THERMO-VISCO-PLASTICITY AND CREEP IN STRUCTURAL-MATERIAL RESPONSE OF FOLDED-PLATE STRUCTURES

Pri projektovanju konstruktivnog elementa podvrgnutog mehaničkim i termalnim opterećenjima potrebno je adekvatno uzeti u obzir ponašanje materijala pod takvim uslovima. Mnogi nacionalni i međunarodni standardi uspostavljeni su za projektovanje/proračun na povišenim temperaturama, kao rezultat iskustva i istraživanja tokom više decenija. Metodi analize koji uključuju termo-viskoplastičnu deformaciju i deformaciju tečenja bili su najpre analitičkog tipa [1]. Kasnije, s napretkom MKE, numerički postupci razvijeni su kako bi se mogli koristiti da realno predvide odzive u veoma složenim konstrukcijama [2]. Ove numeričke procedure uključene su u računarske programe opšte namene, po konačnim elementima, kao što su ADINA, ABAQUS, MARC, PAK, i tako dalje. Numerički algoritam predstavljen u ovom radu oslanja se uglavnom na primenu RDA, koja je razvijena od strane autora za analizu unutrašnje prigušenih neelastičnih konstrukcija [3]. RDA je vrsta neelastične analize, koja transformiše jednu kategoriju veoma komplikovanih materijalno-nelinearnih problema u jednostavnije linearno-dinamičke probleme korištenjem modalne analize [4, 5]. Ovaj koncept ponešto je generalizovan u ovom radu, posebno za termička opterećenja. Razvoj metoda za nelinearnu analizu, naročito u okviru MKE, doveo је do mogućnosti rešavanja veoma komplikovanih inženjerskih problema. Na primer, vlaknasti konačni elementi, greda/stub, na bazi distribuirane plastičnosti pokazali su se veoma uspešnim za modeliranje čelika, betona i kompozitnih materijala [6]. U poslednje dve decenije, razvoj metoda za analizu konstrukcija izloženih uticaju zemljotresa u brzoj je ekspanziji.


INTRODUCTION
In designing structural element subjected to mechanical and thermal loadings it is necessary to adequately take into account material behaviour under such conditions.Many national and international standards have been established for design at elevated temperatures, as a result of experience and research over many decades.Methods of analysis which include thermo-elastic-plastic and creep deformation were first of analytical type [1].Later, with the progress of FEM, numerical procedures have been developed which can be used to realistic predict respone of very complex structures [2].These numerical procedures have been implemented in general purpose finite element programs, like ADINA, ABAQUS, MARC, PAK, etc.The numerical algorithm presented in this paper relies on the main idea of the RDA employed by the author for the analysis of visco-elasto-plastic (VEP) vibrations and their use for the analysis of internally damped inelastic structures [3].The RDA is a type of inelastic analysis, which tranforms one category of very complicated material non-linear problems to simpler linear dynamical problems using modal analysis [4,5].This concept is generalized in this paper, particularly for the thermal loading.

REOLOŠKO-DINAMIČKA ANALOGIJA I TERMALNI EFEKTI
Materijalne mikropukotine prate opterećivanje uzorka.Razmatramo slučaj VEP deformacija u početno geometrijski nepravilnom stubu prikazanom na sl.1(a).U istraživanjima materijala, obe funkcije, napon σ(t) i neelastična deformacija ε * (t)=εve(t)+εvp(t) jesu funkcije vremena.Ако je totalna VEP deformacija ε (t)=εel+ε * (t) predstavljena kao suma od elastične (trenutne), viskoelastične (VE) i viskoplastične (VP) komponente, onda svaki jednovremeni napon-deformacija dijagram za dugi simetričan stub s poprečnim presekom A0 može biti precizno aproksimiran reološkim modelom H-K-(StV|N), koji se sastoji od pet elemenata.Za odgovarajući reološki model -prikazan na sl.1(b) -korišteni su sledeći structures exposed to earthquake actions saw a rapid expansion.An array of alternatives for solving common and complex problems was formed, both in everyday engineering practice, and in scientific research [7].This paper aims at providing a unified frame for quasi-static inelastic buckling and thermal loading of uniformly compressed plate structures using the FSM [3].The first paper on the FSM was published by Cheung in the late 1960s.Since then, many developments have been made toward structural analysis with high efficiency.Milašinović [8,9] developed a harmonic coupled FSM by including the Green-Lagrange strain terms in the formulations, thus bending and membrane are coupled in the geometric stiffness to give more accurate buckling behaviour prediction, especially in large-deflection and elastic postbuckling analysis.The FSM is a variant of the FEM.For its application to folded-plate structures, instead of a finite element discretization, a thin-walled cross-section is discretized into a series of longitudinal strips (or elements).Shape functions in the transverse and longitudinal directions are selected to represent the displacement field.Then, similar to FEM, an eigenvalue problem for buckling analysis can be formulated using different plate theories, but with less degree of freedoms (DOFs).
where EH is the elastic modulus, σY the uniaxial yield stress and Y=σY+H'εvp(t) the VEP yield criterion.
The four properties at fixed step times are: extensional VE viscosity λK, extensional VP viscosity λN, VE modulus EK and VP modulus H'.However, these constants cannot easily be determined in physical experiments, especially Trouton's viscosities λK and λN.
Among the various types of time-dependent stresses, sinusoidal stresses are among the most important, and are consequently most widely investigated in relation to various problems, such as fatigue, dynamic instability, etc.Such stressing conditions are known in rheology as dynamic loading.Let us consider the sinusoidal law ( ) ( ) Sada ćemo definisati dinamički modul (RDA modul) tako da RDA model prigušimo kritično [4] with 0 σ being a constant and A σ a variable component is a load or stress frequency.
Let us now define the dynamic modulus (RDA modulus) along the way required to damp the RDA model critically [4] ( ) 3 ANALIZA IZVIJANJA KORIŠTENJEM METODA KONAČNIH TRAKA

Elastic buckling
Although the formulation of the stiffness matrices may use different plate theories (e.g.Kirchhoff-Love theory versus Mindlin-Reissner theory), a similar eigenvalue problem associated with elastic buckling on a perfect structure considering only geometric nonlinearity may be defined in the same way.
It is well-known that the total potential energy of a strip is defined as the sum of the strain energy, potential energy due to nodal line forces, as well as the additional potential energy due to the initial stress.The formulation of equilibrium equation using the principle of minimum total potential energy yield ˆσ += KqKqQ .
In the semi-analytical FSM, which combines elements of the classical Ritz method and the FEM, the general form of the displacement function can be written as a product of polynomials and trigonometric functions gde su Ym(y) funkcije iz Ritz-ovog metoda, a Nk(x) jesu interpolacione funkcije iz MKE [3].U analizi elastičnog izvijanja, tri uticajna aspekta jesu granični uslovi, izbor trake i mreža.
The most commonly used series are the basic functions (or eigenfunctions) which are derived from the solution of the beam vibration differential equation gde je a dužina grede (trake), a µ je parametar.

Opšti oblik osnovnih funkcija jeste
where a is the length of the beam (strip) and µ is a parameter.

Ravnotežna jednačina za sistem koji pokazuje nelinearno materijalno ponašanje može biti napisana kao
Imperfections can be geometric, material or structural.Purely initial geometric imperfections are most simple, Fig. 1(a), and it implies that only the reference geometry is influenced by imperfection, not the stress state.In this case the initial imperfection may be modelled as the first buckling eigenmode ( )
While the solution of a linear system can be accomplished without difficulty in a direct manner, this is impossible for non-linear systems.A variety of solution schemes have been developed to solve such problems and they can be found in many articles or textbooks.In this paper, a modulus iterative method, which is really the same as the procedure explained in [10]  Eq. ( 15) presents a linear system of equations.Solving these equations, stresses σ (0) and strains ε (0) in all points of a 2D continuum can be obtained.When the je komponenta napona veća od napona na granici elastičnosti, sledeća jednačina može biti korišćena da se dobije bolja aproksimacija [10] stress component is greater than stress at the limit of elasticity the following equation may be used to give a better approximation [10] () () () Međutim, u analizi neelastičnog postizvijajućeg ponašanja ili nelinearnog loma konstrukcija, samo početni naponi i odgovarajuće deformacije moraju biti uključeni.Stoga, sledeći problem svojstvenih vrijednosti može se definisati ovako: However, in the analysis of inelastic post buckling behaviour or non-linear collapse of structures, only the initial stresses and corresponding strains must be included.Thus, the following inelastic eigenvalue problem may be defined () sa ulaznim parametrom za prvu postizvijajuću iteraciju with input parameter for the first post buckling iteration Stoga, novi kritični napon ( ) σ (inelastic buckling stress or yield stress) may be computed according to Eq. ( 17).The corresponding component of strain is

NUMERICAL EXAMPLE
Consider the uniformly compressed rectangular steel plate (a/b=1) presented in Fig. 2, whose all edges are simply supported.
RDA naponi tečenja dobijeni su korišćenjem testpodataka, datim u Tab. 1 i imajući u vidu sledeće RDA module Plate of the following geometrical and elastic properties, t=16 mm, b=1000 mm, EH=210 GPa and 0.3 µ = (Poisson's ratio), was analyzed using the test results for the reduction factors of yield strength and elastic modulus of mild steel a t elevated high temperatures [11].
The RDA yield stresses are obtained using the test data given in Tab. 1 and taking into account the following RDA module where the structural and thermal creep coefficients are respectively Таbela 1. Eksperimentalni i sračunati naponi tečenja usled visokih temperatura i opterećenja kod čelične ploče (a/b=1) od mekog čelika Таble 2.Experimental and computed yield stresses due to high temperatures and loading of steel plate (a/b=1) made of mild steel [11] cr σ [11] [

CONCLUSION
Two source of non-linearity (geometrical non-linearity due to large deflection and material non-linearity due to inelastic behavior) are analyzed for the problem of thermo-visco-plasticity and creep in structural-material response.This paper shows that the critical damage

Dragan D. MILASINOVIC
Many structural parts are exposed to high temperatures and loading.It is then important to have data about material inelastic behaviour under such exploiting conditions.Influence of temperature on mechanical characteristics of a material may be inserted via the creep coefficient in the range of visco-elastoplastic (VEP) strains.This damage parameter is implemented in this paper in conjunction with mathematical material modelling approach named rheological-dynamical analogy (RDA) in order to address structural stiffness reduction due to inelastic material behaviour.The aim of this paper is to define structuralmaterial internal damping based on both the RDA dynamic modulus and modal damping ratio, by modelling critically damped dynamic systems in the steady-state response.These systems are credible base for explanation of the phenomenon of thermo-viscoplasticity and creep in structural-material response due to high temperatures and loading.
Though elastic buckling information for folded-plate structures is not a direct predictor of capacity or collapse behaviour on its own, both the mode and the load (moment) are important proxies for the actual behaviour.In current design codes, such as AISI S100, New Zealand/Australia, and European Union, the design formulae are calibrated through the calculation of elastic critical buckling loads (or moments) to predict the ultimate strength, thus the ability to calculate the associated elastic buckling loads (or moments) has great importance.Moreover, the buckling mode shapes are commonly employed into non-linear collapse modelling as initial geometric imperfections and thermal performance of folded-plate structures in fire.
To examine the buckling behaviour of folded-plate structures, the main n umerical solution methods are used such as the finite element method (FEM) and finite strip method (FSM).This paper aims at providing a unified frame for quasi-static inelastic buckling and thermal loading of uniformly compressed folded-plate structures using the FSM.

ϕ
frekvencija, a ω prirodna frekvencija.Ukupni koeficijent tečenja ϕ u području VEP deformacija, koji uključuje termalni koeficijent tečenja T ϕ , može biti definisan kako sledi: where σ δωω = is the relative frequency and ω is the natural frequency.The total creep coefficient ϕ in the range of VEP strains, which includes the thermal creep coefficient T ϕ may be defined as follows strukturni koeficijent tečenja određen na bazi preseka Euler-ove i RDA krive izvijanja na granici elastičnosti.D H ′ je dinamički VP modul [5].where * ϕ is the structural creep coefficient determined based on the Euler and RDA buckling curves intersection at the elastic limit and D H ′ is the dynamic VP modulus [5].
član je konvencionalna matrica krutosti K sistema, koja zavisi od neelastične konstitutivne matrice C. K ada je reč o 2D problemima primenom MKT, uobičajeno je da se napiše komplians matrica C -1 u oblikuThe non-linear term is the conventional stiffness matrix K of the system, which depends on the inelastic constitutive matrix C. It is often convenient, when dealing with 2D problems using the FSM, to write the compliance matrix is applied.It is based on the closed form solution for the RDA modulus function and stress-strain curve.The RDA modulus iteration starts with the elastic constitutive matrix C .(15) predstavlja linearni sistem jednačina.Rešavanjem ove jednačine, naponi σ (0) i deformacije ε (0) u svim tačkama 2D kontinuuma mogu biti dobijeni.Ako

Slika 3 .
Poređenje napona tečenja usled visokih temperatura i opterećenja čelične ploče a/b=1 napravljene od mekog čelika Figure3.Comparison of yield stresses due to high temperatures and loading of steel plate a/b=1 made of mild steel gde su strukturni i termalni koeficijenti tečenja, redom coefficients are quite similar for the temperature ranging from 22 to 540 C °and very different for temperature ranging from 600 to 940 C °.
Budući da razvoj mikropukotina indukuje smanjenje krutosti materijala, stanje oštećenja može biti okarakterisano varijacijom modula elastičnosti ET.Stoga, kritičnu varijablu oštećenja Dcr karakteriše varijacija Young-ovog modula E(D), kao što sledi Since the development of micro cracks induces a reduction in the stiffness of materials, the damage state can also be charaterized by variation in the elastic modulus ET.Thus, the critical damage variable Dcr is characterized by variation in Young's modulus E(D), as follows