26897 113 110399 54 298 50 628 0 20 293 31195 121 10870 861 187 496 617 20 0 268 10521 103 10370 861 187 496 617 20 10 268 9714 102 10770 861 187 496 617 20 20 268 9372 105 11470 861 187 496 617 10 20 268 10272 098 10970 861 187 496 617 0 20 268 11452 090 103


293 220 446 396 293 0 104 568 742 098 095439 329 447 40 296 0 157 568 906 098 091582 437 448 40 304 0 209 568 932 090 086444 333 445 40 295 0 314 568 743 097 093295 221 442 644 456 0 104 568 524 095 083439 329 45 649 479 0 157 568 740 103 090582 437 448 646 488 0 208 568 743 087 082439 329 45 649 479 0 314 568 593 097 085582 437 448 646 488 0 416 568 639 089 083442 332 447 396 368 0 204 568 914 098 081582 437 448 40 378 0 269 568 986 095 081


Chiew et al [2]40 40 2 337 337 0 82 2612 13634 112 125571 571 14 337 337 0 82 2536 7112 103 11780 80 1 337 337 0 82 2486 3688 096 138

Pavlovcic et al [6]44 39 4 614 562 0 20 3734 56676 118 10944 39 4 614 562 0 60 3734 38485 114 11044 39 4 614 562 0 200 3734 20609 118 121





N

φminAefy+

βmxMx


βmyMy


(14)




N

φxA+

βmxMx





N

φminA+

βmxMx


βmyMy


(16)



fy235radic

and λyfy235radic





(17)


02

04

06

08

1

0 02 04 06 08 1

Nu

P n


S-40S-50S-60S-70

S-80R-40-50R-50-625R-60-75

R-70-875R-80-100ANSIAISC 360-10





N

φminAαfy+

βmxMx


+βmyMy


(18)



5 Conclusions







0

02

04

06

08

1

12

0 02 04 06 08 1

Nu

Nm

MuxMmx + MuyMmy





Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Mrx CmxPrey


Mry CmyPrex






018

027

036

045

054

004 011 018 025 032

λx = 40

λx = 60λx = 80λx = 100

P uP

y

MuxMxy

(a)

018

027

036

045

054

004 011 018 025 032

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

017

024

031

038

045

002 006 01 014 018

P uP

y

MuxMxy

λx = 40

λx = 60λx = 80λx = 100

(c)

017

024

031

038

045

002 006 01 014 018P u

Py

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(d)

017

024

031

038

003 006 009 012

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(e)

017

022

027

032

037

042

002 007 012

P uP

y

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(f)








P

Pc+αxMx

Mxc+αyMy

Mycle 1 (13)

016

025

034

043

001 006 011 016

λx = 40

λx = 60

λx = 80λx = 100

P uP

y

MuxMxy

(a)

016

025

034

043

002 007 012 017

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

015

02

025

03

035

0016 0044 0072 01

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(c)

015

02

025

03

035

0018 0041 0064 0087 011P u

Py

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(d)










295 298 305 853 854 15 0 353 20999 135 109295 298 305 853 854 5 10 353 22462 157 119292 292 305 871 833 5 5 353 25437 181 120439 446 305 572 576 15 0 353 32530 115 098442 442 305 575 573 5 15 353 31224 113 102442 442 305 575 573 15 15 353 28674 121 109524 524 305 488 487 20 0 353 32473 106 091524 524 305 488 487 15 10 353 33872 114 103521 518 305 491 488 10 10 353 34863 111 100642 642 187 644 642 0 10 268 11439 095 096642 637 187 657 634 10 5 268 10818 106 099648 642 187 644 642 15 15 268 8325 098 097759 759 186 548 547 15 0 268 11811 105 110759 753 186 557 547 15 10 268 10924 108 119755 753 186 557 547 15 15 268 10263 106 11986 856 185 487 486 0 20 268 11766 092 13385 861 185 485 488 20 10 268 11530 112 14486 856 185 487 486 15 15 268 11159 090 143403 537 298 51 624 19 0 293 26176 102 096396 54 298 501 633 20 10 293 26037 114 111396 54 298 501 633 20 20 293 23670 113 111399 537 298 502 628 10 20 293 26897 113 110399 54 298 50 628 0 20 293 31195 121 10870 861 187 496 617 20 0 268 10521 103 10370 861 187 496 617 20 10 268 9714 102 10770 861 187 496 617 20 20 268 9372 105 11470 861 187 496 617 10 20 268 10272 098 10970 861 187 496 617 0 20 268 11452 090 103


293 220 446 396 293 0 104 568 742 098 095439 329 447 40 296 0 157 568 906 098 091582 437 448 40 304 0 209 568 932 090 086444 333 445 40 295 0 314 568 743 097 093295 221 442 644 456 0 104 568 524 095 083439 329 45 649 479 0 157 568 740 103 090582 437 448 646 488 0 208 568 743 087 082439 329 45 649 479 0 314 568 593 097 085582 437 448 646 488 0 416 568 639 089 083442 332 447 396 368 0 204 568 914 098 081582 437 448 40 378 0 269 568 986 095 081


Chiew et al [2]40 40 2 337 337 0 82 2612 13634 112 125571 571 14 337 337 0 82 2536 7112 103 11780 80 1 337 337 0 82 2486 3688 096 138

Pavlovcic et al [6]44 39 4 614 562 0 20 3734 56676 118 10944 39 4 614 562 0 60 3734 38485 114 11044 39 4 614 562 0 200 3734 20609 118 121





N

φminAefy+

βmxMx


βmyMy


(14)




N

φxA+

βmxMx





N

φminA+

βmxMx


βmyMy


(16)



fy235radic

and λyfy235radic





(17)


02

04

06

08

1

0 02 04 06 08 1

Nu

P n


S-40S-50S-60S-70

S-80R-40-50R-50-625R-60-75

R-70-875R-80-100ANSIAISC 360-10





N

φminAαfy+

βmxMx


+βmyMy


(18)



5 Conclusions







0

02

04

06

08

1

12

0 02 04 06 08 1

Nu

Nm

MuxMmx + MuyMmy





Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







P

Pc+αxMx

Mxc+αyMy

Mycle 1 (13)

016

025

034

043

001 006 011 016

λx = 40

λx = 60

λx = 80λx = 100

P uP

y

MuxMxy

(a)

016

025

034

043

002 007 012 017

P uP

y

MuyMyy

λx = 40

λx = 60λx = 80λx = 100

(b)

015

02

025

03

035

0016 0044 0072 01

P uP

y

MuxMxy

λx = 40

λx = 60

λx = 80λx = 100

(c)

015

02

025

03

035

0018 0041 0064 0087 011P u

Py

MuyMyy

λx = 40

λx = 60

λx = 80λx = 100

(d)










295 298 305 853 854 15 0 353 20999 135 109295 298 305 853 854 5 10 353 22462 157 119292 292 305 871 833 5 5 353 25437 181 120439 446 305 572 576 15 0 353 32530 115 098442 442 305 575 573 5 15 353 31224 113 102442 442 305 575 573 15 15 353 28674 121 109524 524 305 488 487 20 0 353 32473 106 091524 524 305 488 487 15 10 353 33872 114 103521 518 305 491 488 10 10 353 34863 111 100642 642 187 644 642 0 10 268 11439 095 096642 637 187 657 634 10 5 268 10818 106 099648 642 187 644 642 15 15 268 8325 098 097759 759 186 548 547 15 0 268 11811 105 110759 753 186 557 547 15 10 268 10924 108 119755 753 186 557 547 15 15 268 10263 106 11986 856 185 487 486 0 20 268 11766 092 13385 861 185 485 488 20 10 268 11530 112 14486 856 185 487 486 15 15 268 11159 090 143403 537 298 51 624 19 0 293 26176 102 096396 54 298 501 633 20 10 293 26037 114 111396 54 298 501 633 20 20 293 23670 113 111399 537 298 502 628 10 20 293 26897 113 110399 54 298 50 628 0 20 293 31195 121 10870 861 187 496 617 20 0 268 10521 103 10370 861 187 496 617 20 10 268 9714 102 10770 861 187 496 617 20 20 268 9372 105 11470 861 187 496 617 10 20 268 10272 098 10970 861 187 496 617 0 20 268 11452 090 103


293 220 446 396 293 0 104 568 742 098 095439 329 447 40 296 0 157 568 906 098 091582 437 448 40 304 0 209 568 932 090 086444 333 445 40 295 0 314 568 743 097 093295 221 442 644 456 0 104 568 524 095 083439 329 45 649 479 0 157 568 740 103 090582 437 448 646 488 0 208 568 743 087 082439 329 45 649 479 0 314 568 593 097 085582 437 448 646 488 0 416 568 639 089 083442 332 447 396 368 0 204 568 914 098 081582 437 448 40 378 0 269 568 986 095 081


Chiew et al [2]40 40 2 337 337 0 82 2612 13634 112 125571 571 14 337 337 0 82 2536 7112 103 11780 80 1 337 337 0 82 2486 3688 096 138

Pavlovcic et al [6]44 39 4 614 562 0 20 3734 56676 118 10944 39 4 614 562 0 60 3734 38485 114 11044 39 4 614 562 0 200 3734 20609 118 121





N

φminAefy+

βmxMx


βmyMy


(14)




N

φxA+

βmxMx





N

φminA+

βmxMx


βmyMy


(16)



fy235radic

and λyfy235radic





(17)


02

04

06

08

1

0 02 04 06 08 1

Nu

P n


S-40S-50S-60S-70

S-80R-40-50R-50-625R-60-75

R-70-875R-80-100ANSIAISC 360-10





N

φminAαfy+

βmxMx


+βmyMy


(18)



5 Conclusions







0

02

04

06

08

1

12

0 02 04 06 08 1

Nu

Nm

MuxMmx + MuyMmy





Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









295 298 305 853 854 15 0 353 20999 135 109295 298 305 853 854 5 10 353 22462 157 119292 292 305 871 833 5 5 353 25437 181 120439 446 305 572 576 15 0 353 32530 115 098442 442 305 575 573 5 15 353 31224 113 102442 442 305 575 573 15 15 353 28674 121 109524 524 305 488 487 20 0 353 32473 106 091524 524 305 488 487 15 10 353 33872 114 103521 518 305 491 488 10 10 353 34863 111 100642 642 187 644 642 0 10 268 11439 095 096642 637 187 657 634 10 5 268 10818 106 099648 642 187 644 642 15 15 268 8325 098 097759 759 186 548 547 15 0 268 11811 105 110759 753 186 557 547 15 10 268 10924 108 119755 753 186 557 547 15 15 268 10263 106 11986 856 185 487 486 0 20 268 11766 092 13385 861 185 485 488 20 10 268 11530 112 14486 856 185 487 486 15 15 268 11159 090 143403 537 298 51 624 19 0 293 26176 102 096396 54 298 501 633 20 10 293 26037 114 111396 54 298 501 633 20 20 293 23670 113 111399 537 298 502 628 10 20 293 26897 113 110399 54 298 50 628 0 20 293 31195 121 10870 861 187 496 617 20 0 268 10521 103 10370 861 187 496 617 20 10 268 9714 102 10770 861 187 496 617 20 20 268 9372 105 11470 861 187 496 617 10 20 268 10272 098 10970 861 187 496 617 0 20 268 11452 090 103


293 220 446 396 293 0 104 568 742 098 095439 329 447 40 296 0 157 568 906 098 091582 437 448 40 304 0 209 568 932 090 086444 333 445 40 295 0 314 568 743 097 093295 221 442 644 456 0 104 568 524 095 083439 329 45 649 479 0 157 568 740 103 090582 437 448 646 488 0 208 568 743 087 082439 329 45 649 479 0 314 568 593 097 085582 437 448 646 488 0 416 568 639 089 083442 332 447 396 368 0 204 568 914 098 081582 437 448 40 378 0 269 568 986 095 081


Chiew et al [2]40 40 2 337 337 0 82 2612 13634 112 125571 571 14 337 337 0 82 2536 7112 103 11780 80 1 337 337 0 82 2486 3688 096 138

Pavlovcic et al [6]44 39 4 614 562 0 20 3734 56676 118 10944 39 4 614 562 0 60 3734 38485 114 11044 39 4 614 562 0 200 3734 20609 118 121





N

φminAefy+

βmxMx


βmyMy


(14)




N

φxA+

βmxMx





N

φminA+

βmxMx


βmyMy


(16)



fy235radic

and λyfy235radic





(17)


02

04

06

08

1

0 02 04 06 08 1

Nu

P n


S-40S-50S-60S-70

S-80R-40-50R-50-625R-60-75

R-70-875R-80-100ANSIAISC 360-10





N

φminAαfy+

βmxMx


+βmyMy


(18)



5 Conclusions







0

02

04

06

08

1

12

0 02 04 06 08 1

Nu

Nm

MuxMmx + MuyMmy





Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


N

φminAefy+

βmxMx


βmyMy


(14)




N

φxA+

βmxMx





N

φminA+

βmxMx


βmyMy


(16)



fy235radic

and λyfy235radic





(17)


02

04

06

08

1

0 02 04 06 08 1

Nu

P n


S-40S-50S-60S-70

S-80R-40-50R-50-625R-60-75

R-70-875R-80-100ANSIAISC 360-10





N

φminAαfy+

βmxMx


+βmyMy


(18)



5 Conclusions







0

02

04

06

08

1

12

0 02 04 06 08 1

Nu

Nm

MuxMmx + MuyMmy





Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




N

φminAαfy+

βmxMx


+βmyMy


(18)



5 Conclusions







0

02

04

06

08

1

12

0 02 04 06 08 1

Nu

Nm

MuxMmx + MuyMmy





Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Data Availability




Acknowledgments


References







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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