LIMIT LOAD CAPACITY OF COMPRESSED BEAM WITH IMPERFECTIONS

Konstrukcije napravljene spajanjem ravnih ploča na njihovim podužnim krajevima veoma su česte. Važan podskup tih konstrukcija koje su glavni predmet ovog rada u suštini jesu prizmatične forme, ali one mogu imati i poprečna ukrućenja koja se koriste u sandučastim nosačima, ukrućenim pločama i pločastim nosačima. Opterećenje je uglavnom takvo da su najveći naponi u pravcu dužine, na primer, aksijalno opterećenje ili uzdužno savijanje. Analiza ponašanja pločastih konstrukcija odvijala se na nekoliko različitih načina. Jedan od načina bio je sprovođenje sveobuhvatnog istraživanja jednog tipa pločastih konstrukcija, kao što je sandučasti stub, tako da se testira cela familija modela u laboratoriji [1]. Drugi načini bili su numerički, primenom metoda konačnih elemenata (MKE) koji mogu uključiti komplikovane geometrije konstrukcija [2]. Cilj ovog rada jeste da istraži lom grede s početnim imperfekcijama, pojednostavljenim modelima koji se koriste u mehanici, radi poređenja dobijenih rezultata. Iako je korištenje MKE trenutno dominantno u analizi pločastih konstrukcija, nije tako jednostavno postaviti problem. Za tačnost je poželjno da se koriste manji elementi u zonama gde se čelik izvija lokalno i postaje plastičan, ali nije uvek poznato unapred gde te zone treba de se pojave. Takođe, u analizi ponašanja pločastih konstrukcija može se primeniti i metod konačnih traka (MKT). MKT se zasniva na svojstvenim funkcijama koje su izvedene iz rešenja diferencijalne jednačine


INTRODUCTION
The structures, which are made by joining flat plates at their longitudinal edges are very common.An important sub-set of these structures, and which are the main concern of this paper, are those essentially of prismatic form but which can have some transverse stiffening such as is used in box girders, stiffened plates and plate girders.The loading is generally such that the greatest stresses are in the longitudinal direction, e.g.axial loading or longitudinal bending.
Analysis of the behavior of plate structures has been approached in several different ways.One way of carrying out a comprehensive investigation of a single type of plated structures, such as a box-column, would be to test a whole family of models in the laboratory [1].Other methods were numerically using finite element method (FEM), which may include a complicated geometry of the structures [2].The aim of this paper is to investigate the fracture of beam with initial imperfections simplified models used in mechanics, in order to compare the results.
Although the use of FEM currently dominant in the analysis of plate structures, the problem set is not so simple.For accuracy it is desirable to use smaller elements in the regions where the steel buckles locally and becomes plastic but it is not always known beforehand where these plastic zones will occurs.Also, the analysis of the behavior of plate structures may be approached using the finite strip method (FSM).The FSM is based on eigen functions, which are derived from poprečnih vibracija grede, a pokazao se kao efikasan alat za analiziranje velikog broja konstrukcija kod kojih se i geometrija i svojstva materijala mogu smatrati konstantnim duž podužnog pravca [2].Ipak, ako se geometrija nosača i opterećenje komplikuju -MKE ima prednost.
If we analyze a simple structure, such as a simple beam, simplified models which are present in mechanics are useful because they can provide solutions when the problem is complicated.Such solutions can provide enough information to designers in the design of structures.In this paper we propose a simple and very efficient analysis of the above problem of the limit load capacity regarding inelastic buckling.The starting point is the fact that the thin-walled structures are very sensitive to initial imperfections, and is the premise of their existence.Elastic solution of the problem is well known [1].However, the elastic solution can be applied only to the line of limit load capacity.Solving the problem of the limit load capacity in the inelasticity is achieved by using RDA.Application is simple because RDA transforms a complicated material non-linear problem to a simple linear dynamic problem [3].

DETERMINATION OF LINE OF LIMIT LOAD CAPACITY USING THE RDA
The factors influencing the limit load capacity may divide into two groups.The first group involves the nominal geometry of the compresses member such as cross-section and length, the support conditions, the material properties including the strength, the surrounding climate, the load duration, etc.The second group of factors which influence to the limit load capacity of column involves geometric and material imperfections and their variations.RDA is inelastic theory independent of the theory of plasticity, or non-linear fracture mechanics, which was successfully applied in the study of buckling curves of columns [4].In this paper, RDA is used in research of the limit load capacity of thin-walled beam with initial imperfections.

RDA -a short overview
Micro cracks and deformations in the plastic material are consequence of action of external forces to the carrying member, which leads to its damage or breakage.Consider the case of viscoelastoplastic (VEP) strain of a simple pin-ended column presented in Fig. 1(a).In research of the material, both a stress σ(t) and inelastic strain ε * (t) = εve(t)+εvp(t) are functions of time.
If the total VEP strain ε (t) = εel+ε * (t) is presented as a sum of elastic (instantaneous) εel, viscoelastic (VE) εve(t), and viscoplastic (VP) εvp(t), component, each isochronous stress-strain diagram of a prismatic column (e.g., with a square or circular cross section A0) can accurately be approximated by the rheological model of material H-K-(StV|N), consisting of five elements.The rheological model is shown in Fig. 1(b) using the following symbols: N for the Newton's model, StV for Saint-Venant's model, H for Hooke's model, "" for a parallel connection of models and "" for connection of models in a series.Since a Hooke's model, Kelvin's ov model, Kelvin-ov model (K=H|N) i VP model (StV|N) spojeni u nizu, napon σ(t) u sva tri modela jeste jednak.
Accordingly, the elastic wave propagation constitutes the physical basis for setting up an analogy between two different physical phenomena, rheological and dynamical, called RDA.Based on RDA, a very complicated non-linear problem in the range of VEP strains can be solved as a simple linear dynamic problem.RDA is derived to solve the dynamic problems [4], but can be used also in the analysis of the quasi static problems taking into account the corresponding limit values of derived analytical expressions.For example, each quasi static stress-strain diagram can be obtained by using the RDA modulus function [3], including the compression strength, which is a key parameter for the analysis of fracture energy.

Structural-material constant
The utilisation of the tangent modulus instead of Young modulus (according to the Engesser assumption ) proved to be realistic in the case of inelastic buckling, based on the and experimental researches.In [4], the first author has shown that RDA modulus is equal to the tangent modulus ( Because of this, in consideration of the above problem is introduced RDA modulus function ER.It has already been used for the formulation of the quasi-static stressstrain diagram of the standard concrete cylinder [3], as gde je E(0) modul elastičnosti materijala u inicijalnom, neoštećenom stanju, a KE je strukturalno-materijalna konstanta.Na osnovu (8), sledi kvadratna jednačina where E(0) is the initial elastic modulus in undamaged state, and KE is the structural-material constant.According to Eq. ( 8) the quadratic equation follows Koren jednačine (9), koristeći početne uslove

Tako je
The root of Eq. ( 9), using the initial conditions for the selected deformation ε of stress-strain curve. Thus Na granici elastičnosti nagib je jednak Young-ovom modulu EH (poznata vrednost).Zbog toga, jeste , tako da sledi At the limit of elasticity the slope is equal to the Young modulus EH (a known value).Because of that gdje je ϕ  strukturalno-materijalni koeficijent tečenja na granici elastičnosti.U radu [4] prvi autor na granici elastičnosti odredio je presek Euler-ove i RDA krive izvijanja, iz koga sledi vitkost na granici elastičnosti where ϕ  is the structural-material creep coefficient at the limit of elasticity.In [4], the first author has defined the intersection of the Euler and RDA buckling curves at the limit of elasticity, from which follows the slenderness at the limit of elasticity gde je koristi se za izračunavanje strukturalno-materijalne konstante na granici elastičnosti, kako je pokazano u [3] where Further, Euler's critical stress of a simple pin-ended column is used to calculate the structural-material constant at the limit of elasticity, as explained in [3] Iza granice elastičnosti, koristi se zakon linearne promene kritičnog napona u odnosu na kritični koeficijent tečenja (zakon toka), kako je definisao Milašinović [3] Beyond the limit of elasticity the law of linear changes of the critical stress level in relation to the critical creep coefficient (flow law) is used, as defined in Milašinović [3] 1 Na osnovu svega, funkcija RDA modula glasi Accordingly, the RDA modulus function is as follows gde je kritični koeficijent tečenja where the critical creep coefficient is

Critical damage variable
Since that growth of micro cracks reduces the stiffness of the material, the damaged state of material is described by the variation of Young modulus EH, Lemaitre [5].Hence, the damage variable D is introduced based to the hypotheses of strain equivalence between the damaged and undamaged material, as follows U radu [3] prvi autor ovog rada uveo je pretpostavku da je E(D) jednak RDA modulu, na osnovu čega je definisana varijabla oštećenja D. Ova pretpostavka koristi se i u ovom radu.Kako je tačka C1 na liniji granične nosivosti, koja je ujedno i linija kritičnih napona u neelastičnoj oblasti, varijabla oštećenja koja sledi jeste kritična In [3], the first author of this paper has introduced the assumption that the E(D) is equal to the RDA modulus on the basis of which the damage variable D is defined.This assumption is also used in this paper.As the point C1 is on the line of limit load capacity, which is the line of critical stresses in the inelastic range of strain, damage variable below is the critical gde je where ϕ is the creep coefficient at the point C1, according to the flow law ( 14) jeste relativna frekvencija u kojoj je σ ω kružna frekvencija pobude.U slučaju kvazistatičkog opterećenja sledi δ → 0, pa je kritična varijabla oštećenja is the relative frequency where σ ω is the angular frequency of excitation.In the case of quasi-static loading follows δ → 0, and critical damage variable is Na ovaj način, oštećenje na liniji granične nosivosti u slučaju kvazistatičkog opterećenja opisano je skalarnom veličinom DC1 koja uzima vrednosti između 0 i 1.
Prema RDA, neelastičan ugib grede u slučaju kvazistatičkog opterećenja jeste [3] In this way, the damage at the line of limit load capacity is described by a scalar value DC1, which takes a value between 0 and 1.

Line of limit load capacity obtained using the RDA
Geometric and material imperfections, as well as their variations can significantly affect to the limit load capacity of beams.This is why this problem is very complicated.Purely initial geometric imperfections are considered here.It implies that only the reference geometry is influenced by imperfection, not the stress state.
where W is the elastic section modulus.
The curve line QC-vc is obtained by assuming a series of the initial imperfections a1.This is the line of limit load capacity obtained using the RDA.This line shows normal softening behavior of beam, Fig. 2 (a).

DETERMINATION OF FAILURE LINES
At the beginning of Section 2 are described factors which affect the limit load capacity.However a combination of these factors can lead to a substantially different shape of fracture of the beam, which range from brittle to extremely ductile.The shape of the fracture is closely associated with failure lines that appear in postcritical state, after the limit load capacity is reached.In this section, solutions are given for the failure lines according to the theory of elasticity and plasticity theory in order to compare with solutions obtained by the RDA.

Elastic failure lines
If we analyze the beam with a constant cross-section A0 and initial imperfection a1, elastic failure line (hyperbola) is well known solution for elastic problem of the limit load capacity, Fig. 2(a).The asymptote of solution is Euler's critical load QE, [1] Napon u tački C1 jeste suma od aksijalnog napona i napona savijanja.Mi izjednačavamo ovaj napon s The stress at point C1 is the summation of the axial stress and bending stress.We equates this stress with naponom tečenja σY, pod pretpostavkom plastičnog loma grede yield stress σY, assuming plastic failure of the beam Tako, elastična granična nosivost glasi Thus, the elastic carrying capacity is as follows Hiperbola Q-vc može biti konstruisana izborom niza sila Q. Međutim, treba imati u vidu i to da se kada je reč o realnom materijalu hiperbola može primeniti samo do linije granične nosivosti, dobijene primenom RDA u poglavlju 2.4, odnosno do granične nosivosti QC1.
Hyperbola Q-vc can be constructed by selecting a series of force Q.However, it should be noted that in a real material a hyperbola can be applied only to the line of limit load capacity, which is obtained by the application of RDA in Section 2.4, that is, up to the critical load QC1.
If load is greater than load QC1 then it is possible to appear failure lines based on the new equilibrium states.As mentioned in [1], in the thin-walled structures there are two important reasons for the formation of the failure lines, Fig. 2(a).The first reason is related to the slope of the line of limit load capacity, while the second reason is that in thin-walled structures can not be developed a simple plastic hinges.Simple plastic hinge occurs only at a right angle to the neutral axis of a beam subjected to pure bending.

Plastic failure line for simple plastic hinge
Failure line for simple plastic hinge can be approximated by the method explained in [6], Fig. 3(a).

Failure lines obtained using the RDA
If the load is greater than load QC1, a beam is in the new buckling equilibrium.In the case of normal softening the ultimate carrying capacity QC1 must decreases only, while the deflection increases.Application of RDA starts from the limit of elasticity.Therefore, for the selected initial imperfection a1 a hyperbole may be constructed and the limit load capacity QC1 calculated.For the corresponding stress σC1 = QC1/A0, the slenderness of a beam is Prema radu [4], koeficijent tečenja According to [4], the creep coefficient 1 C ϕ jeste ključni parametar za testiranje omekšavajućeg ponašanja, slika 2(b).Na osnovu zakona toka (14), strukturalno-materijalna konstanta jeste ϕ is a key parameter for the testing of the softening behavior of the material, Fig. 2(b).Based on the flow law (14) the structural-material constant is as follows RDA modul u prvoj iteraciji (1) jeste RDA modulus in the first iteration (1) is Zatim, napon u prvoj iteraciji jeste Then the stress in the first iteration (1) is RDA modul u drugoj iteraciji (2) jeste RDA modulus in the second iteration (2) is gde je where Iterativni postupak objašnjen u [4] nastavlja se sve do konvergencije u rešavanju problema, to jest kada novi RDA modul više ne menja napon.Ugibi u sredini grede moraju se računati putem iteracija -kako sledi Iterative method, which is explained in [4] continues until convergence in solving the problem, i.e. when the new RDA modulus does not change the stress.Deflection in the mid-length of beam must be calculated through iterations as follows

NUMERICAL ANALYSIS
Numerical analysis is carried out in a simple beam with the thin-walled cross-section.The initial imperfections of a1 = 0.1, 1.5 and 5 mm are measured at midheight of beam [1] and beam bending takes place about the stronger axis.Details the cross-section was taken from [1] and is shown in Fig. 4. The beam is made of steel of the following mechanical characteristics of EH = 206 GPa, μ = 0.3 i σY = 250 MPa.It is assumed that bending beam forms a simple plastic hinge in the midlength.
Yield of cross-section starts when stress in the outermost fiber reaches the yield stress σY.Initial yield line, Fig. 6 is calculated according to the Bernoulli-Euler's bending theory.
Fig. 6 shows the failure lines obtained using the RDA, such as occur in a post-critical state of beam, after the limit load capacity is reached.In the case of initial imperfection of a1 = 0.1 mm failure occurs in the elastic zone, because the failure line lies below the initial yield line.Because of this beam behaves brittle although the steel is ductile material.This brittle behavior of the beam is not desirable.In other two cases (a1 = 1.5 and 5 mm) duktilan u elasto-plastičnoj oblasti.Duktilnost je veća kod većih početnih imperfekcija.Linije loma opisuju normalno omekšavajuće ponašanje.To je slučaj negativnih ugiba (dQC/dvc<0), slika 2(a).To znači da se granična nosivost smanjuje, a raste ugib grede.Isti omekšavajući efekat već se dobio putem napondeformacija relacije u radu [7].
the failure of the beam is ductile in the elastic-plastic zone.Ductility is greater at higher initial imperfections.Failure lines describe the normal softening behavior.This is the case of negative slope (dQC/dvc<0), Fig. 2(a).This means that the limit load capacity decreases, while the deflection increases.The same softening effect has already been obtained through the stress-strain relation in the paper [7].Failure curve a1=5.0 mm Initial yield line Slika 6. Uticaj početnih imperfekcija na linije početka tečenja i linije loma Fig. 6.Influence of initial imperfections on the initial yield line and failure lines Slika 7 prikazuje kritične varijable oštećenja, sračunate na liniji granične nosivosti za skup pretpostavljenih početnih imperfekcija.Velike vrednosti kritičnih varijabli oštećenja pokazuju da su tankozidne konstrukcije veoma osetljive na početne imperfekcije, što je dobro poznata činjenica potvrđena eksperimentalno.Fig. 7 shows the critical damage variables calculated at the line of limit load capacity for a set of assumed initial imperfections.Large values of critical damage variables show that the thin-walled structures are very sensitive to the initial imperfections, which is a well known fact proved experimentally.
Pojam granične nosivosti, pod pretpostavkama idealno elastičnog materijala na analiziranom primeru Because that effective load capacity (capacity of undamaged beam)   1 1 C Q • is equal to the limit load capacity QC1, calculations which are made by using the RDA are consistent with the hypothesis of damage mechanics.

CONCLUSINS
The paper was theoretically investigated the problem involved ultimate bearing capacity of thin-walled beam with initial geometrical imperfections under the conditions of elasticity, plasticity and using RDA.RDA is inelastic (viscoelastoplastic) theory and includes both previously mentioned.In order to present applications of RDA the comparison is done with the numerical and experimental results of a steel beam from the literature of Murray [1].Depending on the assumptions about the material the concept of limit load capacity varies and is therefore not possible to give a unique formulation for the ultimate bearing capacity.

Zahvalnost
Ovaj rad predstavlja deo istraživanja koja se sprovode u okviru istraživačkih projekata OI 174027 "Računarska mehanika u građevinskom inženjerstvu" i TR 36017 "Istraživanje mogućnosti primene otpadnih i recikliranih materijala u betonskim kompozitima, sa ocenom uticaja na životnu sredinu, u cilju promocije održivog građevinarstva u Srbiji", uz podršku Ministarstva za nauku i tehnologiju Republike Srbije.initial imperfections shows that the limit load capacity converging to the Euler-critical force.Convergence is slower for the larger the initial imperfections.According to the theory of elasticity the softening effects that observed experimentally under the load close to the critical load can not be obtained.However, the key disadvantage of this model is that in the full sense of the word elastic material does not exist, or it is only hypothetical thought.
The term limit load capacity under the assumption of ideal plasticity provides in the analyzed beam an upper limit load capacity.This theory describes softening effects, since it gives a lower limit load capacity at higher initial imperfections.The key disadvantage of this model is that it does not explain the critical buckling stresses.
The limit load capacity obtained by RDA is always located between the above-described theories.RDA theory describes the critical stresses for inelastic buckling.The reason is that RDA involves inelastic properties of materials in the analysis of buckling.In this paper the quasi-static solution is analyzed only, δ → 0 ), so that the expected additional unexplored effects that RDA theory provides in the problem of dynamic stability under the influence of frequency σ ω (frequency of force).RDA transformed materially nonlinear problem into the linear dynamic problem so that the obtained analytical solutions.
Apart from this, RDA effectively provides explanations in post-critical behavior of the analyzed beam, which show that a beam made from ductile material can break from brittle to extremely ductile, depending on the initial imperfections.Although this a long time established experimentally in this paper is theoretically confirmed.Because that post-critical state is in the field of damage mechanics and nonlinear fracture mechanics, the RDA is successful compared with damage mechanics through the critical variables of damage.