BucklingAnalysisandStabilityofCompressedLow-CarbonSteel ...

Research ArticleBuckling Analysis and Stability of Compressed Low-Carbon SteelRods in the Elastoplastic Region of Materials

Gaioz Partskhaladze 1 Ingusha Mshvenieradze 2 Elguja Medzmariashvili2

Gocha Chavleshvili 1 Victor Yepes 3 and Julian Alcala3

1Engineering and Construction Department Batumi Shota Rustaveli State University Batumi 6010 Georgia2Department of Civil and Industrial Engineering Georgian Technical University Tbilisi 6000 Georgia3Institute of Concrete Science and Technology (ICITECH) Universitat Politecnica de Valencia Valencia 46022 Spain

Correspondence should be addressed to Victor Yepes vyepespcstupves

Received 4 March 2019 Accepted 14 May 2019 Published 29 May 2019

Academic Editor Abdul Aziz bin Abdul Samad

Copyright copy 2019 Gaioz Partskhaladze et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

is paper presents new approaches for solving a problem of the stability of compressed rods in the elastoplastic working region ofmaterials It is known that the columns of buildings supports of engineering devices drill rods of oil and gas extraction industrymay be subjected to significant risk of stability loss Nowadays there are designmethods based on test results defining the relations(eg critical stresses-slenderness) to avoid this risk due to stability loss but the precision and limits of definition are not alwaysknownemain objectives of the study were to develop new approaches that would allow specifying the values of critical stressesof compressed elements beyond the proportional limit e problem of stability of the compressed elements in the elastoplasticregion was studied according to the stability theory e authors suggested an original approach to the issue in particular thedetermination of values of the critical stresses and the finding of the points of the bifurcation were carried out by the tangentestablished by experimental results and by the approximation of the so-called double modulus Comparative analysis showed theadvantage of the proposed approach particularly that the new critical curves were located below the curves of Engesser-Karmanand Shanley and above the critical curves established by building codes A new approach for the determination of critical stressesin the elastoplastic region was developed through which the structural reliability and economic efficiency was increased by almost12 compared to the existing approaches

1 Introduction

For developing modern structural civil mechanical andother fields of engineering it is necessary to work out veryprecise calculating methods for the stability of rod systemsResearchers still have a great interest in the failure ofstructural elements due to the loss of stability

A bulk-forming process from metallic materials such asstamping forging extrusion and spinning is one of themain processes in industrial manufacturing Some forgingtools such as supports punches or ejectors may be sub-jected to significant risk of stability loss (buckling) when theyare used Rods are used for drilling in oil and gas fields estability loss of the drill rod is one of the complicating factorsof the drilling process which is a major problem for oil and

gas extraction industry [1] In the seismic areas steel framesare known as an efficient arrangement for structures Inaddition load-carrying capacity of the columns as the mainparts of the frame systems using limit state design is de-termined by the stability in the elastoplastic region of thematerial [2] erefore steel columns have been widely usedto dissipate seismic energy in seismic regions ese con-siderations are especially relevant in calculating the stabilityof high-rise buildings taking into account the complicatedtechnological load on them [3ndash8]

is paper presents new approaches for solving bucklingproblems for axially compressed columns in the elastoplasticworking region of the material A prismatic (rectangular)steel column was studied as a sample that is depicted inFigure 1 e study of the buckling behavior of prismatic

HindawiAdvances in Civil EngineeringVolume 2019 Article ID 7601260 9 pageshttpsdoiorg10115520197601260

(rectangular) compressed elements is a relevant step towardsunderstanding and assessing the reliability of more complexstructures [9]

In steel columns the issues of buckling can be dividedinto three types overall buckling local buckling and in-teractive buckling [10 11] In prismatic columns withrectangular cross section and medium slenderness ratiooverall buckling is normally observed rst [12]

e buckling appearance (loss of stability) process wasassessed with critical forces (from which the values of re-duction factor are dened) e reduction factor approachdened from critical stresses appears in two alternativemethods in the standards (a) traditional method usingsemiempirical design equations and (b) computer-aidedmethod based on linear buckling analysis [12 13]

It is common knowledge that the physical action causingthe buckling of axially compressed columns is the depletionof its lateral stiness to a value of zero ere are a number ofproblems in actual engineering situations which complicatethe prediction of buckling loads eg initial (geometric)imperfection material nonlinearity and loading eccentric-ity e linear elastic Euler model of a prismatic simplysupported rod is still the fundamental tool for studyingbuckling stability [9 10]

In a similar way several empirical models for instanceEngesserrsquos [7 13 14] are proposed in the literature to

experimentally characterize the elastic-plastic and plasticbuckling Later Shanley [9 15] oered a dierent approachfor dening the buckling process with its semiempiricaldesign equation

In the literature [16 17] researchers presented a nu-merical method for computing the load-deection re-lationships of loaded square columns with various geometricparameters loading eccentricity and column slendernessratios

As mentioned above steel frames are an ecient ar-rangement for structural systems in seismic regions For thisreason they have been the subject of strong research interestin recent years In practice earthquakes generate multidi-rectional ground motions with the occurrence of yieldingand buckling eects [4 6 8] Previous research studies[3 18] indicate how the development of local bucklingrestricts the ability of the members to deform plastically alsohow cross-section geometry inuences strength ductilityand the energy absorption process

In literature [12] the researchers examined the simplysupported steel member with uniform cross section sub-jected to uniform forces e results showed buckling re-sistance for such members and were standardized by thereduction factor based on the AyrtonndashPerry formula [19]and calibrated by strong theoretical and empirical back-ground e local and overall stability of the columnsthrough simulating their geometric and material nonlinearbehavior was investigated [11] A methodology [20] wasdevised to estimate the elastoplastic properties of metallicmaterials from the hardness measurements using the twomost frequently used sharp indenters ie the Vickers in-denter and the Knoop indenter In the study [21] the re-searchers studied overall buckling behavior and design ofhigh-strength steel columns e study results indicated thatwith an increase of the yield strength of high-strength steelcompressive residual stresses ratio to the yield strengthbecomes evidently lower and possesses much less severeeects on the overall buckling behavior Dierent H-shapedsections of the welded steel columns were experimentallyand numerically investigated with their overall bucklingbehavior and their practical application was promoted [22]

e main disadvantage of the previously proposed de-sign methods to avoid risk due to loss of stability in theelastoplastic region was that the accuracy and limits of thedenition were not always known

e main objectives of the study were to (a) develop theapproaches that would allow specifying the values of criticalstresses of compressed elements beyond their proportionallimit (b) approximate and specify maximally the theoreticaland real values of critical loads (c) nd the possible points ofbifurcation in the elastoplastic region and (d) eliminate theexistence of residual deformations in the cross-section andeventually reduce the mass of the structure and increase itsreliability

e problem of the stability of the compressed elementswas studied according to the stability theory Methods basedon the equations of equilibrium of mechanics for slightlyperturbed systems around the initial position were used tostudy the stability

N

l

A

Figure 1 A prismatic (rectangular) pin-ended steel column

2 Advances in Civil Engineering

During the study these steps were followed

(a) e energy method was used(b) e principle of initial imperfections of the rod was

considered(c) Eulerrsquos analytical method was used in (during)

compression which in turn was based on Hookersquoslaw

(d) e process of stability loss (buckling) was studieddepending on the critical stresses and the slendernessratio [12 23ndash26]

e advantage of the proposed approach in determiningstability of the compressed elements compared to theexisting ones is that the critical stresses in the elastoplasticregion are derived analytically and they accurately reect theexperimental data

2 Stability Problem Formulation

21 General Aspects of Understanding Stability e load-carrying capacity of a compressed rod could be depleted as aresult of stability loss or as a result of the deection thatoccurs before the rod fails due to compression e equi-librium of a rigid body could be stable unstable or neutral e same condition occurs in deformable body mechanics e only dierence is that the type of equilibrium dependson the value of the acting load [8 27]

Figure 2 depicts a long thin (slender) cantilevered rodwith axial force loading e rod is compressed by a rela-tively small value of the force N and is present in a stableequilibrium since it assumes a slight deviation from thevertical as a result of any perturbed impact and after theremoval of the force it soon returns to the initial state(Figure 2(a))

e rod gradually returns to the initial position togetherwith the increase of the load and the state of the neutralequilibrium occurs at a certain value of the critical forceNcr e rod obtains equilibrium in the deected position duringthe modest deviation from the vertical position (dashed lineFigure 2(b)) At this time the bifurcation (the coexistence ofcontiguous equilibrium forms) is formed which is char-acterized by an ldquoexchangerdquo of stability between two equi-librium forms of the rod

e rectilinear form ldquolosesrdquo stability and the curvilinearform still ldquotriesrdquo to obtain it As it is known the force thattransfers the rod from the rectilinear stable position to thecurvilinear stable position is a critical force Ncr

e new equilibrium curvilinear form theoreticallybecomes stable at loading the value of which exceedscritical one (Figure 2(c)) is condition is practicallyunacceptable since the rod already works not for purecompression but for compression with bending ie whenthe force reaches its critical value the rectilinear form of therod is already unstable e rod will bend in a less rigidplane and obtain (get) a stable curvilinear form ecurvature of the rod begins to increase rapidly even whenthe load increases slightly and it loses its load-carryingcapacity [2 13]

22 eoretical Analysis and Eulerrsquos Critical Curves in theElastic Region Figure 3(a) depicts the long narrow pin-ended rod on which the compressive force is loadedalong the symmetrical axis Taking into account the principleof initial imperfections of the rod the rod is bent during theload by the axial force (Figure 3(b)) [12 24] In order toreveal the internal bending momentM it is necessary to cutthe column in the middle zone for creating the free body(Figure 3(c))

Moment equilibrium dictates that the bending moment isgiven byM minusNy Here y is the lateral deection From theresistance of materials it is known thatMEI asymp d2ydx2 eabove two relationships provide dening dierential equationfor Euler buckling In addition according to the energymethod the compressive force reaches its critical value whenthe ends of the rod approach each other and the workperformed by external forces is equal to the deformation workof the bending of the compressed rod [6 13 14]

A dierential equation of the curved axis for such rods isshown as follows

d2y

dx2+N

EIy 0 (1)

By solving it the formula for calculating critical forcestress was developed by Leonard Euler [9 12]

Ncr π2EImin

l20

σcr π2Eλ2

(2)

whereNcr is the critical force σcr is the critical stress E is themodulus of elasticity (Youngrsquos modulus) λ l0imin is the

N lt Ncr

(a)

N = Ncr

(b)

N gt Ncr

(c)

Figure 2 Equilibrium types of deformed body-cantilevered col-umn under axial force (a) Stable equilibrium (b) neutral equi-librium (c) unstable equilibrium

Advances in Civil Engineering 3

slenderness ratio that is equal to the ratio of the effectivelength of the column to the least of radius gyration l0 μl isthe effective length of the rodcolumn μ is the reductionratio which depends on the boundary conditions at the endsof the rodcolumn the radius of gyration is equal toimin

IminA

1113968 Imin is the minimummoment of inertia (the

second moment of area) of the cross section and A is thearea of the cross section of the column σ0 NA

e geometrical and material characteristics of thecarbon steel (S235) columns with various grades are pre-sented in Table 1 A diagram depicting the stress-strainrelationship for carbon steel is constructed in Figure 4(a)according to Table 1 and a diagram depicting the slen-derness ratio of critical stresses (equation (2)) for carbonsteel is constructed in Figure 4(b) Curve 1 [9 15 21 28]

Equation (2) is valid for such rods that work in the elasticregion Critical stresses (σcr) are less than the proportionallimit (σpl) σcr lt σpl in this region (Figure 4(a)) and theslenderness ratio (λ) is higher than the critical slendernessratio (λcr) λgt λcr for carbon steel For example the diagramconstructed on Figure 4(b) depicts that the elastic regioncorresponds to λisin[102 200] the zone of slenderness forsteel-S235 Here λcr π

Eσpl

1113969 where σpl is the pro-

portional limit of the material e constant modulus ofelasticity (Youngrsquos modulus) E spreads in the whole crosssection of the rod A diagram constructed in the zone of thelarge slenderness is called Eulerrsquos quadratic hyperbola(Figure 4(b) Curve 1) [9] e points of bifurcation arelocated at the points existing on the hyperbola diagram Itshows the coexistence of the neighboring equilibrium formsfor long flexible rods (ie columns with slender pro-portions) working in the elastic region

23 Critical Curves and eoretical Analysis in the Elasto-plastic Region When a perfectly straight column is

subjected to an axial force the state of internal stresses nearthe midpoint of the rodcolumn is uniform compressionFor a crooked column (the principle of initial imperfec-tions) the state of internal stress near the midpoint is notuniform and there arises the net tension stress on the outercurve of the column [12 24]

Figure 5 depicts a blowup of the middle slenderness ratioof the column and the stress distribution in the cross sectionof the middle zone (Figure 5(a)) e tensile zone ie theunloaded zone (area A1 in Figure 5(b)) together with thecompressed zone (area A2 in Figure 5(b)) arises in the crosssection during buckling in which (tensile zone) constantmodulus of elasticity E will exist because of the relativelysmall value of tensile stresses and Gerstnerrsquos law (see thezone where σ lt σel Figure 4(a)) [9 27]

ere is no increase in compressive force during thebuckling in the elastoplastic region in accordance with thefirst theory of calculating the stability of compressed ele-ments e compressive stresses will transfer from theunloaded zone to the compressed zone due to the crook-edness of the column [18 29] Consequently the stresses willbe increased in the compressed zone and instead of theconstant modulus of elasticity E the smaller value of thetangent modulus Et will exist (Figure 5(b))us in the crosssection there will be simultaneously two elastic moduli emodulus E will exist in the tensile zone and the tangentmodulus Et will exist in the compressed zone (Figure 5(b))

Only the constant modulus of elasticity E is no longeruseful in the elastoplastic working region of the materialthat is the zone of slendernessmdashλ isin [40 102] for the steelS235 in the cross section because the stress-strain re-lationship in this region becomes nonlinear (Figures 4(a)and 5(b) compressed zone) [22 30]

e formula for calculating critical stresses for theelastoplastic region is identical to the calculation formula forthe elastic region ere is just one difference between themeg instead of the constant elastic modulus E the reducedmodulus T of Engesser-Karman appears which has thefollowing form in the case of a rectangular cross section[13 27]

N

l

x

y

(a)

NΔl

f

y

x

l

x

y

(b)

Nf

y

x

N

M

x

y

(c)

Figure 3 e long slender pin-ended rodcolumn (a) Designscheme of a compressed pinned rod (b) buckling of a long slenderrod (c) free body for Euler buckling model

Table 1 Design parameters of carbon steel with various grades

ε times 108 σ (kNcm2) Et times 10minus4(kNcm2) T times 10minus4 (kNcm2) λT λt

095 20 206 206 102 10210 21 142 172 90 81811 22 099 139 79 66612 228 067 105 676 5413 234 046 085 59 4414 238 026 054 475 32715 239 013 033 37 23116 24 006 019 28 15718ndash40 24 0 0 0 045 241 002 007 17 915 242 004 013 23 1276 247 005 015 245 1428 2575 005 015 24 13810 2685 005 015 236 13612 28 005 015 23 135


T 4EEtE

radic+Etradic

( )2 (3)

In this case when using equation (3) Eulerrsquos formulawill be equal to the following

σcr π2Tλ2 (4)

Curve 2 in Figure 4(b) depicts the dependency diagram[σcr minus λ] in the zone of medium slenderness λ isin [40 102]constructed by the use of equation (4)

If the sum of the moments of the internal forces will beequal to the external moment then the formula of thebending moment for any cross section with allowance forthe so-called tangent modulus will be equal to thefollowing

M intA1

σ1ydA + intA2

σ2ydAhArrM intA1

E

ρy2dA

+ intA2

Et

ρy2dAhArrM

E

ρintA1

y2dA +Et

ρintA2

y2dA

(5)

Hence

M 1ρEI1 + EtI2( ) (6)

where I1 and I2 are the moments of inertia of both partsof the cross section towards the neutral axis (Figure 6) e tangent modulus Et is obtained as constant (anaverage)

As commonly known the formula will be obtained usingthe elastoplastic modulus Tlowast instead of the elastic modulus Ein the formula M EIρ of the bending moment

M TlowastI

ρ (7)

whereM is the bending moment Tlowast is the double modulusI is the moment of inertia and ρ is the local bending radius(the radius of bending at the current section)

e formula for the double modulus of elasticity isderived by equating equations (6) and (7) and by solving Tlowast[7] as follows

Tlowast 1IEI1 + EtI2( ) (8)

As is obvious from equation (8) the double modulusdepends on the shape of the cross section of the rod eneutral axis divides the cross section into parts havingvarious moduli that allow nding the reduced stiness(Tlowast middot I) of the rod as the product of the moment of inertia (I)of the rod on the reduced modulus (Tlowast)

According to the second approach for calculatingthe stability of compressed elements (Shanleyrsquos theory)there will not be the unloaded zone in the cross sec-tion in the elastoplastic region of the material sincethe compressive force increases along with the bending[13]

At the beginning of stability loss the loading speed withthe longitudinal forces of the rod will be much more than theunloading speed of the bent bers and the stresses will in-crease throughout the section ie a small bendwill not lead tounloading erefore above the limits of elasticity the criticaltangential force will be the tangent according to modulus Et

Elastic zone

E

Elastoplastic zone

Et σy

σel

times108

0

5

10

15

20

25

30St

ress

σ (k

Nc

m2 )

2 4 6 8 10 120Strain

(a)

3

64

5

2

1

Short columns Columns with middleslender proportions Columns with

slender proportions

Eulerrsquos quadratichyperbola

σy

σel

σcr = π2Eλ2

λcr

20 40 60 80 100 120 140 160 180 2000Slenderness ratio λ

0

5

10

15

20

25

30

Criti

cal s

tress

σcr

(kN

cm

2 )

(b)

Figure 4 Blown up middle zone of a long slender rod (a) Free body for Euler buckling model (b) combined axial and bending stresses in acrooked column


[7] is assumption greatly simplies the calculation of thecompressed rods on critical forces especially the dicult tasksof the stability of slabs and shells [15]

en the parabolic equation proposed by Shanley andproven by Stelmakhrsquos tests can be used for the tangentmodulus of various steels at the elastoplastic stage

Et E 1minusσcrt minus σelσy minus σel

( )2

(9)

where σcrt is the critical stress in the elastoplastic region σelis the elastic limit and σy is the yield strength

On the contrary the critical stress calculated by Engessercan be expressed with the dependence of the slenderness λand the tangent modulus Et In this case Eulerrsquos formulawill be as follows [14 27]

σcr π2Et

λ2 (10)

Curve 3 in Figure 4(b) depicts the dependency graph[σcr minus λ] in the zone of medium slenderness λ isin [40 102]constructed by equation (10) and according to Table 1

3 Problem Solution

31 Choice of Design Loads and Criteria for Stability Inaddition to the methods presented in the introduction whenstudying the rod deformations small deviations comparedto the length of the rod are adopted or when studying roddeformation it is assumed that the rod is in a geometricallylinear form e stress-strain relation is known Despite allthis the problem includes many diculties e rst di-culty is to determine the points of bifurcation e seconddiculty is to determine which of the loads the tangentmodulus (Et) or the reduced modulus (Engesser-Karman-T) must be chosen for the calculation e third diculty ishow to eliminate residual deformations in the cross sectionIt is necessary to obtain such a dependence which by a singleapproach will determine such critical stresses which will be

I2 I1 b

I

h1

h

y

h2

Compressedzone

Tensilezone

Neutralaxis

Figure 6 e cross section of the rod

y

EtE

ndashyy

1

A1 A2

h2h1

b

h

y

1ndash1

M

N

e

1

x

Compressedzone

Tensilezone

E

Nf

y

x

N

(a) (b)

M

x

y

Et

f

σ1 = +Eyρσ2 = ndashEt yρ

σ0 = NA

Figure 5 Euler diagram with a real elastoplastic behavior using a specic variation of stresses for carbon S235 steel (a) Stress-straindiagram (b) dependence of the critical stresses and the slenderness ratio (1) Eulerrsquos quadratic hyperbola (2) Engesser-Karmanrsquos model (3)Shanleyrsquos model (4) authors approach (5) model of building codes (6) experimental data


close to the real values of critical loadsey will also be closeto the experimental data

In order for the critical curves to be determined not onlyin the elastoplastic region on the basis of the experimentaldata the authors proposed a new approach to the analyticalsolution of the formula for calculating critical stresses

32 Proposed New Approach for Obtaining Critical Stresses inthe Elastoplastic Region and Its eoretical Analysis ecalculation methods based on the results of the experimentalresearch have been the most common up to now ereforevarious authors have proposed formulas for calculating thelongitudinal bending beyond the limits of elasticity [12]

e authors suggested an original approach to the issuein particular the determination of values of the criticalstresses and the finding of the points of the bifurcation werecarried out by the tangent (Et) established by experimentalresults and by the approximation of the so-called doublemodulus (Tlowast)

If the modulus of elasticity E is replaced by the doublemodulus Tlowast in the Eulerrsquos formula the critical stress willhave the following form

σcr π2Tlowast

λ2 (11)

In this study the authors substituted the value of thedouble modulus (Tlowast) from equation (8) into the formula ofthe critical stress (equation (11)) and it was assumed thatEulerrsquos critical stress is valid until the critical slenderness(λcr)

e cross section was divided into the tensile andcompressed zones e constant modulus of elasticity E willbe used for the tensile zone and the tangent modulus Et fromequation (9) established by Shanley and proven by Stel-makhrsquos tests in the compressed zone

e joint work of both moduli in the cross section(modulus of elasticity E in the tensile zone and the tangentmodulus Et in the compressed zone) was used in the doublemodulus (Tlowast) (equation (8)) to obtain the followingequation

σcr π2Tlowast

λ2

π2 middot 1( 1113857I EI1 + EtI2( 1113857

λ2π2E

λ2cr

I1

I+π2Et

λ2t

I2

I

(12)

From equation (12) the following equation is obtained

σcr σcrprimeI1I

+ σcrtI2I

(13)

where σcrprime is the critical stress when the critical slendernesswithin the elasticity reaches the limit value (λcr 102) andλt le λcr is the critical slenderness that has low value incomparison with λcr (Figure 4(b))

σcrt is the critical stress for the tangent modulus (in theelastoplastic region) and is equal to the following

σcrt π2Et

λ2t (14)

Input the value of Et from (9) into equation (14) In thiscase the formula of σcrt will be as follows

σcrt π2E

λ2t1minus

σcrt minus σelσy minus σel

1113888 1113889

2⎡⎣ ⎤⎦ (15)

e following quadratic equation is obtained by solvingσcrt from equation (15)

σ2crt minus 2σel minusσy minus σel1113872 1113873

2

π2Eλ2t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦σcrt + σy 2σel minus σy1113872 1113873 0 (16)

e value of the critical stress-σcrt was solved fromquadratic equation (16) and substituted in equation (13) andthus the adjusted formula for calculating the critical stresseswas obtained in the elastoplastic working region of thematerial

σcr π2E

λ21minus

I2I

1113874 1113875 + σel minusλ2t σy minus σel1113872 1113873

2

2π2E⎛⎝ ⎞⎠

I2I

+I2

I

σel minusλ2t σy minus σel1113872 1113873

2

2π2E⎛⎝ ⎞⎠minus σy 2σel minus σy1113872 1113873

11139741113972

(17)

33 Numerical Analysis According to equation (17) of thecritical stresses obtained by the authors the dependencebetween the slenderness and the critical stresses in the regionof medium slenderness is constructed as Curve 4 inFigure 4(b)

e following data were used for obtaining the numericalvalues

e material carbon steel (S235) proportional limitσpl 196 kNcm2 elastic limit σel 20 kNcm2 yielding limitσy 24 kNcm2 modulus of elasticity E 21000 kNcm2design resistance Ry 235 kNcm2 slenderness ratio 40leλt le 102 critical slenderness ratio λcr asymp 102

At the elastoplastic region and middle slenderness ratioλ isin [40 102] from Figure 4(b) it is seen that the curve ofcritical stresses constructed according to the reducedmodulus (T) of Engesser-Karman equation (4) is above allthe graphs

Curve 3 of the critical stresses (equation (10)) de-termined by the tangent modulus (Et) of Shanley is locatedbelow Curve 2 (Figure 4(b))

Curve 4 of the critical stresses (equation (17)) obtainedby approximating the double modulus proposed by authorsis located below the curve of Shanley which is very close tothe results obtained by the experimental data (Figure 4(b)asterisks-6) It confirms the accuracy of the new approach(Figure 4(b))

Curve 5 depicts values of critical stresses defined bybuilding codes for the rectangular cross section(Figure 4(b))

On the basis of theoretical and numerical analysis thefollowing results were obtained


(1) A new approach to the determination of criticalstresses in the elastoplastic region was developed as aresult of which the values of critical stresses with ahigh accuracy were determined

(2) e locations of possible points of the bifurcation-buckling in the elastoplastic region were establishedese bifurcation points are on Curve 4 (authorsapproach) because this curve is the closest to the realbifurcation points ie the experimental data

(3) e approach for the elimination of the presence ofresidual deformations in the cross section was de-veloped because Curve 4 in Figure 4(b) is belowCurve 2 and Curve 3 and closer to the experimentaldata

(4) Approximation and accuracy of theoretical and realcritical loads were arisen because Curve 4 is theclosest to the real critical loads ie the experimentaldata

(5) e results obtained from the proposed new ap-proaches indicate that the structure reliability wasenhanced by almost 10ndash12 in comparison withexisting approaches

(6) Economic efficiency was increased by almost 12ndash14in comparison with the building codes Since the newcritical curve is located above the curves determinedby building codes which is defined by using a safetyfactor gained empirically

(7) e study of the buckling behavior of prismatic(rectangular) compressed elements is a relevant steptowards understanding and assessing the reliabilityof more complex structures For the column ofother cross sections eg circular and open-sectionthe formula for calculating critical stress (equation(17)) remains the same only some geometricalcharacteristics need to be changed eg moments ofinertia

4 Conclusions

From the results of this study the following conclusions canbe made

(1) It is recommended to use the tangent modulus-Et(equation (9)) compared with the reduced modulus-T (equation (3)) in determining the critical stressesin the elastoplastic region

(2) It is recommended to use the new approach pro-posed by the authors to determine the stability of thecompressed rod in the elastoplastic region where thevalues of the critical stresses (equation (17)) aredetermined by approximation of the doublemodulusTlowast (equation (8)) and the tangent modulus Et(equation (9)) established by the results of experi-mental studies

(3) A new approach to the determination of criticalstresses proposed by the authors increases reliabilityby almost 10ndash12

(4) In rods (columns) of medium slenderness the pointsbelow the critical curve obtained as a result of newapproaches will not experience stability loss and thedevelopment of residual plastic deformations andthe bifurcation points will be located on this Curve 4(Figure 4(b))

(5) A new approach to the determination of criticalstresses proposed by the authors achieves economicefficiency by almost 12ndash14

(6) Reduction of the mass of the structure respectivelyreduces CO2 emissions

Data Availability

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

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is research was financially supported by the ErasmusMundus Action 2 Project ldquoElectra Enhancing Learning inENPI Countries through Clean Technologies and Researchrelated Activitiesrdquo (project ELEC1400294) and the SpanishMinistry of Economy and Competitiveness along withFEDER funding (project BIA2017-85098-R)

References

[1] D J Braun ldquoOn the optimal shape of compressed rotating rodwith shear and extensibilityrdquo International Journal of Non-Linear Mechanics vol 43 no 2 pp 131ndash139 2008

[2] B Rossi and K J R Rasmussen ldquoCarrying capacity ofstainless steel columns in the low slenderness rangerdquo Journalof Structural Engineering vol 139 no 6 pp 1088ndash1092 2013

[3] X Cheng Y Chen L Niu and D A Nethercot ldquoExperi-mental study on H-section steel beam-columns under cyclicbiaxial bending considering the effect of local bucklingrdquoEngineering Structures vol 174 pp 826ndash839 2018

[4] Y Goto M Muraki and M Obata ldquoUltimate state of thin-walled circular steel columns under bidirectional seismicaccelerationsrdquo Journal of Structural Engineering vol 135no 12 pp 1481ndash1490 2009

[5] M J Leamy ldquoWave based analysis of buckling in columns andframesrdquo in Proceeding of the 22nd Reliability Stress Analysisand Failure Prevention Conference 25th Conference on Me-chanical Vibration and Noise International Design EngineeringTechnical Conferences and Computers and Information in En-gineering Conference vol 8 ASME Portland OR USA 2013

[6] J Lu B Wu and Y Mei ldquoBuckling mechanism of steel coreand global stability design method for fixed-end buckling-restrained bracesrdquo Engineering Structures vol 174 pp 447ndash461 2018

[7] A G Razdolsky ldquoRevision of Engesserrsquos approach to theproblem of Euler stability for built-up columns with battenplatesrdquo Journal of Engineering Mechanics vol 140 no 3pp 566ndash574 2014


[8] D G Zapata-Medina L G Arboleda-Monsalve andJ D Aristizabal-Ochoa ldquoStatic stability formulas of aweakened Timoshenko column effects of shear de-formationsrdquo Journal of Engineering Mechanics vol 136no 12 pp 1528ndash1536 2010

[9] A Ziołkowski and S Imiełowski ldquoBuckling and post-bucklingbehaviour of prismatic aluminium columns submitted to aseries of compressive loadsrdquo Experimental Mechanics vol 51no 8 pp 1335ndash1345 2010

[10] P Li X Liu and C Zhang ldquoInteractive buckling of cable-stiffened steel columns with pin-connected crossarmsrdquoJournal of Constructional Steel Research vol 146 pp 97ndash1082018

[11] L Yang G Shi M Zhao and W Zhou ldquoResearch on in-teractive buckling behavior of welded steel box-section col-umnsrdquo in-Walled Structures vol 115 pp 34ndash47 2017

[12] F Papp ldquoBuckling assessment of steel members throughoverall imperfection methodrdquo Engineering Structuresvol 106 pp 124ndash136 2016

[13] P D Simatildeo ldquoInfluence of shear deformations on the bucklingof columns using the generalized beam theory and energyprinciplesrdquo European Journal of MechanicsmdashASolids vol 61pp 216ndash234 2017

[14] X-F Li and K Y Lee ldquoEffects of Engesserrsquos and Haringxrsquoshypotheses on buckling of Timoshenko and higher-ordershear-deformable columnsrdquo Journal of Engineering Me-chanics vol 144 no 1 article 04017150 2018

[15] J Becque ldquoInelastic plate bucklingrdquo Journal of EngineeringMechanics vol 136 no 9 pp 1123ndash1130 2010

[16] M Ahmed Q Q Liang V I Patel and M N S HadildquoNonlinear analysis of rectangular concrete-filled double steeltubular short columns incorporating local bucklingrdquo Engi-neering Structures vol 175 pp 13ndash26 2018

[17] Y-L Long and L Zeng ldquoA refined model for local buckling ofrectangular CFST columns with binding barsrdquo in-WalledStructures vol 132 pp 431ndash441 2018

[18] C D Moen A Schudlich and A von der Heyden ldquoEx-periments on cold-formed steel C-section joists withunstiffened web holesrdquo Journal of Structural Engineeringvol 139 no 5 pp 695ndash704 2013

[19] J Szalai ldquoComplete generalization of the Ayrton-Perry for-mula for beam-column buckling problemsrdquo EngineeringStructures vol 153 pp 205ndash223 2017

[20] C Zhang F Li and B Wang ldquoEstimation of the elasto-plasticproperties of metallic materials from micro-hardness mea-surementsrdquo Journal of Materials Science vol 48 no 12pp 4446ndash4451 2013

[21] H Ban and G Shi ldquoOverall buckling behaviour and design ofhigh-strength steel welded section columnsrdquo Journal ofConstructional Steel Research vol 143 pp 180ndash195 2018

[22] T-Y Ma Y-F Hu X Liu G-Q Li and K-F ChungldquoExperimental investigation into high strength Q690 steelwelded H-sections under combined compression and bend-ingrdquo Journal of Constructional Steel Research vol 138pp 449ndash462 2017

[23] A Kervalishvili and I Talvik ldquoModified procedure forbuckling of steel columns at elevated temperaturesrdquo Journal ofConstructional Steel Research vol 127 pp 108ndash119 2016

[24] T Tankova J P Martins L Simotildees da Silva L MarquesH D Craveiro and A Santiago ldquoExperimental lateral-torsional buckling behaviour of web tapered I-section steelbeamsrdquo Engineering Structures vol 168 pp 355ndash370 2018

[25] N Tullini A Tralli and D Baraldi ldquoBuckling of Timoshenkobeams in frictionless contact with an elastic half-planerdquo

Journal of Engineering Mechanics vol 139 no 7 pp 824ndash8312013

[26] B Xie J Hou Z Xu and M Dan ldquoComponent-based modelof fin plate connections exposed to fire-part I plate in bearingcomponentrdquo Journal of Constructional Steel Researchvol 149 pp 1ndash13 2018

[27] J D Aristizabal-Ochoa ldquoStability of columns with semi-rigidconnections including shear effects using Engesser Haringxand Euler approachesrdquo Engineering Structures vol 33 no 3pp 868ndash880 2011

[28] F M Mitenkov V G Bazhenov V K Lomunov andS L Osetrov ldquoEffects of elasticity plasticity and geometricalnonlinearity in problems of static and dynamic bending ofplatesrdquo Doklady Physics vol 56 no 12 pp 622ndash625 2011

[29] J Bielski and B Bochenek ldquoOn a compressed elasticndashplasticcolumn optimized for post-buckling behaviourrdquo EngineeringOptimization vol 40 no 12 pp 1101ndash1114 2008

[30] O Fergani I Lazoglu A Mkaddem M El Mansori andS Y Liang ldquoAnalytical modeling of residual stress and theinduced deflection of a milled thin platerdquo InternationalJournal of Advanced Manufacturing Technology vol 75no 1ndash4 pp 455ndash463 2014


International Journal of

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(rectangular) compressed elements is a relevant step towardsunderstanding and assessing the reliability of more complexstructures [9]

In steel columns the issues of buckling can be dividedinto three types overall buckling local buckling and in-teractive buckling [10 11] In prismatic columns withrectangular cross section and medium slenderness ratiooverall buckling is normally observed rst [12]

e buckling appearance (loss of stability) process wasassessed with critical forces (from which the values of re-duction factor are dened) e reduction factor approachdened from critical stresses appears in two alternativemethods in the standards (a) traditional method usingsemiempirical design equations and (b) computer-aidedmethod based on linear buckling analysis [12 13]

It is common knowledge that the physical action causingthe buckling of axially compressed columns is the depletionof its lateral stiness to a value of zero ere are a number ofproblems in actual engineering situations which complicatethe prediction of buckling loads eg initial (geometric)imperfection material nonlinearity and loading eccentric-ity e linear elastic Euler model of a prismatic simplysupported rod is still the fundamental tool for studyingbuckling stability [9 10]

In a similar way several empirical models for instanceEngesserrsquos [7 13 14] are proposed in the literature to

experimentally characterize the elastic-plastic and plasticbuckling Later Shanley [9 15] oered a dierent approachfor dening the buckling process with its semiempiricaldesign equation

In the literature [16 17] researchers presented a nu-merical method for computing the load-deection re-lationships of loaded square columns with various geometricparameters loading eccentricity and column slendernessratios

As mentioned above steel frames are an ecient ar-rangement for structural systems in seismic regions For thisreason they have been the subject of strong research interestin recent years In practice earthquakes generate multidi-rectional ground motions with the occurrence of yieldingand buckling eects [4 6 8] Previous research studies[3 18] indicate how the development of local bucklingrestricts the ability of the members to deform plastically alsohow cross-section geometry inuences strength ductilityand the energy absorption process

In literature [12] the researchers examined the simplysupported steel member with uniform cross section sub-jected to uniform forces e results showed buckling re-sistance for such members and were standardized by thereduction factor based on the AyrtonndashPerry formula [19]and calibrated by strong theoretical and empirical back-ground e local and overall stability of the columnsthrough simulating their geometric and material nonlinearbehavior was investigated [11] A methodology [20] wasdevised to estimate the elastoplastic properties of metallicmaterials from the hardness measurements using the twomost frequently used sharp indenters ie the Vickers in-denter and the Knoop indenter In the study [21] the re-searchers studied overall buckling behavior and design ofhigh-strength steel columns e study results indicated thatwith an increase of the yield strength of high-strength steelcompressive residual stresses ratio to the yield strengthbecomes evidently lower and possesses much less severeeects on the overall buckling behavior Dierent H-shapedsections of the welded steel columns were experimentallyand numerically investigated with their overall bucklingbehavior and their practical application was promoted [22]

e main disadvantage of the previously proposed de-sign methods to avoid risk due to loss of stability in theelastoplastic region was that the accuracy and limits of thedenition were not always known

e main objectives of the study were to (a) develop theapproaches that would allow specifying the values of criticalstresses of compressed elements beyond their proportionallimit (b) approximate and specify maximally the theoreticaland real values of critical loads (c) nd the possible points ofbifurcation in the elastoplastic region and (d) eliminate theexistence of residual deformations in the cross-section andeventually reduce the mass of the structure and increase itsreliability

e problem of the stability of the compressed elementswas studied according to the stability theory Methods basedon the equations of equilibrium of mechanics for slightlyperturbed systems around the initial position were used tostudy the stability

N

l

A

Figure 1 A prismatic (rectangular) pin-ended steel column

















d2y

dx2+N

EIy 0 (1)


Ncr π2EImin

l20

σcr π2Eλ2

(2)


N lt Ncr

(a)

N = Ncr

(b)

N gt Ncr

(c)




IminA





Eσpl









N

l

x

y

(a)

NΔl

f

y

x

l

x

y

(b)

Nf

y

x

N

M

x

y

(c)




095 20 206 206 102 10210 21 142 172 90 81811 22 099 139 79 66612 228 067 105 676 5413 234 046 085 59 4414 238 026 054 475 32715 239 013 033 37 23116 24 006 019 28 15718ndash40 24 0 0 0 045 241 002 007 17 915 242 004 013 23 1276 247 005 015 245 1428 2575 005 015 24 13810 2685 005 015 236 13612 28 005 015 23 135


T 4EEtE

radic+Etradic

( )2 (3)


σcr π2Tλ2 (4)



M intA1

σ1ydA + intA2

σ2ydAhArrM intA1

E

ρy2dA

+ intA2

Et

ρy2dAhArrM

E

ρintA1

y2dA +Et

ρintA2

y2dA

(5)

Hence

M 1ρEI1 + EtI2( ) (6)



M TlowastI

ρ (7)







Elastic zone

E

Elastoplastic zone

Et σy

σel

times108

0

5

10

15

20

25

30St

ress

σ (k

Nc

m2 )

2 4 6 8 10 120Strain

(a)

3

64

5

2

1


slender proportions


σy

σel

σcr = π2Eλ2

λcr


0

5

10

15

20

25

30

Criti

cal s

tress

σcr

(kN

cm

2 )

(b)






( )2

(9)



σcr π2Et

λ2 (10)


3 Problem Solution


I2 I1 b

I

h1

h

y

h2

Compressedzone

Tensilezone

Neutralaxis


y

EtE

ndashyy

1

A1 A2

h2h1

b

h

y

1ndash1

M

N

e

1

x

Compressedzone

Tensilezone

E

Nf

y

x

N

(a) (b)

M

x

y

Et

f


σ0 = NA








σcr π2Tlowast

λ2 (11)




σcr π2Tlowast

λ2

π2 middot 1( 1113857I EI1 + EtI2( 1113857

λ2π2E

λ2cr

I1

I+π2Et

λ2t

I2

I

(12)


σcr σcrprimeI1I

+ σcrtI2I

(13)



σcrt π2Et

λ2t (14)


σcrt π2E

λ2t1minus


1113888 1113889

2⎡⎣ ⎤⎦ (15)



2

π2Eλ2t



σcr π2E

λ21minus

I2I


2

2π2E⎛⎝ ⎞⎠

I2I

+I2

I


2


11139741113972

(17)

















4 Conclusions








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

















d2y

dx2+N

EIy 0 (1)


Ncr π2EImin

l20

σcr π2Eλ2

(2)


N lt Ncr

(a)

N = Ncr

(b)

N gt Ncr

(c)




IminA





Eσpl









N

l

x

y

(a)

NΔl

f

y

x

l

x

y

(b)

Nf

y

x

N

M

x

y

(c)




095 20 206 206 102 10210 21 142 172 90 81811 22 099 139 79 66612 228 067 105 676 5413 234 046 085 59 4414 238 026 054 475 32715 239 013 033 37 23116 24 006 019 28 15718ndash40 24 0 0 0 045 241 002 007 17 915 242 004 013 23 1276 247 005 015 245 1428 2575 005 015 24 13810 2685 005 015 236 13612 28 005 015 23 135


T 4EEtE

radic+Etradic

( )2 (3)


σcr π2Tλ2 (4)



M intA1

σ1ydA + intA2

σ2ydAhArrM intA1

E

ρy2dA

+ intA2

Et

ρy2dAhArrM

E

ρintA1

y2dA +Et

ρintA2

y2dA

(5)

Hence

M 1ρEI1 + EtI2( ) (6)



M TlowastI

ρ (7)







Elastic zone

E

Elastoplastic zone

Et σy

σel

times108

0

5

10

15

20

25

30St

ress

σ (k

Nc

m2 )

2 4 6 8 10 120Strain

(a)

3

64

5

2

1


slender proportions


σy

σel

σcr = π2Eλ2

λcr


0

5

10

15

20

25

30

Criti

cal s

tress

σcr

(kN

cm

2 )

(b)






( )2

(9)



σcr π2Et

λ2 (10)


3 Problem Solution


I2 I1 b

I

h1

h

y

h2

Compressedzone

Tensilezone

Neutralaxis


y

EtE

ndashyy

1

A1 A2

h2h1

b

h

y

1ndash1

M

N

e

1

x

Compressedzone

Tensilezone

E

Nf

y

x

N

(a) (b)

M

x

y

Et

f


σ0 = NA








σcr π2Tlowast

λ2 (11)




σcr π2Tlowast

λ2

π2 middot 1( 1113857I EI1 + EtI2( 1113857

λ2π2E

λ2cr

I1

I+π2Et

λ2t

I2

I

(12)


σcr σcrprimeI1I

+ σcrtI2I

(13)



σcrt π2Et

λ2t (14)


σcrt π2E

λ2t1minus


1113888 1113889

2⎡⎣ ⎤⎦ (15)



2

π2Eλ2t



σcr π2E

λ21minus

I2I


2

2π2E⎛⎝ ⎞⎠

I2I

+I2

I


2


11139741113972

(17)

















4 Conclusions








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



IminA





Eσpl









N

l

x

y

(a)

NΔl

f

y

x

l

x

y

(b)

Nf

y

x

N

M

x

y

(c)




095 20 206 206 102 10210 21 142 172 90 81811 22 099 139 79 66612 228 067 105 676 5413 234 046 085 59 4414 238 026 054 475 32715 239 013 033 37 23116 24 006 019 28 15718ndash40 24 0 0 0 045 241 002 007 17 915 242 004 013 23 1276 247 005 015 245 1428 2575 005 015 24 13810 2685 005 015 236 13612 28 005 015 23 135


T 4EEtE

radic+Etradic

( )2 (3)


σcr π2Tλ2 (4)



M intA1

σ1ydA + intA2

σ2ydAhArrM intA1

E

ρy2dA

+ intA2

Et

ρy2dAhArrM

E

ρintA1

y2dA +Et

ρintA2

y2dA

(5)

Hence

M 1ρEI1 + EtI2( ) (6)



M TlowastI

ρ (7)







Elastic zone

E

Elastoplastic zone

Et σy

σel

times108

0

5

10

15

20

25

30St

ress

σ (k

Nc

m2 )

2 4 6 8 10 120Strain

(a)

3

64

5

2

1


slender proportions


σy

σel

σcr = π2Eλ2

λcr


0

5

10

15

20

25

30

Criti

cal s

tress

σcr

(kN

cm

2 )

(b)






( )2

(9)



σcr π2Et

λ2 (10)


3 Problem Solution


I2 I1 b

I

h1

h

y

h2

Compressedzone

Tensilezone

Neutralaxis


y

EtE

ndashyy

1

A1 A2

h2h1

b

h

y

1ndash1

M

N

e

1

x

Compressedzone

Tensilezone

E

Nf

y

x

N

(a) (b)

M

x

y

Et

f


σ0 = NA








σcr π2Tlowast

λ2 (11)




σcr π2Tlowast

λ2

π2 middot 1( 1113857I EI1 + EtI2( 1113857

λ2π2E

λ2cr

I1

I+π2Et

λ2t

I2

I

(12)


σcr σcrprimeI1I

+ σcrtI2I

(13)



σcrt π2Et

λ2t (14)


σcrt π2E

λ2t1minus


1113888 1113889

2⎡⎣ ⎤⎦ (15)



2

π2Eλ2t



σcr π2E

λ21minus

I2I


2

2π2E⎛⎝ ⎞⎠

I2I

+I2

I


2


11139741113972

(17)

















4 Conclusions








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


T 4EEtE

radic+Etradic

( )2 (3)


σcr π2Tλ2 (4)



M intA1

σ1ydA + intA2

σ2ydAhArrM intA1

E

ρy2dA

+ intA2

Et

ρy2dAhArrM

E

ρintA1

y2dA +Et

ρintA2

y2dA

(5)

Hence

M 1ρEI1 + EtI2( ) (6)



M TlowastI

ρ (7)







Elastic zone

E

Elastoplastic zone

Et σy

σel

times108

0

5

10

15

20

25

30St

ress

σ (k

Nc

m2 )

2 4 6 8 10 120Strain

(a)

3

64

5

2

1


slender proportions


σy

σel

σcr = π2Eλ2

λcr


0

5

10

15

20

25

30

Criti

cal s

tress

σcr

(kN

cm

2 )

(b)






( )2

(9)



σcr π2Et

λ2 (10)


3 Problem Solution


I2 I1 b

I

h1

h

y

h2

Compressedzone

Tensilezone

Neutralaxis


y

EtE

ndashyy

1

A1 A2

h2h1

b

h

y

1ndash1

M

N

e

1

x

Compressedzone

Tensilezone

E

Nf

y

x

N

(a) (b)

M

x

y

Et

f


σ0 = NA








σcr π2Tlowast

λ2 (11)




σcr π2Tlowast

λ2

π2 middot 1( 1113857I EI1 + EtI2( 1113857

λ2π2E

λ2cr

I1

I+π2Et

λ2t

I2

I

(12)


σcr σcrprimeI1I

+ σcrtI2I

(13)



σcrt π2Et

λ2t (14)


σcrt π2E

λ2t1minus


1113888 1113889

2⎡⎣ ⎤⎦ (15)



2

π2Eλ2t



σcr π2E

λ21minus

I2I


2

2π2E⎛⎝ ⎞⎠

I2I

+I2

I


2


11139741113972

(17)

















4 Conclusions








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





( )2

(9)



σcr π2Et

λ2 (10)


3 Problem Solution


I2 I1 b

I

h1

h

y

h2

Compressedzone

Tensilezone

Neutralaxis


y

EtE

ndashyy

1

A1 A2

h2h1

b

h

y

1ndash1

M

N

e

1

x

Compressedzone

Tensilezone

E

Nf

y

x

N

(a) (b)

M

x

y

Et

f


σ0 = NA








σcr π2Tlowast

λ2 (11)




σcr π2Tlowast

λ2

π2 middot 1( 1113857I EI1 + EtI2( 1113857

λ2π2E

λ2cr

I1

I+π2Et

λ2t

I2

I

(12)


σcr σcrprimeI1I

+ σcrtI2I

(13)



σcrt π2Et

λ2t (14)


σcrt π2E

λ2t1minus


1113888 1113889

2⎡⎣ ⎤⎦ (15)



2

π2Eλ2t



σcr π2E

λ21minus

I2I


2

2π2E⎛⎝ ⎞⎠

I2I

+I2

I


2


11139741113972

(17)

















4 Conclusions








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







σcr π2Tlowast

λ2 (11)




σcr π2Tlowast

λ2

π2 middot 1( 1113857I EI1 + EtI2( 1113857

λ2π2E

λ2cr

I1

I+π2Et

λ2t

I2

I

(12)


σcr σcrprimeI1I

+ σcrtI2I

(13)



σcrt π2Et

λ2t (14)


σcrt π2E

λ2t1minus


1113888 1113889

2⎡⎣ ⎤⎦ (15)



2

π2Eλ2t



σcr π2E

λ21minus

I2I


2

2π2E⎛⎝ ⎞⎠

I2I

+I2

I


2


11139741113972

(17)

















4 Conclusions








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









4 Conclusions








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


BucklingAnalysisandStabilityofCompressedLow-CarbonSteel ...

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