Strain Energy and Entropy Based Scaling of Buckling Modes
Abstract
A new utilization of entropy in the context of buckling is presented. The novel concept of connecting the strain energy and entropy for a pin-ended strut is derived. The entropy of the buckling mode is extracted through a surrogate model by decomposing the strain energy into entropy and virtual temperature. This concept rationalizes the ranking of buckling modes based on their strain energy under the assumption of given entropy. By assigning identical entropy to all buckling modes, they can be ranked according to their deformation energy. Conversely, with identical strain energy assigned to all the modes, ranking according to entropy is possible. Decreasing entropy was found to represent the scaling factors of the buckling modes that coincide with the measurement of the initial out-of-straightness imperfections in IPE160 beams. Applied to steel plane frames, scaled buckling modes can be used to model initial imperfections. It is demonstrated that the entropy (scale factor) for a given energy roughly decreases with the inverse square of the mode index. For practical engineering, this study presents the possibility of using scaled buckling modes of steel plane frames to model initial geometric imperfections. Entropy proves to be a valuable complement to strain energy in structural mechanics.
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Citation: Kala, Z. Strain Energy and Entropy Based Scaling of Buckling Modes. Entropy 2023,25, 1630. https://doi.org/10.3390/e25121630 Academic Editor: Tatiana Morosuk Received: 12 November 2023 Revised: 30 November 2023 Accepted: 4 December 2023 Published: 6 December 2023 Copyright: © 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). entropy Article Strain Energy and Entropy Based Scaling of Buckling Modes Zdenˇek Kala Institute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic; [email protected].cz Abstract: A new utilization of entropy in the context of buckling is presented. The novel concept of connecting the strain energy and entropy for a pin-ended strut is derived. The entropy of the buckling mode is extracted through a surrogate model by decomposing the strain energy into entropy and virtual temperature. This concept rationalizes the ranking of buckling modes based on their strain energy under the assumption of given entropy. By assigning identical entropy to all buckling modes, they can be ranked according to their deformation energy. Conversely, with identical strain energy assigned to all the modes, ranking according to entropy is possible. Decreasing entropy was found to represent the scaling factors of the buckling modes that coincide with the measurement of the initial out-of-straightness imperfections in IPE160 beams. Applied to steel plane frames, scaled buckling modes can be used to model initial imperfections. It is demonstrated that the entropy (scale factor) for a given energy roughly decreases with the inverse square of the mode index. For practical engineering, this study presents the possibility of using scaled buckling modes of steel plane frames to model initial geometric imperfections. Entropy proves to be a valuable complement to strain energy in structural mechanics. Keywords: strain energy; entropy; buckling; structural mechanics; structural engineering; steel structures; initial imperfections 1. Introduction The investigation of buckling modes is of great importance in the field of structural engineering and structural mechanics. Understanding the stability behaviour of structures under various loads and conditions is essential to ensuring their safety and reliability [ 1 , 2 ]. Throughout history, the study of structural stability has been intertwined with the evolution of engineering and mathematics [ 3 ]. The concept of buckling, where slender columns or beams, which are subjected to axial loads, deform and potentially fail under compression, has captured the attention of engineers and mathematicians for centuries [ 4 ]. The history of buckling analysis can be traced back to the 18th century in the works of Euler, who has made significant contributions to the understanding of the stability of columns [ 5 ]. As theory advances, so does our ability to investigate the structural behaviour of frame structures [ 6 , 7 ]. The emergence of computer-based analysis and numerical methods have enabled engineers to explore the intricacies of buckling modes in various configurations. Initially, perfectly straight bars were modelled, although real geometries were always identified as imperfect. A significant advancement has been the introduction of finite element methods and geometrically and materially non-linear solutions [ 8 ] capable of analysing the load-bearing capacity of structures with initial imperfections; see, e.g., [ 9 – 11 ]. Generally, imperfections can be categorized into three main groups: geometrical imperfections, material imperfections, and structural imperfections [12]. Currently, the design of structural steel employs a systematic design approach known as the Direct Design Method [ 13 ], which explicitly accounts for material and geometric nonlinearities, residual stresses, and the presence of initial imperfections. The shape of these initial imperfections should be defined to take into account their influence on the designs’ load-carrying capacity while respecting the real-world measures from experiments; Entropy 2023,25, 1630. https://doi.org/10.3390/e25121630 https://www.mdpi.com/journal/entropy
Entropy 2023,25, 1630 2 of 17 see, e.g., [ 14 – 16 ]. Typically, the first buckling mode shape is used to represent these initial imperfections, assuming it is the most critical [ 17 , 18 ]. However, in cases where the critical buckling loads of two distinct buckling modes coincide, the sensitivity to imperfections increases significantly [ 19 , 20 ]. In these cases, the buckling mode shapes may be more important than the magnitude of the critical force when modelling initial imperfections. As most steel structures are unique, data on the initial imperfections from experiments are limited. One approach to modelling initial imperfections is using the superposition of the first several buckling modes and the probabilistic characteristics of their amplitudes [ 13 , 21 ]. Buckling modes have shapes of trigonometric functions with indeterminate amplitudes, which are a subject of research. This article is based on the heuristic argument that if two distinct buckling modes coincide, the system tends to follow the buckling mode with lower energy and higher entropy. Entropy can provide new information about the stability behaviour of slender structures, both in the context of Euler buckling and geometrically nonlinear solutions with initial imperfections. Since the entropy of the buckling mode has not yet been studied, the concept of a surrogate model is proposed based on thermodynamics, where entropy naturally occurs. The entropy measure used in this study is defined by decomposing the strain energy of the buckling mode into entropy and virtual temperature. Thus, a new concept of entropy and virtual temperature, based on the equivalence of strain and heat energy in the surrogate model, is introduced. The surrogate model is based on an unconventional connection between classical theories: entropy [ 22 ], heat energy [ 22 ], strain energy [ 23 ], and buckling [ 1 ]. Entropy is presented as a new indicator of stability and resistance to buckling. This study follows a method for modelling initial geometric imperfections using a linear combination of buckling modes scaled by entropy. The research progresses from a simple strut to more complex structures, such as steel plane frames. 2. Surrogate Entropy Model of Buckling Structural mechanics uses the energies in the system, following the fundamental principles of Lagrangian mechanics, but does not consider entropy. The question is how to calculate entropy for a deformed structure. One of the approaches may involve a surrogate model based on thermodynamics and the isothermal process in ideal gases; see Figure 1. Entropy2023,25,xFORPEERREVIEW2of17 Currently,thedesignofstructuralsteelemploysasystematicdesignapproach knownastheDirectDesignMethod[13],whichexplicitlyaccountsformaterialandgeometricnonlinearities,residualstresses,andthepresenceofinitialimperfections.The shapeoftheseinitialimperfectionsshouldbedefinedtotakeintoaccounttheirinfluence onthedesigns’load-carryingcapacitywhilerespectingthereal-worldmeasuresfromexperiments;see,e.g.,[14–16].Typically,thefirstbucklingmodeshapeisusedtorepresent theseinitialimperfections,assumingitisthemostcritical[17,18].However,incaseswhere thecriticalbucklingloadsoftwodistinctbucklingmodescoincide,thesensitivitytoimperfectionsincreasessignificantly[19,20].Inthesecases,thebucklingmodeshapesmaybe moreimportantthanthemagnitudeofthecriticalforcewhenmodellinginitialimperfections. Asmoststeelstructuresareunique,dataontheinitialimperfectionsfromexperimentsarelimited.Oneapproachtomodellinginitialimperfectionsisusingthesuperpositionofthefirstseveralbucklingmodesandtheprobabilisticcharacteristicsoftheiramplitudes[13,21].Bucklingmodeshaveshapesoftrigonometricfunctionswithindeterminateamplitudes,whichareasubjectofresearch. Thisarticleisbasedontheheuristicargumentthatiftwodistinctbucklingmodes coincide,thesystemtendstofollowthebucklingmodewithlowerenergyandhigher entropy.Entropycanprovidenewinformationaboutthestabilitybehaviourofslender structures,bothinthecontextofEulerbucklingandgeometricallynonlinearsolutions withinitialimperfections.Sincetheentropyofthebucklingmodehasnotyetbeenstudied,theconceptofasurrogatemodelisproposedbasedonthermodynamics,whereentropynaturallyoccurs.Theentropymeasureusedinthisstudyisdefinedbydecomposing thestrainenergyofthebucklingmodeintoentropyandvirtualtemperature. Thus,anewconceptofentropyandvirtualtemperature,basedontheequivalenceof strainandheatenergyinthesurrogatemodel,isintroduced.Thesurrogatemodelisbased onanunconventionalconnectionbetweenclassicaltheories:entropy[22],heatenergy [22],strainenergy[23],andbuckling[1].Entropyispresentedasanewindicatorofstabilityandresistancetobuckling.Thisstudyfollowsamethodformodellinginitialgeometricimperfectionsusingalinearcombinationofbucklingmodesscaledbyentropy.The researchprogressesfromasimplestruttomorecomplexstructures,suchassteelplane frames. 2.SurrogateEntropyModelofBuckling Structuralmechanicsusestheenergiesinthesystem,followingthefundamental principlesofLagrangianmechanics,butdoesnotconsiderentropy.Thequestionishowto calculateentropyforadeformedstructure.Oneoftheapproachesmayinvolveasurrogate modelbasedonthermodynamicsandtheisothermalprocessinidealgases;seeFigure1. (a)(b) Figure 1. Surrogate model with ideal gas. (a) State before buckling. (b) State after buckling; a pair of pistons has a resultant non-zero force. Entropy can be viewed as a characteristic closely related to energy [ 24 , 25 ]. In thermodynamics, heat energy is generated as a result of changes in the entropy of a system [ 22 ]. In
Entropy 2023,25, 1630 3 of 17 modern theories, this entropy can be described using a relationship based on the energy balance [26]. F·∆x=T·∆S. (1) Equation (1) relates mechanical work to thermodynamic work. The left-hand side of the equation is the mechanical work performed by force Fon path ∆ xas the body moves. The right-hand side of Equation (1) is the heat energy produced by the system at temperature Tdue to changes in the entropy of the system ∆S; see, e.g., [22]. Figure 1illustrates the relationship between the strain energy of the lateral deformation of the strut and the heat energy of a surrogate model based on an ideal gas. The calculation of strain energy is grounded in linear elasticity theory, following Hooke’s law. The static load action of the critical force is taken into account. Heat energy is modelled using the work of pistons, where all pistons are identical, with an ideal gas, with constant mass and constant temperature, surrogating the original strut model. The pistons undergo deformation caused by buckling. The change in piston energy is equivalent to the change in energy due to bending in a buckled strut. Both energies depend on an indeterminate constant derived from Euler’s solution for buckling. With the exception of adopting buckling energy, the surrogate model, based on the isothermal process of the gas, does not exchange energy with its surroundings. The entropic approach can be employed in novel applications, illustrated by modelling the initial imperfections through the utilization of scaled buckling modes. 3. Buckling of a Pin-Ended Strut This section illustrates a pin-ended strut as a specific case for explaining the entropy problem. The strain energy is derived and decomposed into entropy and virtual temperature using a surrogate model. The section concludes with case studies of scaled buckling modes, showcasing the application of maximum entropy and minimum energy principles in the establishment of initial imperfections. Consider a slender strut with pin-ended supports subjected to an axial compressive load P. The strut remains ideally straight until the load reaches its critical value P cr and buckling occurs [ 1 , 2 ]. The critical load places the strut in a state of unstable equilibrium, where, in addition to equilibrium on the straight strut, there also exists equilibrium on the deflected strut; see Figure 2. Entropy2023,25,xFORPEERREVIEW3of17 Figure1.Surrogatemodelwithidealgas.(a)Statebeforebuckling.(b)Stateafterbuckling;apairof pistonshasaresultantnon-zeroforce. Entropycanbeviewedasacharacteristiccloselyrelatedtoenergy[24,25].Inthermodynamics,heatenergyisgeneratedasaresultofchangesintheentropyofasystem[22]. Inmoderntheories,thisentropycanbedescribedusingarelationshipbasedontheenergy balance[26]. S T xF .(1) Equation(1)relatesmechanicalworktothermodynamicwork.Theleft-handsideof theequationisthemechanicalworkperformedbyforceFonpathΔxasthebodymoves. Theright-handsideofEquation(1)istheheatenergyproducedbythesystemattemperatureTduetochangesintheentropyofthesystemΔS;see,e.g.,[22]. Figure1illustratestherelationshipbetweenthestrainenergyofthelateraldeformationofthestrutandtheheatenergyofasurrogatemodelbasedonanidealgas.The calculationofstrainenergyisgroundedinlinearelasticitytheory,followingHooke’slaw. Thestaticloadactionofthecriticalforceistakenintoaccount.Heatenergyismodelled usingtheworkofpistons,whereallpistonsareidentical,withanidealgas,withconstant massandconstanttemperature,surrogatingtheoriginalstrutmodel.Thepistonsundergo deformationcausedbybuckling.Thechangeinpistonenergyisequivalenttothechange inenergyduetobendinginabuckledstrut.Bothenergiesdependonanindeterminate constantderivedfromEuler’ssolutionforbuckling.Withtheexceptionofadoptingbucklingenergy,thesurrogatemodel,basedontheisothermalprocessofthegas,doesnot exchangeenergywithitssurroundings. Theentropicapproachcanbeemployedinnovelapplications,illustratedbymodellingtheinitialimperfectionsthroughtheutilizationofscaledbucklingmodes. 3.BucklingofaPin‐EndedStrut Thissectionillustratesapin-endedstrutasaspecificcaseforexplainingtheentropy problem.Thestrainenergyisderivedanddecomposedintoentropyandvirtualtemperatureusingasurrogatemodel.Thesectionconcludeswithcasestudiesofscaledbuckling modes,showcasingtheapplicationofmaximumentropyandminimumenergyprinciples intheestablishmentofinitialimperfections. Consideraslenderstrutwithpin-endedsupportssubjectedtoanaxialcompressive loadP.ThestrutremainsideallystraightuntiltheloadreachesitscriticalvaluePcrand bucklingoccurs[1,2].Thecriticalloadplacesthestrutinastateofunstableequilibrium, where,inadditiontoequilibriumonthestraightstrut,therealsoexistsequilibriumonthe deflectedstrut;seeFigure2. Figure2.Theflexuralbucklingofthepin-endedstrut. Forastrutundercriticalload,bendingdeformationcanbedescribedaccordingto theEuler–Bernoullibeamtheory,usingthefollowingdifferentialequation: 0 2 2 y IE P dx xdy ,(2) wherey(x)isthelateraldeflection,EisYoung’smodulus,andIisthesecondmomentof thearea.Thisequationisalinear,nonhomogeneous,differentialequationofthesecond orderwithconstantcoefficients.Uponsubstitutingα2=P/(E·I),theparticularsolutionof thisdifferentialequationcanbeexpressedintheform Figure 2. The flexural buckling of the pin-ended strut. For a strut under critical load, bending deformation can be described according to the Euler–Bernoulli beam theory, using the following differential equation: dy2(x) dx2+P E·Iy=0, (2) where y(x) is the lateral deflection, Eis Young’s modulus, and Iis the second moment of the area. This equation is a linear, nonhomogeneous, differential equation of the second order with constant coefficients. Upon substituting α2=P/(E·I), the particular solution of this differential equation can be expressed in the form y(x)=c1·sin(α·x)+c2·cos(α·x), for x∈[0, L]. (3) From the boundary conditions, it can be calculated that c 2 = 0 for y(0) = 0 and c 1· sin( α· L) = 0 for y(L) = 0, where Lis the strut length and c 1 is the indeterminate amplitude. By using c 1 > 0, it can be obtained that sin( α· L) = 0, and thus α =i ·π /L, where irepresents a natural
Entropy 2023,25, 1630 4 of 17 number of the buckling mode. The corresponding buckling modes can be derived from Equation (3) as eigenmodes, resulting in deformation patterns that are characterized by sine functions. y(x)=c1·siniπ·x L, for x∈[0, L],i=1, 2, . . . , (4) where iis the number of half-sine curvatures that occur lengthwise. Buckling modes can be described as the shapes the strut assumes during buckling, with the sine function playing a crucial role in their formulation; see Figure 3. Entropy2023,25,xFORPEERREVIEW4of17 xcxcx y cossin 21 ,for],0[ Lx .(3) Fromtheboundaryconditions,itcanbecalculatedthatc2=0fory(0)=0and c1·sin(α·L)=0fory(L)=0,whereListhestrutlengthandc1istheindeterminateamplitude. Byusingc1>0,itcanbeobtainedthatsin(α·L)=0,andthusα=i·π/L,whereirepresentsa naturalnumberofthebucklingmode.Thecorrespondingbucklingmodescanbederived fromEquation(3)aseigenmodes,resultingindeformationpatternsthatarecharacterized bysinefunctions. L x icxy sin 1,for],0[ Lx ,i=1,2,..,(4) whereiisthenumberofhalf-sinecurvaturesthatoccurlengthwise.Bucklingmodescan bedescribedastheshapesthestrutassumesduringbuckling,withthesinefunctionplayingacrucialroleintheirformulation;seeFigure3. Figure3.Thebucklingmodesofthepin-endedstrut. Usingthepreviouslydefinedequationα2=P/(E·I),thecriticalloadsare 2 22 ,L IE iP icr ,fori=1,2,..,(5) Equation(5)providesthestandardcriticalloadatwhichbucklingoccursinapinendedstrut.Thisprovidesinsightintothebehaviourofslenderstrutsunderaxialcompression,andvaluableunderstandingofthedesignandanalysisofstructuralsystems. InEulerbuckling,therankofthebucklingmodesisdeterminedbytherankingof thesmallestcriticalforcetothelargest.Thisisthemostcommonapproachforranking bucklingmodes,withthefirstoneconsideredthemostdangerous[27]. 3.1.StrainEnergy Theestimationofentropyisbasedonthedecompositionofthestrainenergyofa bucklingmodeintoentropyandvirtualtemperature.Inthefirststep,ananalyticalsolutionofthestrainenergyisderived. Thediscretizationofy(x)isperformedatthecentroidsoffiniteelements.Byintroducing(j−0.5)/Ninsteadofx/L,Equation(4)canberewritteninthediscreteform N j icfij 5.0 sin 1 ,forj=1,2,…N,(6) wherej=1,2,...NandNisthenumberofbeamelements.Theelementsareconsideredto havethesamelength. Initsdifferentialform,thetotalinternalpotentialenergy(strainenergy)ofthei-th bucklingmodecanbeobtainedas 3 42 1 4 0 2 2 1 0 2 2 2 4 sin 2 1 2 1 L c IEidx L x i L icIEdx dx xdy IE LL i .(7) ThevalueofΔΠirepresentsthedifferencebetweenthezero-strainenergyoftheunloadedstrutandthestrainenergyofthebuckledstrut. Figure 3. The buckling modes of the pin-ended strut. Using the previously defined equation α2=P/(E·I), the critical loads are Pcr,i=i2·π2E·I L2, for i=1, 2, . . . , (5) Equation (5) provides the standard critical load at which buckling occurs in a pinended strut. This provides insight into the behaviour of slender struts under axial compression, and valuable understanding of the design and analysis of structural systems. In Euler buckling, the rank of the buckling modes is determined by the ranking of the smallest critical force to the largest. This is the most common approach for ranking buckling modes, with the first one considered the most dangerous [27]. 3.1. Strain Energy The estimation of entropy is based on the decomposition of the strain energy of a buckling mode into entropy and virtual temperature. In the first step, an analytical solution of the strain energy is derived. The discretization of y(x) is performed at the centroids of finite elements. By introducing (j−0.5)/Ninstead of x/L, Equation (4) can be rewritten in the discrete form fij =c1·siniπ·(j−0.5) N, for j=1, 2, . . . N, (6) where j= 1, 2, . . . Nand Nis the number of beam elements. The elements are considered to have the same length. In its differential form, the total internal potential energy (strain energy) of the i-th buckling mode can be obtained as ∆Πi=1 2 L Z0 E·Idy2(x) dx22 dx =1 2 L Z0 E·Ic1·iπ L2 ·siniπ·x L2 dx =i4·E·I·c2 1·π4 4·L3. (7) The value of ∆Πi represents the difference between the zero-strain energy of the unloaded strut and the strain energy of the buckled strut. It can be noted that the square of the second derivative can be replaced by the product of the function value and the fourth derivative, E · I · (y ′′ ) 2 =y · E · I · y ′′ ′′ . In this expression, the term involving the fourth derivative, E · I · y ′′ ′′ , represents a fictitious transverse load action. The total potential energy is the sum of ∆Πi and ∆Πe . According to the principle of conservation of mechanical energy, the strain energy from the internal forces ∆Πi (from bending moment) is equal to the potential energy of the external forces ∆Πe (from load action Pcr,i).
Entropy 2023,25, 1630 5 of 17 ∆Πe=1 2·Pcr,i· L Z0dy(x) dx 2 dx =1 2·i2·π2E·I L2· L Z0c1·iπ L·cosiπ·x L2dx =i4·E·I·c2 1·π4 4·L3. (8) The discrete form of Equation (7) is used to introduce discrete entropy. By introducing L/Ninstead of dx and j/Ninstead of x/L, Equation (7) can be rewritten in its discrete form as ∆Πi= N ∑ j=1 ∆Πij =1 2 N ∑ j=1 E·I·c1·iπ L2 ·siniπ·(j−0.5) N2L N, for N>i, (9) when N>iand iis the index of the buckling mode. The discrete form is suitable for use in the finite element method. One member of the series ∆Πij is the strain energy of a single element of the strut. The square of the sine function in Equation (9) converges to the value of N/2. N ∑ j=1siniπ·(j−0.5) N2 =N 2, for N>i. (10) Let each buckling mode have its own scale factor using amplitude c i . By substituting Equation (10) into Equation (9), the sum of the discrete strain energy is equal to the value calculated from Equation (7). ∆Πi=i4·E·I·c2 i·π4 4·L3, (11) where iis the buckling mode number, Eis the elastic Young’s modulus, Iis the second moment of the area, Lis the length of the specimen, and c i is the amplitude of the half-sine curvature that occurs lengthwise. The strain energy is a function of the indeterminate amplitude c i , as seen, for example, in Equation (11). However, if it is possible to link energy with entropy, ranking based on the strain energy can be justified. 3.2. Entropy in the Surrogate Model The strain energy of the deformed structure can be decomposed into temperature and entropy using a surrogate model. For this purpose, the strain energy is transformed into gas energy. In investigating the characteristics of a surrogate model, the terms entropy ∆ S i and virtual temperature Tiare used. Following Equation (1), it can be written as ∆Πi=Ti·∆Si, (12) where ∆Πi is the strain energy of the buckled strut from the previous section, and the right-hand side is the heat energy associated with the gas surrogate model from the following section. When the virtual temperature T i is constant, the change in entropy ∆ Scan be expressed as the work of an ideal gas with constant mass mduring an isothermal process. Under constant T i , the internal energy of the gas remains unchanged. Thus, the heat absorbed during pressure change equals the work done by the gas; see, e.g., [ 28 ]. The entire system is isolated, meaning there is no exchange of particles or energy with the surrounding environment, i.e., the number of particles and energy remain constant. The change in entropy of an isothermal process for gases, where pressure varies as a function of volume, can be expressed as ∆S=m·R Mmln ρ1 ρ2 =m·R Mmln V2 V1 , (13)
Entropy 2023,25, 1630 6 of 17 where ρ1 is the initial pressure, ρ2 is the final pressure, V 1 is the initial volume, V 2 is the final volume, M m is the molar mass of the gas, and R= 8.314 Jmol −1 K −1 is the molar gas constant; see, e.g., [29]. The change in entropy in the surrogate model can be described using the change in gas volume. The buckling causes gas compression (volume reduction) on the deflection side and expansion (volume increase) on the opposite side of the strut; see Figure 4. The pistons are located at the centroids of the finite elements in the direction of deformation, and the mesh of the finite elements is uniform. Entropy2023,25,xFORPEERREVIEW6of17 systemisisolated,meaningthereisnoexchangeofparticlesorenergywiththesurroundingenvironment,i.e.,thenumberofparticlesandenergyremainconstant. Thechangeinentropyofanisothermalprocessforgases,wherepressurevariesasa functionofvolume,canbeexpressedas 1 2 2 1lnln V V M Rm M Rm S mm ,(13) whereρ1istheinitialpressure,ρ2isthefinalpressure,V1istheinitialvolume,V2isthe finalvolume,Mmisthemolarmassofthegas,andR=8.314Jmol−1K−1isthemolargas constant;see,e.g.,[29]. Thechangeinentropyinthesurrogatemodelcanbedescribedusingthechangein gasvolume.Thebucklingcausesgascompression(volumereduction)onthedeflection sideandexpansion(volumeincrease)ontheoppositesideofthestrut;seeFigure4.The pistonsarelocatedatthecentroidsofthefiniteelementsinthedirectionofdeformation, andthemeshofthefiniteelementsisuniform. Figure4.Surrogatemodelforthetransformationofthebucklingmodeintoentropy. ThechangeinentropyΔSijcanbewrittenonthebasisofEquation(13).Thegasloadingofthej-thelement(compressionandexpansion)hasamassofm/N,whereNisthe numberofbeamelements.Themassofthegasinonepistonis0.5·m/N.Forthej-thelement,theequationforthechangeinentropyononesideofthestrut(compressionor expansion)canbeobtainedvia h h MN Rm hA hA MN Rm V V MN Rm Sij m ij m ij m ij ln 2 ln 2 ln 21 ,(14) whereVijisthefinalgasvolumeandhijisthefinalpistonheight.ThevolumeVisexpressedastheproductoftheareaAandtheheighth.TheentropyΔSij,dependingontwo indices,issimilartothevolumesofthepistonsVij.Thechangeinvolumecanbeexpressed asthechangeintheheighthofthepistoninthegasvessel,wherehisaconstant. Figure4presentsthegasanalogywithvirtualforcesthatstraightentheelasticstrut releasedbytheaxialforce.Thechangeintheentropyofthegasonbothsidesofthebeam elementcanbeexpressedasthedifferencebetweentwoentropiescausedbyexpansion(h +Δhij)/handcompression(h−Δhij)/h. h h MN Rm h hh h hh MN Rm SSS ij m ijij m C ij E ijij lnln 2,(15) wheredisplacementΔhijrepresentsasmallchangeinthepistonheight.Thecharacteristic Δhij/hcanbeinterpretedasananalogytothestrainεinHooke’slaws.ThesmalldisplacementsareastandardconditionintheEuler–Bernoullibeamtheoryinstructuralmechanics.ThemagnitudeofΔSijcanbeexpressedas Figure 4. Surrogate model for the transformation of the buckling mode into entropy. The change in entropy ∆ S ij can be written on the basis of Equation (13). The gas loading of the j-th element (compression and expansion) has a mass of m/N, where Nis the number of beam elements. The mass of the gas in one piston is 0.5 · m/N. For the j-th element, the equation for the change in entropy on one side of the strut (compression or expansion) can be obtained via ∆Sij =m·R 2·N·Mmln Vij V1 =m·R 2·N·Mmln A·hij A·h=m·R 2·N·Mmln hij h, (14) where V ij is the final gas volume and h ij is the final piston height. The volume Vis expressed as the product of the area Aand the height h. The entropy ∆ S ij , depending on two indices, is similar to the volumes of the pistons V ij . The change in volume can be expressed as the change in the height hof the piston in the gas vessel, where his a constant. Figure 4presents the gas analogy with virtual forces that straighten the elastic strut released by the axial force. The change in the entropy of the gas on both sides of the beam element can be expressed as the difference between two entropies caused by expansion (h+∆hij)/hand compression (h−∆hij)/h. ∆Sij =∆SE ij −∆SC ij =m·R 2·N·Mmln h+∆hij h−ln h−∆hij h≈m·R N·Mm · ∆hij h, (15) where displacement ∆ h ij represents a small change in the piston height. The characteristic ∆ h ij /hcan be interpreted as an analogy to the strain ε in Hooke’s laws. The small displacements are a standard condition in the Euler–Bernoulli beam theory in structural mechanics. The magnitude of ∆Sij can be expressed as ∆Sij ≈m·R h·Mm ·1 N·∆hij=k1·1 N·∆hij, where ∆hij << h. (16) This equation expresses that there is an entropy change in the direction of deformation ∆ h ij . The constant k 1 replaces the characteristics of the gas in the pistons, and in the context of the buckling of the strut, it represents lateral deformation stiffness. The change in entropy is linearly dependent on displacement.
Entropy 2023,25, 1630 7 of 17 3.3. Entropy and Virtual Temperature During the i-th buckling mode, the virtual temperature T i is constant in the isothermal process, and the strain energy can be summed over all beam elements, where T i is constant across all elements. In Equation (12), entropy and strain energy are additive quantities, so their values can be obtained by the summation of all elements. After introducing ∆ h ij =f ij , the summation of all j-th elements leads to i4·E·I·c2 i·π4 2·L3·1 N N ∑ j=1siniπ·(j−0.5) N2 | {z } ∆Πi =Ti·k1·1 N· N ∑ j=1ci·siniπ·(j−0.5) N | {z } ∆Si , (17) where c i > 0 is a scale factor of the i-th buckling mode. Equation (17) creates a connection between the products of the second derivative (bending moment) on the left-hand side and the product of deformation on the right-hand side of the equation. The sum of the absolute value of the sine function in Equation (17) converges to the value of 2·N/π. lim N→∞ N ∑ j=1ci·siniπ·(j−0.5) N=2·N·ci π. (18) For i<N< ∞ , Equation (18) takes the form of an approximate relationship. By substituting Equations (10) and (18) into Equation (17), Equation (17) can be simplified to the form of Equation (19). i4·E·I·c2 i·π4 4·L3 | {z } ∆Πi =Ti·2·k1·ci π | {z } ∆Si , (19) where the left-hand side is the change in strain energy ∆Πi and the right-hand side T i·∆ S i takes into account the change in entropy. By separating the entropic term from Equation (20), an expression for entropy can be written. ∆Si=2·k1·ci π, (20) where mis the mass of the gas in all pistons and m,R,M m ,hare constants. Equation (20) expresses the entropy of the ideal gas in the surrogate model. Due to the specific shape of the sine functions of the buckling modes, the entropy of the ideal gas is a function of only the amplitude ciand constant k1. The virtual temperature T i of the gas calculated from the surrogate model of the pin-ended strut can be written using Equation (19), as Ti=i4·ci·E·I·π5 8·k1·L3. (21) The decomposition of ∆Πi into T i and ∆ S i introduced both T i and ∆ S i as dependent on ci, but only the virtual temperature Tiis a fourth power function of the index i. Equation (19) can be written using Pcr,i, Pcr,1 z }| { π2E·I L2·i4·c2 i·π2 4·L | {z } ∆Πi = Pcr,1 z }| { π2E·I L2·i4·ci·π3 8·k1·L | {z } Ti ·2·k1·ci π | {z } ∆Si . (22)
Entropy 2023,25, 1630 8 of 17 Equation (22) introduced the decomposition of the strain energy ∆Πi into terms of virtual temperature T i and entropy ∆ S i . The utilization of Equation (22) can be presented in two fundamental cases. In thermodynamics, any equilibrium state can be characterized either as a state of maximum entropy for a given energy or as a state of minimum energy for a given entropy [ 30 ]. In structural mechanics, the strain energy and the principle of minimum total potential energy exist in many applications [ 31 – 40 ], but the principle of maximum entropy is not commonly considered. Equation (22) establishes a link between structural mechanics and thermodynamics. In the surrogate model, the principle of maximum entropy can be applied using the right-hand side of Equation (22). The case of constant energy, ∆Πi = constant. If c i =c 1 /i 2 is introduced, then the strain energy remains constant across all buckling modes, and entropy decreases with the square of index i. The first buckling mode, which has the highest entropy at a constant energy, is realized first. In the case of constant entropy, ∆ S i = constant: if c i =c 1 is introduced as a constant for all buckling modes, then the strain energy increases with the fourth power of the index i, and entropy remains constant across all buckling modes. The first buckling mode, which has the lowest strain energy at constant entropy, is realized first. 3.4. The Case Study This case study considers a pin-ended IPE240 steel member with the length of L= 3 m. The member has a Young’s modulus of E= 210 GPa, and its second moment of area is I= 2.83 · 10 −6 m 4 . Euler’s critical load is P cr,1 = 651.7 kN. The results in Table 1are obtained with the assumption of c i = 1 m and k 1 = 1000 Jm −1 K −1 . Table 1presents a scenario of constant entropy, as entropy does not depend on the buckling mode index. Table 1. Ranking of buckling modes according to strain energy, assuming constant entropy. Buckling Mode Index Critical Force Pcr,i [kN] ci Strain Energy ∆Πi[MJ] Virtual Temperature Ti[K] Entropy ∆Si[J·K−1] 1 651.7 1 0.536 842.0 636.62 2 2606.9 1 8.576 13,472.7 636.62 3 5865.5 1 43.418 68,200.3 636.62 4 10,427.6 1 137.221 215,546.7 636.62 5 16,293.1 1 335.013 526,237.0 636.62 6 23,462.0 1 694.683 1,091,205.0 636.62 In Equation (22), considering the decomposition of strain energy into entropy, virtual temperature is the measure by which energy is evaluated in terms of entropy. The criterion that the equilibrium state can be characterized as a state of minimum energy for given entropy can be applied. The minimal strain energy occurs for the first buckling mode, which occurs first; see Table 1. Subsequent buckling modes are ranked in ascending order, just as if according to critical forces. It holds that ∆Πi=i4·∆Π1. The same conclusion can be obtained using the criterion that the equilibrium state can be characterized as a state of maximum entropy for a given energy. By using Equation (22), the amplitude (scale factor) needs to be set as decreasing across buckling modes, c i =c 1 /i 2 , in order to keep the strain energy constant across all modes. Using this new scale, the entropy of the buckling modes is computed; see the last column in Table 2. In Table 2, the criterion of maximum entropy for a given energy is applied. The first buckling mode, which occurs first, corresponds to the maximum entropy. Subsequent buckling modes are ranked in descending order, inversely to their ranking by critical forces.
Entropy 2023,25, 1630 9 of 17 Table 2. Ranking of buckling modes according to entropy, assuming constant strain energy. Buckling Mode Index Critical Force Pcr,i [kN] ci Strain Energy ∆Πi[MJ] Virtual Temperature Ti[K] Entropy ∆Si[J·K−1] 1 651.7 1 0.536 1.0000 636.62 2 2606.9 0.25 0.536 15.3229 159.15 3 5865.5 0.111 0.536 25.0961 70.74 4 10,427.6 0.063 0.536 58.2375 39.79 5 16,293.1 0.04 0.536 72.6713 25.46 6 23,462.0 0.028 0.536 130.7333 17.68 The scale factor c i in Table 2decreases approximately as intensively as the mean values of the scale factors of the first three buckling modes for I-sections in [ 21 ]. The article [ 21 ] presents data in its Table 1from measurements of the initial out-of-straightness of nine IPE 160 columns, performed at the Polytechnic University of Milan and published by the ECCS 8.1 committee [ 41 ]. In addition, the article [ 21 ] uses the results from the measurements of 428 samples [ 42 ]. The scale factors c i in Table 2, obtained from the analysis of strain energy and entropy, are practically the same (differences are minimal) as the scale factors from those experiments [21,41]. The case in Table 2is a realistic reflection of the buckling problem, and the scale factor c i represents the imperfection measure. Thus, the initial imperfection can be introduced as a linear combination of the scaled buckling modes using their amplitudes c i =c 1 /i 2 , where Iis the index of the buckling mode and c 1 is the amplitude of the first buckling mode. The amplitudes cialso express the (virtual) entropy for a given strain energy. The ranking of buckling modes by strain energy in Table 1and by entropy in Table 2 is the same. In the case study, both the criteria of minimum energy and maximum entropy lead to the same ranking of the buckling modes. 4. Buckling of a Cantilever It can be assumed that the entropy in Equation (16) is applicable to other patterns of the buckling modes of other compression columns with different boundary conditions. The entropy of the i-th buckling mode can be obtained by summing the terms from Equation (16). ∆Si=k1·1 N· N ∑ j=1∆hij, (23) where k 1 is constant and h ij is a deflection from the eigenvalue vector of the i-th buckling mode. Each eigenvector is dimensionless, with an indeterminate scale factor c i . For example, the deformation of the i-th buckling mode of a cantilever can be expressed by the function cy(x)=ci·1−cos(2i−1)π·x 2·L, for x∈[0, L],i=1, 2, . . . , (24) where c i > 0 and the prefix cin c y(x) stands for cantilever. By substituting j/Nfor x/L, the continuous function can be written in its discrete form as cfij =ci·1−cos(2i−1)π·(j−0.5) 2·N, for j=1, 2, . . . N. (25) For ∆ h ij = c f ij , the entropy of the i-th buckling mode of the cantilever can be written as c∆Si=k1·1 N· N ∑ j=1ci·1−cos(2i−1)π·(j−0.5) 2·N. (26)
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