post critical behaviour of shells: a sequential limit analysis approach

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Post Critical Behaviour of Shells:

a Sequential Limit Analysis Approach

Tesi presentata per ilconseguimento del titolo di Dottore di Ricerca

Politecnico di MilanoDipartimento di Ingegneria Strutturale

Dottorato in Ingegneria delle Strutture - XIII Ciclo

diNicola Panzeri

Dicembre, 2001



Post Critical Behaviour of Shells:a Sequential Limit Analysis Approach

Tesi di Dottorato dell’Ing. Nicola Panzeri

Relatori: Prof. Arch. Leone CorradiProf. Ing. Carlo Poggi

Dicembre 2001

Dottorato in Ingegneria delle Strutturedel Politecnico di Milano

Collegio dei Docenti:

Prof L. BiolziProf S. BittantiProf.ssa G. BolzonProf. L. CedolinProf.ssa C. ComiProf. R. ControProf. A. CoriglianoProf. L. Corradi

Prof. M. Di PriscoProf. A. FranchiProf. P GambarovaProf. G. Maier (Coordinatore)Prof. F. MolaProf.ssa A. PandolfiProf. A. PavanProf. U. PeregoProf. C. PoggiProf. A. QuarteroniProf. G. SacchiProf. S. SirtoriProf. A. Taliercio



Summary

In the present work, a method for the study of the post-collapse behaviour of shell structures, that buckle in the plastic range, is presented. The material isassumed to be rigid-plastic, thus the elastic deformations are neglected. Themethod consists of a sequence of limit analyses solved by means of the finite ele-ment method.The following parts can be identified in the thesis:

• an introductory part (chapter 1) presenting the main argument of the thesis.Moreover, some hystorical remarks on the limit analysis and on the methodadopted, in order to study the collapse behaviour of shell structures, arerecalled;

• a second part consisting of chapters 2 and 3, where the theoretical argumentsare explained. In particular the elastic plastic constitutive relations and thelimit analysis theorems are presented in chapter 2. The method used, for thesolution of the limit analysis problem, is introduced in the same chapter.Its implementation, by means of the finite element method, is explainedexhaustively in chapter 3. Firts of all a general description is presented,therefore two elements are formulated: the first is a two nodes axisymmetricshell element, based on the Kirchhoff hypotheses, the second is a three nodes

shell element capable to deal with shear strains. This element can be usedwith thin and medium thick shells too;

• some analyses performed on different shell structures are presented in thethird part of the thesis (chapter 4 and 5) . In particular, chapter 4 dealswith the validation of the proposed method by comparing different resultsobtained for symply supported plates. In chapter 5 some results concerninga cylinder, a frusta and a square box column will be discussed pointingout the capabilities of the method when used to predict the post-collapsebehaviour of shell structures;

• in the appendixes, A, B and C, some numerical and theoretical aspects, forsake of clearness not reported in chapters 2 and 3, are presented in detail.



Sommario

Il presente lavoro propone un metodo per lo studio del collasso di strutture aguscio in cui le deformazioni elastiche sono trascurabili. Questo metodo si basasu di una sequenza di analisi limite la cui risoluzione e implementata mediante ilmetodo degli elementi finiti.Nell’ambito della tesi si possono individuare le seguenti parti:

• una parte introduttiva (capitolo 1) in cui viene illustrato l’argomento af-

frontato.`E inoltre proposta una breve revisione storica dell’analisi limite edel metodo utilizzato per studiare il comportamento a collasso di strutture

a guscio;

• una seconda parte, costituita dai capitoli 2 e 3, nella quale vengono affrontatigli argomenti teorici. In particolare, i fondamenti della teoria della plasticitae i teoremi dell’analisi limite sono richiamati nel capitolo 2. Sempre nellostesso capitolo viene introdotto il metodo risolutivo, con approccio cine-matico, del problema di analisi limite. La sua implementazione, mediante ilmetodo degli elementi finiti, e descritta nel capitolo 3. Dopo una trattazionegenerale vengono formulati due differenti elementi finiti di guscio: il primoe un elemento rettilineo a due nodi, assialsimmetrico, con formulazione allaKirchhoff, mentre il secondo e un elemento triangolare a tre nodi, formulato

in modo da poter essere applicato anche per la modellazione di gusci mediospessi;

• nella terza parte, composta dai capitoli 4 e 5, vengono presentate alcuneanalisi svolte su strutture a guscio di diversa tipologia. In particolare la vali-dazione del metodo proposto, illustrata nel capitolo 4, e effettuata mediantel’analisi del collasso di piastre semplicemente appoggiate. Nel capitolo 5vengono invece studiati un cilindro, un cono ed un tubo a sezione quadrata,ponendo in evidenza i vantaggi ottenibili applicando il metodo dell’analisilimite sequenziale allo studio del collasso di strutture a guscio;

• nelle appendici A, B e C, vengono approfonditi alcuni aspetti teorici e nu-merici non riportati nei capitoli 2 e 3 per non appesantirne eccessivamente

la lettura.



. . . alla fine delle mie fatiche come studente di dottorato, e giunto il momento di scri-

vere qualche riga di ringraziamento per tutti coloro che hanno accompagnato la mia

crescita scientifica e professionale in questi tre anni. Come dimenticare i momenti di

frenetica attivita prima di partire per un congresso, o i momenti di sconforto quando

una soluzione che sembrava a portata di mano si e p oi rivelata sbagliata? Indimenticabili

sono soprattutto i momenti trascorsi scherzando e ridendo, come amici di lungo corso

che si ritrovano dopo tanto tempo a discutere del piu e del meno.

I miei ringraziamenti vanno dunque al prof. Carlo Poggi, che un p omeriggio di quattro

anni fa mi convinse a proseguire gli studi. In questo periodo ne e stata fatta di strada.

Un doveroso grazie al prof. Leone Corradi al quale, nel corso della stesura della presente

tesi, mi sono sempre potuto rivolgere con la certezza di ricevere qualche utile suggeri-

mento.

Desidero, ovviamente, ringraziare anche tutti i docenti del collegio e il prof. Giulio Maier,

coordinatore del Dottorato di Ricerca in Ingegneria delle Strutture, per le possibilita for-nitemi di approfondire le mie conoscenze nei vari campi dell’ingegneria strutturale.

Ringrazio, inoltre, l’ing. Valter Carvelli, prodigo di consigli, e l’ing. Stefano Mariani per

avermi risparmiato molta fatica avviandomi alla scrittura di questo lavoro in LAT E X.

Mi e impossibile nominare tutti coloro che dovrei ringraziare, cosı come non e mia in-

tenzione ridurre questa pagina ad un semplice elenco di nomi. Vorrei quindi dire, a tutti

coloro che non ho ringraziato esplicitamente, che la mia riconoscenza va anche a loro, al

di la di queste poche righe, scritte in fretta, ma con la consapevolezza di aver ricevuto

piu di quanto ho dato.



to my parents . . .



Contents

1 Introduction 1

1.1 Objectives and organization of the thesis . . . . . . . . . . . . . . . 1

1.2 Limit analysis of structures . . . . . . . . . . . . . . . . . . . . . . 2

1.3 The sequential limit analysis . . . . . . . . . . . . . . . . . . . . . 5

2 Limit analysis and computation of the limit load 7

2.1 Theory of plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Limit analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Limit theorems . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.3 Energy dissipation . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 The limit problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Finite element solution of the limit problem of shells 17

3.1 Finite element modelling . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Mesh Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 The axisymmetric shell element . . . . . . . . . . . . . . . . . . . . 23

3.4.1 Nodal variables . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4.2 Rigid-body motions and natural modes . . . . . . . . . . . 24

3.4.3 Natural strains in the element . . . . . . . . . . . . . . . . . 26

3.4.4 The element dissipation . . . . . . . . . . . . . . . . . . . . 27

3.5 The general shell triangular element . . . . . . . . . . . . . . . . . 29

3.5.1 Nodal variables . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5.2 Natural and cartesian strains . . . . . . . . . . . . . . . . . 31

3.5.3 Rigid-body motions and natural modes . . . . . . . . . . . 33

3.5.4 Natural strains in the element . . . . . . . . . . . . . . . . . 37

3.5.5 The element dissipation . . . . . . . . . . . . . . . . . . . . 43

3.5.6 Nodal equivalent loads . . . . . . . . . . . . . . . . . . . . . 45

3.5.7 The minimization procedure . . . . . . . . . . . . . . . . . . 46

i



4 Test example: simply supported plate 49

4.1 Geometry and mechanical characteristics . . . . . . . . . . . . . . . 50

4.2 Mechanism model . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2.1 Thin plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2.2 Thick plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Finite element limit analysis . . . . . . . . . . . . . . . . . . . . . . 544.3.1 Mesh variation . . . . . . . . . . . . . . . . . . . . . . . . . 544.3.2 Thickness variation . . . . . . . . . . . . . . . . . . . . . . . 55

4.4 Post collapse behaviour . . . . . . . . . . . . . . . . . . . . . . . . 594.4.1 Simply supported plate . . . . . . . . . . . . . . . . . . . . 594.4.2 Simply supported plate with in plane renstraints . . . . . . 62

5 Test examples: some shells 65

5.1 Cylinder S1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.1.1 Axisymmetric mechanism approach . . . . . . . . . . . . . . 68

5.1.2 The present approach . . . . . . . . . . . . . . . . . . . . . 705.2 Conical shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2.1 The TICC5 cone . . . . . . . . . . . . . . . . . . . . . . . . 775.3 Square tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.3.1 Axial load . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6 Conclusions 91

6.1 A critical survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.2 Future developments . . . . . . . . . . . . . . . . . . . . . . . . . . 92

A Strains from drilling modes 93

B Details on the compilation of the energy dissipation 99

C Details on the minimization procedure 109

C.1 Analytical integration to obtain K . . . . . . . . . . . . . . . . . . 109C.2 Minimization respect to χ . . . . . . . . . . . . . . . . . . . . . . . 114

ii



Chapter 1

Introduction

Plasticity is an important branch of structural engineering and many researchershave dealt with it in the twentieth century. For a long time theoretical deve-lopments, experiments and discussions have been led and now the topic is wellsettled. The applicability fields of the limit analysis have been defined, and inmany applications the method is used in order to predict the limit carrying ca-pacity of structures, or their behaviour respect to a limit state. Extensions tomaterials different from steel, such as alluminium alloy or anisotropic materials,have also been studied.The knowledge of the collapse load or at least of a more or less good approxi-mation has been a great step ahead in the fifties, but nowadays many structuresneed a design that can not leave out of consideration the behaviour after collapse.Examples of such structures are bumpers, energy absorber and any structure that

can be subject to impact and develops large deformations in the plastic field. Of-ten the design of such structures is strictly connected to the safety aspect, a topicthat is becoming always more important. In the last decade many publicationshave dealt with this topic, whose field of application is very wide. The readercan refer to some publications whose refences are reported in the bibliography:in particular [68] deals with the crashworthiness of vehicles, Tabiei and Wu pre-sented in [117] a study on the roadside safety structures, while in [101] Ravalardet al. applied the prototypage virtuel to the crash of railway automotives.

1.1 Objectives and organization of the thesis

The aim of this work is to provide a useful tool for the prediction of the limit loadof shell structures and of their behaviour after the collapse. In order to obtainthis goal a finite element program based on the method of limit analysis has beendeveloped using the approach introduced by Capsoni and Corradi in 1995 [29].A sequence of limit analysis is carried-out in order to obtain the post-collapse

1



Chapter 1. Introduction

behaviour of the shell structure.After a brief review of the limit analysis of structures and of the sequential limit

analysis method presented in this chapter, the fundamentals theorems of plasticityand the approach used are explained in chapter 2. Chapter 3 deals with the for-mulation of the two shell elements used: an axisymmetric and a general triangularshell element based on the TRIC element introduced by Argyris in 1997. Finallysome examples and calculations are presented in chapter 4 for simply supportedplates and in chapter 5 for various shells geometries, compared to experimentaldata when available. Some conclusions and future developments are indicated inchapter 6 while some details of calculations are reported in the appendixes. Inthe bibliography many references to articles and books on the subject of plasticityand limit analysis are reported. According to the author the knowledge of theplasticity theory development can be very useful in the study of new and moreefficient methods of analysis. These references are not exhaustive of the subjectbut can provide a wide background on the topic of plasticity, limit analysis and

on the numerical methods used up to now.

1.2 Limit analysis of structures

In plasticity the yield criterion plays a central role: it is with this tool that thedesigner can know if a structure, or a part of it, develops plastic strains. The firstyield condition for metals was proposed in a series of papers from 1864 to 1872by Tresca, who stated that a metal yields plastically when the maximum shearstress attains a critical value. In the following years the theory was formulatedby St. Venant and Levy who introduced the basic constitutive relations for rigidperfectly plastic materials in plane stress and in three dimensions. The flow rule

was also introduced. In 1913 von Mises published a paper where his widely usedpressure-insensitive yield criterion was described. In 1924, Prandtl extended theSt. Venant-Levy-von Mises equation for the plane continuum problem to includethe elastic component of strain, and Reuss in 1930 carried out this extension tothree dimension. In 1928, von Mises generalized his previous work to include ageneral yield function and discussed the relation between the direction of plasticstrain rate and the regular yield function, thus introducing the concept of yield

function as a plastic potential . In the same period Prandtl attempted to formu-late general relations for hardening behaviour, and Melan, in 1938, generalizedthe foregoing concepts of perfect plasticity and gave incremental relations for har-dening solids with regular yield surface.In an independent way, in 1949, Prager arrived at a general framework, similar to

that proposed by Melan, in which the yield function and the loading-unloadingconditions were precisely formulated. In 1958 this framework was extended byPrager to include thermal effects by allowing the yield surface to change its shapewith temperature.In 1951, Drucker proposed the material stability postulate, a significant concept

2



1.2. Limit analysis of structures

of work hardening that permitted to treat in an unified manner the plastic stress-strain relations and many other fundamental aspects of the subject.

Before these developments, an important contribution to the topic was suppliedby the Steel Structure Research Committee (UK), set up in 1929 in order to bringsome order into the design of steel structures. Two years after the Committeeformation, a Code of Practice was published in the First Report that formed thebasis of the British Standard Specification n. 449. In the 1939 the Committeeconcluded that the method of design inherent in the Code of Practice was almost

entirely irrational and therefore incapable of refinement. It has to be noted thatthe BS449 rules gave, in many cases, a safe method of design, based upon thosetheorems of the plastic theory well known nowadays but not clearly articulatedin the ’30s.J. Baker, Technical Officer in the Committee until 1936, Professor at Bristol Uni-versity from 1933 until 1943 and in Cambridge since 1943, was at the head of one of the more active and important research groups, operating in Cambridge

and that had members like J.W. Roderick and M.R. Horne. Their first completeexperimental analysis of the single-bay rectangular portal frame was publishedin 1950, but the accompanying theory had been done on a mechanical approach,without the guidance of any general principle. The sophisticated, but clear, con-cept of virtual work had not yet found its application into such analysis. Thischanged slowly from the publication in 1948 of a work by Freiberg (in Russian)who stated the basic principles that are needed. It appears however that theearliest reference to the theorems of limit analysis was probably due to Gvozdev(1936). Precise formulations of the two fundamentals theorems of limit analysiswere given by Drucker, Greenberg and Prager in 1951-52 for an elastic-perfectlyplastic material and by Hill (1951-52) from the point of view of rigid-ideally pla-stic materials.

The application of the upper and lower bound theorems has been for a long timerestricted to very simple structures, in most cases only one-dimensional problems.The subsequent introduction of the discretization techniques and the simultane-ous growth of the calculations tools led to an impressive development of numericalmethods for the limit analysis of structures. Mathematical programming [83] wassoon recognized as a suitable tool for the solution of rigid-plastic problems, andthe analysis of beams and frames was formulated for a long time in terms of lin-ear programming theory [85, 91, 116, 118, 126]. For continuous solids the use of this method is subject to a piece-wise linearization of the yield condition thatcan introduce more or less meaningful errors, and the great number of variablesintroduced by linearization reduces the effectivenes of the approach.On the contrary, the linearization can be avoided and the resulting problems canbe solved with major accuracy by means of the nonlinear programming theory.Some finite elements based on mixed functionals have been proposed [67] andthe favorable ratio accuracy/computation charge played an important role in thesuccess of this approach.In the last decade new methods have been proposed for the solution of limit

3




problems based on the kinematic theorem: Sloan and Kleemen [111] obtainedvery good solutions for many engineering interesting cases using the mathemati-

cal programming and finite elements that can account for non-continuous velocityfields. Jiang has recently proposed a different method that lead the formulationof the limit problem to the optimization of a regular functional solved using aniterative procedure already applied to visco-elastic problems [67].In 1991, Huh and Yang [61] proposed a new method for the calculus of limit solu-tions of a plane stress problem, but their method can also be applied to the moregeneral three dimensional problems. Using a duality theorem they defined con-vergence from above and from below to the exact solution. The non-smoothnessof the original problem was avoided by introducing a small quantity δ such aswhen δ → 0 the solution of the original problem is recovered. The robustness andthe rate of the convergence of the computational algorithm has been theoreticallydemonstrated [21,127] and succesfully tested [61].Finally Capsoni and Corradi [29] and Liu et al. [81] separately proposed two ap-

proaches very similar: their are based on a dissipation function expressed directlyin terms of the plastic strains while the normality rule can be considered as acostraint between the components. This allows to obtain the collapse multiplieras the free minimum of a convex function and the solution is obtained by meanof a iterative procedure.The approaches proposed by Huh & Yang and by Capsoni & Corradi differ fromthe method used to deal with the non-smoothness of the dissipation functional.In [61] a small positive coefficient δ was introduced so that the original problemcorresponds to δ → 0, while Capsoni & Corradi proposed to set as rigid the e-lements that do not exhibit plastic flow. This approach avoids the outer iterationon δ needed in [61] and permits to reduce the problem size when rigid elementsare imposed. Unfortunately an efficient way to impose these constraints has not

yet been found and in some cases the efficiency of the method can be reduced.In the author’s knowledge these new methods have been used in most cases toobtain the limit load of simple structures, like plate or plane strain/stress pro-blems [33]. Recently Corradi and Vena published a paper related to the limitanalysis of anisotropic materials using the Tsai-Wu criterion [34, 44, 119], butthe analyses remain confined to the achievement of the collapse load while thepost-collapse behaviour of the structure remain unknown. An attempt to ob-tain the post-collapse behaviour of rigid-plastic structures has been proposed bySeitzberger and Rammerstorfer: in [109,110] they proposed an older idea [53,59]consisting in a sequence of limit analyses applied to a mesh updated, consequentlyto the velocity field obtained in the previous solution. Their procedure was appliedto axysimmetric structures only, while for more complex geometries an iterativetechnique was used.

4



1.3. The sequential limit analysis

1.3 The sequential limit analysis

Despite its semplicity, sequential limit analysis has not been widely used for thestudy of the post-collapse behaviour of structures. The method is based on asequence of limit analyses carried out on updated meshes: being the first limitanalysis performed on the original mesh, as a results the limit load and the kine-matic solution are obtained. The velocity field can be seen as instantaneous nodalvelocities of the structure in a Lagrangian coordinate system, therefore it can beintegrated over a small time interval to obtain a small displacement vector. Thesuperposition of these displacements to the original mesh provides an updatedgeometry that can be used for the next iteration. This process is repeted to forma sequence leading to the solution of a large displacement problem.Sequential limit analysis has been originally proposed for the study of framesby Horne and Merchant [59]. Some applications by Gavarini, Horne and Morriscan be found in [53, 60]. Recently the method has received a growing interest,

it has been applied by Yang [128] in order to study the behaviour of some trussstructures, while Seitzberger and Rammerstofer employed it to simulate the largedeformation crushing behaviour of axisymmetric shells [110]. Since in limit ana-lysis elastic strains are neglected, only the rigid-plastic collapse curve is obtainedbut this is a minor limitation in the study of structures that undergo large defor-mations and for which the elastic contribution is negligible.Different methods can be employed to predict the post-collapse behaviour: themost complete simulation is provided by large displacement, incremental analysis

performed keeping track of all types of informations, elastic and plastic strains.This method, implemented in various commercial F.E. packages, has high com-puting cost and often suffers of numerical instability problems. Other methodslike mechanism analysis can be employed to predict the post-collapse curve. They

follow the evolution of elementary mechanism consisting of circumferential hingeswhere bending dissipation is concentrated, while the regions in between experiencemembrane flow only [2, 3, 43]. However, such a procedure only considers axisym-metric collapse modes and axisymmetric geometric imperfections. Furthermore,the change in shape of the mechanism can hardly be followed to great accuracy.Moreover some structures do not collapse in axisymmetric way but could exhibitdifferent collapse modes such diamond shape [86–88,90,99,100] or circumferentialwaves [105].At the end, the sequential limit analysis seems to be a good balance between com-putationally efficiency, numerical stability and accuracy of results. It by-passesthe tedious and computationally demanding task of keeping track of stresses andstrains, typical of the incremental approach, remaining globally stable whatevermaterial and geometry is considered. The difficulties that arise during the elasto-

plastic incremental analyses, when dealing with imperfection sensitivity structuresor with softening materials, do not affect the stability of the sequential limit ana-lysis method, allowing it to be more efficient and to obtain the post-collapse curvewith a minor number of increments.

5




6



Chapter 2

Limit analysis and

computation of the limit

load

From the structural point of view strength and ductility are the most importantmaterial characteristics. When the stress in a point of the material reaches limitstrength, then rupture or fracture can occur (brittle materials), or some plasticstrains can develop (ductile materials). Structures built with ductile materialscan deform in a considerable way before collapse and often their crisis is dueto the impossibility to achieve an equilibrium configuration rather than materialrupture. This leads to the important consideration that the collapse load, in many

real situations is independent from the load history and its value can be obtainedin a direct way by means of limit analysis. The aim of this chapter is to presentthe theoretical bases and assumptions on which the calculations are performed.In section 2.1 some hypotheses on the theory of plasticity are recalled and themain theorems are illustrated. Particularization to the limit analysis of structuresis presented in section 2.2 while in the last section of this chapter a formulation of the limit problem suitable for a finite element analysis is presented. The solutionstrategy and the finite element adopted are discussed in chapter 3.

2.1 Theory of plasticity

Let’s consider a body subject to volume forces F in Ω and surface pressures f on∂ f Ω; the surface ∂ uΩ is fixed. The following hypotheses are assumed:

• material is isotropic;

• quasi-static response of the structure is considered so that any dynamic

7



Chapter 2. Limit analysis and computation of the limit load

effects are neglected;

•temperature is ignored;

• elastic properties are not influenced by the load story;

• displacement gradient are small so that kinematic relations are linear.

Under previous assumptions the strains in a body can be regarded as the sum of the elastic and plastic part while the stresses depends from elastic strains only:

ij = eij + pij (2.1a)

σij = Dijklekl (2.1b)

where Dijkl is the elastic stiffness tensor. Latin subscripts are employed to identifythe tensor components and run over the numbers 1,2,3 related to the cartesianaxes.Analogous relationships can be written using rate terms:

ij = eij + ˙ pij (2.2a)

σij = Dijklekl (2.2b)

where a dot over the relevant symbol stands for a time derivative. In order toobtain an analytical formulation of the problem some definitions are needed:

• an elastic instantaneous domain, or yield surface that can recognize stressstates related to elastic or plastic situations;

• a flow rule governing the plastic strain rates.

The elastic domain is defined by means of one or more yield functions as follow:

φα (σij, χh) ≤ 0 , α = 1, . . . , Y (2.3)

where χh are internal variables governing the hardening of the material.Strain rates are assumed to be normal to the elastic domain by means of thefollowing flow rule:

˙ pij =

Y α=1

∂φα

∂σijλα (2.4)

The admissible alternatives are:

λα ≥ 0 if φα = 0 and φα = 0 ; λα = 0 otherwise (2.5)

From the above equations it can be concluded that:

• if relations (2.3) are satisfied as strict inequalities, the material behaviourpurely is elastic, and the stress state is described by a point into the elasticdomain;

8



2.1. Theory of plasticity

• plastic strains can develop only when the stress state corresponds to a pointon the boundary of the elastic domain (2.3);

• equations (2.4) and (2.5) describe plastic strain rates normal to the elasticdomain at the stress state point. The direction of the normal is uniquelydefined if only one yield function is equal to zero; otherwise plastic strainrates ˙ pij have to be contained in the cone defined by the normals in thepoint identifying the stress state.

Relations (2.4) and (2.5) can be obtained assuming Drucker’s inequality:σij − σ∗ij

˙ pij ≥ 0 ∀ σ∗ij:φα

σ∗ij, χh

≤ 0 (2.6)

where σij is the stress on the yield surface generating the plastic strain rate ˙ pijand σ∗ij is an admissible stress state.Power dissipation can now be defined as:

D ( ˙ pij) = σij ˙ pij (2.7)From simple geometrical considerations it can be seen that the dissipation dependson the plastic strain rates ˙ pij only. In fact if the surface (2.4) is stricly convex,a given normal direction defines uniquely the associated stress point. Otherwiseif the elastic domain has a flat portion of the boundary, then the stress stateassociated to the plastic strain rate is not uniquely defined, but his projectionon the normal direction is the same within the whole flat domain and thus thedissipation is still uniquely defined.It remains to be decided what yield function to use: in the case of an isotropicmaterial it depends only on the stress invariants:

φ (I 1, I 2, I 3) = 0 (2.8)

where I j are the invariants of the stress tensor defined as:

I 1 = σii (2.9a)

I 2 =1

2

I 21 − σijσij

(2.9b)

I 3 =1

6

2σijσjkσki − 3I 1σijσij + I 31

(2.9c)

In (2.9) the Einstein’ summation convention on repeated indexes is adopted.Plastic deformations do not produce volume changes in metal materials thereforeequation (2.10) must be satisfied for any admissible velocity field:

˙ pii = 0 (2.10)

As a consequence the yield surface has to be independent from the mean hydro-static tension σii/3 and it is natural to suppose that the yield function dependson the deviatoric stress tensor sij expressed by:

sij = σij − 1

3δijI 1 (2.11)

9




where δij is the Kronecker’s tensor.Subject to this hypothesis the yield function for an isotropic material can be

expressed in terms of the deviatoric stress tensor invariants J 2 and J 3:φ (J 2, J 3) = 0 (2.12)

where

J 2 =1

2sijsij (2.13a)

J 3 =1

3sijsjkski (2.13b)

One of the most used and accurate yield function for metals is described by thevon Mises criterion: plastic strains can evolve when the octahedral shear stress

τ oct =

23 J 2 reaches the limit value

23 τ 0. The von Mises yield function is

expressed by:

φ (J 2) = 3J 2 − σ0 (2.14)where σ0 =

√3τ 0 is the uniaxial yield stress.

2.2 Limit analysis

In section 2.1 the relations governing an elastic-plastic problems have been pre-sented. It is clear that the solution of a such problem requires an incrementalprocedure due to the irreversible nature of elastic-plastic behaviour. Such analysiscan be very complicated, and the computational effort is not negligible. In manyengineering practical applications the limit load of the structure under conside-

ration is the most important information required by the designer: its value canbe obtained by means of direct methods such as limit analysis. It provides, in adirect way, the value of the collapse load avoiding the computationally demandingprocedure of the elasto-plastic incremental method. According to the approachused, statically or kinematically admissible, the value obtained corresponds tothe lower bound or to the upper bound multiplier of the base loads applied. Be-cause of the hypotheses assumed, and reported in subsection 2.2.1, the multipliercalculated by means of the limit analysis is not the actual collapse load of thereal structure but an approximation. Neverthless the limit analysis provides auseful tool in order to predict the ultimate carrying capacity of a structure andits collapse mechanism.In the following the main theorems of the limit analysis are remembered.

2.2.1 Basic assumptions

As mentioned before, the limit or collapse load is defined as the plastic collapseload of an idealized structure for which some assumptions are made:

10



2.2. Limit analysis

• plastic deformations can increase without limits;

•no hardening or softening is considered (ideal or perfect plasticity);

• changes in the geometry are negligible, hence the geometrical description of the structure remains unchanged during the deformation at the limit load.

The small deformations assumption allows the use of the virtual work principle(2.15):

∂ fΩ

f T

u∗dx +

Ω

FT u∗dx =

Ω

σT ∗dx (2.15)

where the ()∗ quantities form a compatible set while f , F and σ form an equili-brated set. Engineering notation is used, therefore:

σ = σx, σy, σx, τ xy, τ yz , τ xzT (2.16a)

= x, y, x, γ xy, γ yz , γ xzT

(2.16b)

The virtual work principle, and its rate form obtained substituting the finitequantities with their rates, may be used to obtain the theorems of limit analysispresented in section 2.2.2

2.2.2 Limit theorems

In the mechanics of deformable solids three basic relations must be satisfied fora valid solution of a problem, namely the equilibrium equations, the constitutiverelations and the compatibility equations. Three solution sets can be associated tothe previous relations: a statically admissible set S , the constitutively admissibleset C = Cσ

∩C and the kinematically admissible set K . The set Cσ contains

all the solutions satisfying equation (2.3) while in C the solutions have to satisfyequations (2.4) and (2.5). If the intersection of the sets S , C and K is empty, thereis no solution. If it consists of a single point, the solution is unique. Otherwisethere is a set of feasible solutions, one of which may be the most preferred. Thelimit analysis considers only two sets at a time and the search of the optimalsolution is facilitated by an objective function. In the primal formulation of alimit analysis problem, the optimal solution corresponds to an extreme point inS ∩C σ. All the non-optimal solutions furnish a multiplier lower than the limitone. On the contrary, the optimal solution of the dual formulation correspondsto an extreme point in K ∩C . In this case the non-optimal solutions furnish amultiplier greater than the limit one.

Statically admissible stress field and lower bound theorem

A statically admissible field is characterized by the following properties:

• the equilibrium relations are satisfied;

11




• the stress boundary conditions are also satisfied;

•no part of the body violates the yield condition.

Load associated to a statically admissible field cannot be greater than the collapseload, therefore the lower bound theorem may be stated as follows:

Theorem 2.2.1. If an equilibrium distribution of stresses σE can be found which

balances the body forces αF in Ω and the surface tractions α f in ∂ f Ω and it is

everywhere internal to the elastic domain, φ

σEij ≤ 0, then the body subject to

loads αF and α f will not collapse.

Obviously α ≤ s where α is the multiplier of the static admissible problemand s the exact solution of the limit problem.

Kinematically admissible velocity field and upper bound theorem

A kinematically admissible velocity field ∗ is characterized by the following pro-perties:

• strain rate and velocity compatibility equations are satisfied;

• the homogeneous velocity boundary conditions are also verified;

Equating the external rate of work Π (u∗) to the internal dissipation power D (∗)a base load multiplier β can be found so that applied loads are always greater orequal to the actual limit load:

β Π (u

∗

) = D (∗

) (2.17)where

D (∗) =

Ω

σT ∗dx (2.18a)

Π (u∗) =

Ω

FT u∗dx +

∂ fΩ

f T u∗dx (2.18b)

The upper bound theorem may be stated as follows:

Theorem 2.2.2. If a compatible mechanism of plastic rate deformations ∗ and

velocities u ∗ is assumed which satisfies the kinematics boundary conditions on

∂ uΩ, then the loads β F and β f determined equating the rate of external work to

the rate of internal dissipation will be either greater or equal to the actual limit

load.

12



2.2. Limit analysis

The kinematic multiplier β is always greater than the limit multiplier s exceptfor the optimal solution that furnishes β = s.

It must be noted that the lower bound theorem considers only the equilibrium andthe yield conditions while the upper bound theorem considers only the velocityfield and the energy dissipation. A suitable choice of stress and velocity fieldsallows to obtain sufficiently close bounds of the limit load for the problem underconsideration.

2.2.3 Energy dissipation

The use of the upper bound theorem requires the definition of the energy dissi-pation of the structure for a given strain rate field. The specific dissipation isdefined as:

D = σT (2.19)

where the stress tensor σ satisfies the yield condition

φ (σ) = f (σ) − σ0 = 0 (2.20)

f and σ0 in (2.20) depend on the yield criterion used while, because of the rigid-plastic assumption, represents the plastic strains rate, normal to the yield boundby means of the flow rule.In the following the von Mises criterion is assumed: the yield function is thereforestrictly convex, continuously differentiable and the associated flow rule permitsto describe the plastic strain rates as functions of plastic flow. In particular theyield function is expressed in terms of the second deviatoric stress invariant (eq.(2.13a)):

f (s) = 3J 2 = 12 sT Rs (2.21)

where R is the diagonal constant matrix defined in (2.22) and s the deviatoricstress tensor.

R = diag 3, 3, 3, 6, 6, 6 (2.22)

Plastic deformations occur only when the stress state point is on the boundaryof the elastic domain (2.3) and the flow rule (2.4) imposes the normality betweenthe boundary and the plastic strain rate vector. Therefore:

φ (s) = 0 ⇔ f (s) = σ0 (2.23)

and:

=∂φ

∂ s

λ = nλ (2.24)

where represent the deviatoric strain tensor.It is easy to verify using eq. (2.23) that the normal n can be expressed as:

n =1

2σ0Rs (2.25)

13




A finite element solution of the rigid-plastic problem by means of the upper boundtheorem requires the expression of the dissipation power in term of plastic strains.

In order to achieve this expression some operations are performed using eq. (2.24).Let Θ be a symmetric matrix, therefore a quadratic form in the plastic strainscan be written:

T Θ =λ2

4σ20

sT RΘRs (2.26)

If a matrix Θ can be found satisfying the condition

sT RΘRs = 4σ20 ∀ s such that φ (s) = 0 (2.27)

one obtains:λ2 = T Θ (2.28)

The condition of plasticity (2.23) implies sT Rs = 2σ20 and this is verified for any

matrixΘ

such that RΘR = 2R (2.29)

The solution of equation (2.29) leads to the following relation:

Θ = 2R−1 =1

3diag 2, 2, 2, 1, 1, 1 (2.30)

and the dissipation power can be expressed in terms of plastic strains only byvirtue of eq. (2.23) and eq. (2.28):

D = σT ∂φ

∂ σλ = σT ∂f

∂ σλ = σ0λ = σ0

T Θ (2.31)

The expression (2.31) for the dissipation is the base for the limit analysis by means

of the finite element method.The use of the deviatoric stress tensor bypasses the problem of a singular matrixin the definition of Θ, a problem solved in a different way in [31].

2.3 The limit problem

As seen in section 2.2 the collapse load multiplier can be obtained as the maximumof the lower bound solutions, or as the minimum of the upper bound multipliers.In a finite element approach as that used in this work, the upper bound theorem ismore useful than the lower bound one because it is easier to impose a kinematicallyadmissible velocity field than an equilibrated stress field. Moreover, an expression

of the dissipation power in terms of plastic strain rates has been obtained insubsection 2.2.3 and this can be used to obtain the multiplier in an upper boundapproach. In this section an alternative form of the upper bound theorem will bepresented and a proof of the statement will be provided.Considering the upper bound theorem reported in subsection 2.2.2 it can be noted

14



2.3. The limit problem

that both the dissipation power and the rate of work of external load dependlinearly on the mechanism amplitude; hence this is arbitrary and it can be fixed

imposing Π (u) = 1.Moreover plastic deformations develop at constant volume, therefore the limitproblem can be expressed as follows:

s = min,u

Ω

D () dx (2.32a)

subject to = su in Ω (2.32b)

µT = 0 in Ω (2.32c)

u = 0 on ∂ uΩ (2.32d)

Π (u) =

Ω

FT udx +

∂ fΩ

f T udx = 1 (2.32e)

where D () is defined in eq. (2.31) and

µ = 1, 1, 1, 0, 0, 0T (2.33)

Equations (2.32b), (2.32c) and (2.32d) represent respectively the compatibilityequations, the incompressibility and the boundary velocity conditions. The pro-blem (2.32) is convex, but where no plastic strains occur, D () is not differentiableand as a consequence the functional is not stationary at solution.In order to prove the statement (2.32) the Lagrangean functional L of the problemis introduced:

L (, u,σ, P , α) = Ω D () − σT ( − su) + P µT dx − α [Π (u) − 1] (2.34)

The functional (2.34) is differentiable respect to all variables but . Its optimalitycondition reads:

δL ≥ 0 subject to u = 0 on ∂ uΩ (2.35)

Relation (2.35) must hold an equality when variations respect σ, P and α areconsidered as required by (2.32).Stationarity respect to u leads to the following expression:

δuL =

Ω

σT δ (su) dx − α

Ω

FT δudx +

∂ fΩ

f T δudx

= 0

∀δu = 0 on ∂ uΩ

(2.36)

If variation respect to the plastic strains rate is considered the following relationis obtained:

δL =

Ω

δD () − (σ − P µ)

T δ

dx ≥ 0 ∀ δ (2.37)

15




In order to calculate δD () it is necessary to distinguish between the regionΩP , where plastic strains occur, and the rigid part ΩR: in ΩP the dissipation is

differentiable and eq (2.37) holds as an equality leading to

σT = sT + P µT (2.38)

In fact δD =

∂ D/∂ T

δ and

∂ D

∂

T

= σ0T Θ T Θ

= σ0T Θ

λ= σ0nT Θ (2.39)

and recalling equations (2.25) and (2.30):

∂ D

∂ T

= sT

(2.40)

Relation (2.37) therefore holds as an equality if equation (2.38) is satisfied in anypoint of Ω.In ΩR the dissipation is zero and its variation due to a plastic strain rate is equalto the dissipation calculated in the deformed state:

δD = D () = sT δδ (2.41)

and the following relation is obtained:

[sδ − (σ − P µ)]T

δ ≥ 0 ∀ δ such that µT δ = 0 in ΩR (2.42)

If the lagrangean multiplier fields σ (x) and P (x) are interpreted as stresses andhydrostatic pressure distributions, equation (2.36) imposes the equilibrium of thebody subject to the basic loads multiplied by α.Moreover equation (2.38) has been obtained imposing the normality rule (2.4)therefore the stress tensor σ is everywhere contained within the limit domainin ΩP . Equation (2.42) compared with the Drucker’s postulate (2.6) impliesφ (σ) ≤ 0 in ΩR.Therefore the optimal value of α is both statically and kinematically admissiblebeing, for eq. (2.32e):

α∗ = α∗Π (u) =

Ω

σT ∗ ∗dx =

Ω

D (∗) dx = β ∗ (2.43)

where the subscript ∗ refers to the solution values.

16



Chapter 3

Finite element solution of

the limit problem of shells

3.1 Finite element modelling

Let us consider a discretization of the body Ω in ne finite elements and nn nodes.Each element is connected to nne nodes and in each node nv nodal variables(displacement or rotations) are unknown. These nodal variables can be defined inthe global reference system (GRS) or in a local reference system (LRS) associatedto the element into consideration.Let us define:

Definition 1. (O,x,y,z) the global reference system (GRS) of the structure;

Definition 2. (G, x, y, z) the local reference system (LRS) of the finite element

considered.

The displacements can be expressed in either reference system by means of the following relation:

v = Tv (3.1)

where the vectors v and v represent the displacements in the LRS and GRSrespectively and T is the transformation matrix. Because of the ortogonality of the reference systems T−1 = TT and

v = TT v (3.2)

Let us collect the nodal variables of the whole structure in the vector u if referredto the GRS or in the vector u if referred to the LRS. If only nodal variablesrelated to an element e are considered they are grouped in the vectors ue and u

e

17



Chapter 3. Finite element solution of the limit problem of shells

referred to the GRS and to the LRS respectively.Let us consider now an element e with all its nodal variables collected in a vector

u

e: the displacement over the element surface (or axis) can be expressed by meansof a matrix of shape functions N (x):

ue (x) = N (x) u

e (3.3)

Strains over the element are obtained with the hypothesis of small displacement,so ij = 1

2(si,j + sj,i). When the derivation process is applied to the matrix

N (x) one obtains:e (x) = Be (x) u

e (3.4)

Local coordinates have been used because calculations are generally easier in theLRS than in the GRS.In computing the dissipation only modes that entail plastic strains are useful,while rigid body motions have only to be considered from a kinematical point of view. The natural formulation proposed in the sixties by Argyris is appropriatebecause it is based on the separation of rigid-body motions from natural modes.The sum of rigid-body motions and natural modes has to be equal to the numberof nodal variables of the element and they can be collected in the vector ρe:

ρe =

ρ0

ρN

(3.5)

where the sub-vector ρ0 contains the rigid-body motions and the sub-vector ρN the natural modes.Normally natural modes are referred to a natural reference system (NRS) definedaccordingly to the element used: for example for a triangular shell element thenatural axes are coincident with the sides of the element it self. If strains areexpressed in the NRS there exist a matrix T so as

e (x) = T−1e e (x) (3.6)

and they can be obtained from the natural modes:

e (x) = be (x)ρN (3.7)

In all the expressions a bar over a symbol denote a quantity in the NRS. Supposethat the relation between the natural modes ρN and the nodal variables u

e isknown, therefore plastic strains (3.4) can be expressed in terms of the naturalmodes only. Using equations (3.6) and (3.7) one obtains:

e (x) = be (x)ρN (3.8)

where the vector of natural strains

ρN = Ceu

e (3.9)

18



3.2. Solution procedure

has s = m−nr entries, being m the number of nodal variables and nr the numberof rigid-body motions of an element.

Matrix be (x

) can be obtained from the natural counterpart multiplying it by thetransformation matrix T:be (x) = Tebe (x) (3.10)

3.2 Solution procedure

Keeping in mind the formulation (2.32) of the limit problem the power dissipationcan be obtained in terms of natural strains. Recalling equations (2.28) and (3.8)and observing that the matrix Θ has to be invariant respect to a coordinatetransformation being the material isotropic, one obtains:

λ = ρT N bT

e (x) TT

e ΘTebe (x) ρN = ρT N bT

e (x) Γebe (x) ρN (3.11)

where a kinematically admissible velocity field has been considered. The matrix

Γe = TT

e ΘTe depends only on the element position while matrix be, describingthe strains in the element, depends on the coordinates x.The dissipation of the whole structure can be found adding together the energydissipation of each element:

D =

nee=1

Ωe

σ0

ρT N b

T

e (x) Γebe (x) ρN dx (3.12)

Problem (2.32) can now be formulated in terms of finite element quantities:

s = minρN ,u

nee=1

Ωe

σ0

ρT N b

T

e (x) Γebe (x) ρN dx (3.13a)

subject to ρN = Ceu e = 1, . . . , ne (3.13b)

pT u = 1 (3.13c)

where:

• the incompressibility condition has not be written because it will be con-sidered in the obtaining of the appropriate matrix Θ;

• nodal variables are eliminated on the constrained boundary;

• vector p is the vector of equivalent nodal loads.

As suggested in section 2.3 equations (2.32) define the collapse multiplier as theminimum of a convex, but not everywhere differentiable function. The finiteelement domain has to be split in two part: a region E P where the elements

19




develop plastic strain (ρN = 0), and a region E R where the elements are onlysubject to rigid-body motions. In the first region the elements contribute to the

dissipation power and the function De is differentiable, while in E R the rigiditiesimpose some constraints to the nodal variables. These constraints can be seen asa set of linear equations:

u = G0u0 (3.14)

where the vector u0 collects all the independent nodal variables in the globalreference system.Natural straining modes can be obtained as follows:

ρN = Ceu

e = CeTeue = C

eTeG0eu0e = C0eu0e (3.15)

where the pedix e refers to a single element e. After substitution of ρN in eq.(3.12) the dissipation power becomes:

D =e

Ωe

σ0

uT

0eCT 0eb

T

e (x) Γebe (x) C0eu0edx

=e

Ωe

σ0

uT

0eΛ0eu0edx

(3.16)

and the discrete form of the limit problem can be written as:

s = minu0

D (u0) ∀ e in E P (3.17a)

subject to pT G0u0 = 1 (3.17b)

The lagrangean function reads:

L (u0; α) = D (u0) − α

pT G0u0 − 1

(3.18)

In equations (3.17) and (3.18) all the elements in E P have to be considered.Between the element nodal variables and all the nodal variables there exist arelation due to the assemblage operation simbolically written as:

ue = Leu (3.19)

For an illustration purpose of the solution procedure let us suppose that all theelements undergo plastic flow so that E R = 0 and G0 = I. Under this hypothesisthe lagrangean function is differentiable everywhere and must be stationary at

solution. Therefore the following non-linear problem is obtained:

∂L

∂ u0=

nee=1

Ωe

σ0Λ0e (x)

uT 0 Λ0e (x) u0

dx

u0 − αGT 0 p = 0 (3.20)

20



3.3. Mesh Updating

Equation (3.20) is solved iteratively with this procedure:

Hj =

nee=1

Ωe

σ0Λ

0e(x)

uT 0 Λ0e (x) u0

dx (3.21a)

u∗ = H−1j GT

0 p ; αj+1 =1

pT G0u0; u0(j+1) = αj+1u∗ (3.21b)

This procedure is ended whenαj+1 − D

u0(j)

D

u0(j)

≤ kd (3.22)

being kd an appropriate value defined by the user.In the procedure just illustrated the possibility of one or more rigid elements has

not been considered. If during the solution process one or more elements aredetected as rigid, and this can be done using the following check:

De

V e≤ ke (3.23)

the elements are eliminated from the E P domain and the constraints imposed bythe rigidities are applied changing in a suitable way matrix G0.In equation (3.23) De and V e are respectively the dissipation and the volume of the element e while the constant ke is an appropriate value defined by the user.It would be important to check the solution, to ensure that the minimum of thedissipation power is attained. Unfortunately, such a test can hardly be devised,due to a well-known property of rigid plastic collapse: a corollary to the limitanalysis theorems states that the stress state associated with a mechanism isdefined in the plastic portion only, and no information on stresses is available inthe rigid region. Therefore the solution should be considered as a kinematicallyadmissible value, bounding from above the plastic multiplier. However, numericaltests performed with different starting vectors produced the same final mechanismsuggesting the correctness of the procedure implemented.

3.3 Mesh Updating

The method proposed is based on a sequence of limit analyses performed onupdated meshes. This updating is done by multiplying the collapse mechanism,obtained solving the limit problem, by a constant k such as:

max |vj | ≤ umax ∀ j ∈ [1, nn] (3.24)

where vj is the displacement vector in the node j, umax a value defined by theuser and nn the number of nodes in the discretization.

21




Let us collect the nodal coordinates of the node j of the original mesh in thevector sj , therefore the coordinates of the updated mesh are:

supj = sj + kvj ∀ j ∈ [1, nn] (3.25)

A new limit analysis is then performed on the updated mesh and the procedureiterates up to the maximum number of steps defined by the user.

22



3.4. The axisymmetric shell element

3.4 The axisymmetric shell element

Axisymmetric shell structures, subject to axisymmetric loads and boundary con-ditions, can be analyzed with axisymmetric elements, but some limitations areimplicit in the procedure. As illustrated in [43, 46, 96] an axisymmetic approachcan only deal with axisymmetric collapse while other collapse mode, such as di-amond mode, cannot be recognized. Moreover only axisymmetric imperfectionscan be considered. In spite of these drawbacks, the use of axisymmetric elementspermits to reduce a shell structure to a mono-dimensional one, decreasing in aconsiderable way the number of the elements to be used and as a consequence thetime required to obtain the solution.Therefore an axisymmetric element, labelled AX2P , has been developed for theanalysis of collapse of rigid plastic shells, and in this section its main characteri-stics will be presented.

3.4.1 Nodal variables

The AX2P element is a 2 node linear axisymmetric shell element with 3 degreesof freedom per node formulated in natural coordinates in order to separate therigid body motions from the natural modes.Let us consider a global reference system (O,x,z) and a local reference system(G, x, z) where G is the centroid of the element, x axis is coincident with theshell axis and the direction of x is the same as the shell meridian (see fig. 3.1).Let le be the length of the element e and ζ a dimensionless coordinate such that

l

2

1

xz

x`,ζ

z`φ

G

w `¡

u `¡

β ¡ `

w `¢

u `¢

β2`

Figure 3.1: the AX2P element and the coordinate systems

−12

≤ ζ ≤ 12

. Due to the simplicity of the element the natural reference systemcan be taken coincident with the local reference system.The three degrees of freedom per node correspond to two displacements (axial

23




and radial) and to one rotation as reported in figure 3.2. These nodal variables

2

1

x

z

w

u β2

w ¡

u¡

β1

Figure 3.2: nodal variables of the AX2P element

are collected in the vector ue:

ue

(6×1)

= u1, w1, β 1, u2, w

2, β 2T (3.26)

or, if expressed in the global reference system:

ue(6×1)

= u1, w1, β 1, u2, w2, β 2 (3.27)

Vectors ue and ue are connected by the following relation:

ue =

T

T

ue = Teue (3.28)

where

T =

cos φ sin φ 0− sin φ cos φ 0

0 0 1

(3.29)

3.4.2 Rigid-body motions and natural modes

The axisymmetric shell element AX2P has six degrees of freedom but only five

modes imply dissipation, while one mode corresponds to the rigid motion in ver-tical direction. The degrees of freedom are grouped in the vector ρc:

ρc6×1

=

ρ01

ρN 5×1

24




where ρ01 is the rigid-body motion and the sub-vector ρN represents the naturalmodes.

In the following a mathematical definition of rigid-body motion and natural modeswill be given.

Rigid-body motion

In an axisymmetric problem the rigid body motion corresponds to a displacementin the vertical direction as pictured in figure 3.3.

ρ01 =u1 + u2

2=

u1 + u22

cos φ − w1 + w

2

2sin φ (3.30)

Figure 3.3: rigid-body motion of AX2P element

Natural straining modes

The five natural straining modes collected in the vector ρN are pictured in figure3.4 and their definition is given by equations (3.31).

q1 =w1 + w2

2=

u1 + u22

sin φ +w

1 + w2

2cos φ (3.31a)

q2 =−w1 + w2

2=

−u1 + u22

sin φ +−w

1 + w2

2cos φ (3.31b)

q3 =−u1 + u2

2

=−u1 + u2

2

cos φ

−−w

1 + w2

2

sin φ (3.31c)

q4 =β 1 + β 2

2(3.31d)

q5 =−β 1 + β 2

2(3.31e)

25




(a) natural mode q1

(b) natural mode q2

(c) natural mode q3

(d) natural mode q4

(e) natural mode q5

Figure 3.4: natural straining modes of AX2P element

3.4.3 Natural strains in the element

In order to obtain the strains into the element, Kirchhoff’s hypotheses will beconsidered. Therefore the element is intended to be used with thin shells whereshear strains are negligible.As introduced in subsection 3.4.1 the natural and local reference system are co-incident, therefore no relation between them has to be provided. Strains in theelement can be easily written if the nodal variables are expressed in the LRS. Thedisplacements in the elements are:

u

(ζ ) = u1 1

2 − ζ + u2 1

2 + ζ (3.32a)

w (ζ ) =w

1 + w2

2− β 1 − β 2

2+

5

4(w

2 − w1) ζ +

β 1 − β 22

ζ 2 + (w2 − w

1) ζ 3

(3.32b)

26




The membranal strains and curvatures are obtained as follows:

es =

1

le

du (ζ )

dζ (3.33a)

eθ =w (ζ )cos φ + u (ζ )sin φ

r (ζ )(3.33b)

χs = − 1

l2e

d2w

dζ 2(3.33c)

χθ = − sin φ

r (ζ ) le

dw (ζ )

dζ (3.33d)

Substituting equations (3.32) in (3.33) one obtains:

es = − 1

leu1 +

1

leu2 (3.34a)

eθ =12

− ζ sin φr (ζ )

u1 +12

− 54

ζ − ζ 3 cos φr (ζ )

w1 +ζ 2

2− 1

2 cos φ

r (ζ )β 1+

1

2+ ζ

sin φ

r (ζ )u2 +

1

2+

5

4ζ + ζ 3

cos φ

r (ζ )w

2 −

ζ 2

2− 1

2

cos φ

r (ζ )β 2

(3.34b)

χs = − 1

l2e

(−6ζw1 + β 1 + 6ζw

2 − β 2) (3.34c)

χθ = − sin φ

r (ζ ) le

−5

4− 3ζ 2

w

1 + ζβ 1 +

5

4+ 3ζ 2

w

2 − ζβ 2

(3.34d)

or in a more compact way: (ζ ) = Be (ζ ) u

e (3.35)

where (ζ ) =

es eθ χs χθ

T (3.36)

Taking into consideration the thickness of the shell the strains can be obtainedfrom the midplane quantity:

s (z, ζ ) = es + zχs (3.37a)

θ (z, ζ ) = eθ + zχθ (3.37b)

In the following the strains are collected in the vector (z, ζ ).

3.4.4 The element dissipation

The dissipation can be computed in terms of strains as obtained in equation(2.31):

D =e

De =e

Ωe

σ0λdx (3.38)

27



3.5. The general shell triangular element

3.5 The general shell triangular element

The analysis of shell structures more general than axisymmetric ones requires theuse of a different element able both to mesh every shell geometry and to accuratelydescribe plastic strains in all directions. As seen in section 3.1 a formulation innatural coordinates can be very useful and it has been verified that such elementshave a good behaviour in describing also complicated phenomena [15,19]. More-over, in order to have the widest field of application, the shell element has to beable to describe every shell geometry and this has led to the choice of a triangularelement.The element presented in this section is an evolution of the TRIC element al-ready formulated in natural coordinates by the Argyris work group. The TRIC isa 3-node shear deformable flat shell element developed in the late ’90s by Argyrisand his co-workers to simulate arbitrary isotropic and laminated composite shells.It represents the natural evolution of three previous elements, the TRUMP, the

TRUNC and the LACOT [7–16, 18, 19]. The main characteristics of the TRIC are the formulation based on the natural approach, the elimination of the lockingphenomena and its robustness and computational effectiveness compared to con-ventional isoparametric elements. Recently the convergence requirements havebeen theoretically proved be means of the non-consistent formulation proposedby Bergan and co-workers [22,24,25,51]. The original description of the elementcan be found in [18], while in the following the modified formulation used to adaptthe TRIC element to the rigid-plastic analysis will be presented accordingly tothe symbols introduced in sections 3.1 and 3.2.

3.5.1 Nodal variables

The TRIC element is a triangular shell element with 3 nodes and 6 degrees of freedom per node. The particular formulation allows the separation of the six rigidbody motions from the twelve natural modes that involve constant and higherorder deformations. These modes are the only responsible of the dissipation.Let us consider a triangular element, its vertex are numerated from one to threewhile the sides are called α, β , γ . Side α is opposite to vertex number 1, side β to vertex number 2 and side γ to vertex number 3. The length of each side isindicated by li where i = α,β,γ . Accordingly to section 3.1 the global referencesystem is referred to as (O,x,y,z) while the local reference system, associatedto every element is referred to as (G, x, y, z) where G is the centroide of theelement. The z axis is normal to the element surfaceThe six degrees of freedom per node correspond to three displacements and three

rotations as reported in figure 3.6. These nodal variables are collected in thevector u

e:

ue

(18×1)

= u1, v1, w1, θ1, φ1, ψ

1, u2, v2, w2, θ2, φ2, ψ

2, u3, v3, w3, θ3, φ3, ψ

3T (3.43)

29




1

2

3

∠α

∠γ

∠ββ

α

γ

y

x

zx’

y’

z’

G

Figure 3.5: the TRIC element and the local coordinate system

¡

¢

£ ¤

¦ ¤

¨ ¤

θ

φ ψ

Figure 3.6: nodal variables of the TRIC element

or in the global reference system:

ue(18×1)

= u1, v1, w1, θ1, φ1, ψ1, u2, v2, w2, θ2, φ2, ψ2, u3, v3, w3, θ3, φ3, ψ3T (3.44)

Vectors ue and ue are connected by the following relation:

ue =

T

T

T

T

T

T

ue = Teue (3.45)

where

T =

cos xx cos xy cos xzcos yx cos yy cos yzcos zx cos zy cos zz

(3.46)

30




3.5.2 Natural and cartesian strains

In this section the relations between natural and cartesian strains will be provided.

The natural reference system is referred to as ( α,β,γ ) and it is defined by thethree sides of the corresponding element.Strains in the global reference system are defined as:

ij =1

2(vi,j + vj,i) i, j = x,y,z (3.47)

where v is the cartesian displacements vector. The consistent strain and stressvectors are:

(6×1)

= xx, yy , zz , γ xy, γ xz, γ yz (3.48)

σ(6×1)

= σxx, σyy , σzz , σxy, σxz, σyz (3.49)

where γ ij are the engineering shear strains.

Membrane strains

Definition 3. Natural membrane strains m are defined as the three strains

measured in the natural coordinate system.

Membrane strains defined in the local and natural reference system are groupedin the vectors m and m:

m(3×1)

= xx , yy , γ xy (3.50)

m(3×1)

=

mα, mβ, mγ

(3.51)

In order to obtain the relations between cartesian and natural strains we get atriangle in the non deformed and deformed configuration:

∆s2(1 + mα)2 =∆x2(1 + xx)2 + ∆y2(1 + yy)

2

− 2∆x(1 + xx)∆y(1 + yy) cos(π

2+ γ xy)

but ∆x = ∆s cos α and ∆y = ∆s sin α, therefore with some simplifications:

(1 + mα)2 =(1 + xx)2 cos2 α + (1 + yy)

2 sin2 α

− 2(1 + xx)(1 + yy)cos α sin α cos(π

2+ γ xy)

Subject to hypotheses of small strains the value of γ xy is small and it can beassumed that cos(π2

+ γ xy) −γ xy . The above relation becomes:

(1 + mα)2 =(1 + xx)2 cos2 α + (1 + yy)

2 sin2 α

+ 2γ xy(1 + xx)(1 + yy)cos α sin α

31




£

¤

¥

α

∆ ¦

∆§

∆¨

£ ©

¤ ©

¥ ©

∆ ε§

∆ ε

# #

∆ ε¦

&

α

' ( )

γ 1 2 3 2

Figure 3.7: the TRIC element and the local coordinate system

If the second and third order terms are neglected a simple relation between carte-sian and natural strains can be obtained:

1 + 2mα = cos2 α + 2 cos2 αxx + sin2 α + 2 sin2 αyy + 2γ xy cos α sin α

Following the same procedure for the β and γ sides one obtains:mα

mβ

mγ

=

c2αx s2

αx cαxsαx

c2βx s2

βx cβxsβx

c2γx s2

γx cγxsγx

xx

yyγ xy

⇔ m = T−1m m (3.52)

where cαx = cos(αx) and sαx = sin(αx) are the cosine and sine of the anglebetween the side α and the local axis x. Matrix Tm corresponds to the membranalpart of the transformation matrix introduced in equation (3.6):

Tm =

sβxsγx

d1

sγxsαx

d2

sαxsβx

d3cβxcγx

d1

cγxcαx

d2

cαxcβx

d3

−sβxcγx+cβxsγx

d1− sγxcαx+cγxsαx

d2−sαxcβx+cαxsβx

d3

(3.53)

where:

d1 = (cβxsαx − cαxsβx) (cγxsαx − cαxsγx)

d2 = (cγxsβx − cβxsγx) (cαxsβx − cβxsαx)

d3 = (cαxsγx − cγxsαx) (cβxsγx − cγxsβx)

Transversal strains

Definition 4. Natural transversal strains s are defined as the three strains

measured transversally to the natural coordinate system.

32




εsα

α

α

x’

y’

εsα

εy’x’

εx’z’

Figure 3.8: the transverse strains

It is supposed that a face of the hypothetical triangular base prism deformsbut all the other faces’ angles remain right angles. From simple trigonometricconsiderations the following relations can be obtained:

γ sαγ sβ

γ sγ = cαx sαxcβx sβx

cγx sγxγ xz

γ y

z ⇔

s = T−1s s (3.54)

It is important to observe that three transverse natural strains depend on only twolocal shear strains and that the superscript −1 does not denote an inverse matrix,but it has been reported only for congruency with the symbology introduced insection 3.1.

3.5.3 Rigid-body motions and natural modes

The shell element TRIC has 18 degrees of freedom but only 12 modes implydissipation. The degrees of freedom are grouped in vector ρc:

ρc18×1

= ρ0

6×1

ρN 12×1

where the sub-vector ρ0 contains the rigid-body motions and the sub-vector ρN the natural modes.In the following a brief mathematical definition of rigid-body motions and naturalmodes will be given, for a more detailed definition see [16].

Rigid-body motions

In the 3D-space three translations and three rotations are allowed.

ρ06×1

=

ρ01 ρ02 ρ03 ρ04 ρ05 ρ06

T In the local reference system the translation modes are easily detected as the

33




¡

£ ¡

¤ ¡

¦ §

¡

£ ¡

¤ ¡

§

¡

£ ¡

¤ ¡

§

¡

£ ¡

¤ ¡

θ§

ρ

© ¡

£ ¡

¤ ¡

φ§

ρ !

¡

£ ¡

¤ ¡

ψ §

ρ "

Figure 3.9: rigid-body motions of TRIC element

centroid displacement:

ρ01

=u1 + u2 + u3

3ρ

02=

v1 + v2 + v3

3ρ

03=

w1 + w

2 + w3

3(3.55a)

Rotation motions in x and y can be obtained writing the equation of the planefor three points and using partial derivatives. For the definition of the rotationin the z axis see [17].

ρ04 =−x2w

1 + x3w1 + x1w

2 − x3w2 − x1w

3 + x2w3

2Ω(3.56a)

ρ05 =−y2w

1 + y3w1 + y1w

2 − y3w2 − y1w

3 + y2w3

2Ω(3.56b)

ρ06 =

−

u1xα + u2xβ + u3xγ + v1yα + v2yβ + v3yγ

4Ω

(3.56c)

where Ω is the area of the element, ui, vi, wi the displacements of vertex i in the

local reference system and xi, yi the projections of the side i onto the local axesx and y.

34




Natural straining modes

The 12 natural straining modes are subdivided into four categories:

ρN 12×1

= q0m qS qA qd

Axial modes The axial modes are grouped in the sub-vector q0

m and they aredue to the stretching of the corresponding side (see fig. 3.10). The straining iscalculated as the elongation of the side scaled by its length. If ui and vi are the

1/2 l∆ β

1/2 l∆ β

1/2 l∆ γ

1/2 l∆ γ

1/2 l∆ α

1/2 l∆ α

qmβqmα qmγ

Figure 3.10: natural axial straining modes of TRIC element

local displacement in x and y directions at node i the axial straining modes are:

q0mα

q0mβ

q0mγ

= 0 0 −cαx

lα

−sαxlα

cαxlα

sαxlα

cβx

lβ

sβx

lβ 0 0

−cβx

lβ

−sβx

lβ−cγx

lγ

−sγx

lγ

cγx

lγ

sγx

lγ0 0

u1v1u

2v2u3v3

(3.57)

Symmetric bending modes The symmetric bending modes are grouped inthe vector qS and are calculated from the symmetric part of the sides bending(see fig. 3.11). The nodal rotations are projected on the direction perpendicularto the side and then the symmetric part is evaluated.

If θi and ϕi are the rotations of the node i in the x and y directions the

straining modes are:

qSαqSβqSγ

= 0 0 sαx −cαx −sαx cαx−sβx cβx 023 024 sβx −cβx

sγx −cγx −sγx cγx 0 0

θ1

ϕ

1θ2ϕ

2

θ3ϕ

3

(3.58)

35




qSbq

Sa qSg

Figure 3.11: natural symmetric bending straining modes of TRIC element

Antisymmetric bending and shearing modes The antisymmetric bendingand shearing modes are grouped in the vector qA and they are calculated both

from the antisymmetric part of the bending and from the shearing part of thesides. The nodal rotations are projected on the direction perpendicular to theside and then the antisymmetric part is evaluated eliminating the rigid rotationsρ04 and ρ05.

q b

Aaq

b

Ab q b

Ag

Figure 3.12: natural antisymmetric bending straining modes of TRIC element

qAαqAβqAγ

=

0 0 sαx −cαx sαx −cαxsβx −cβx 0 0 sβx −cβxsγx −cγx sγx −cγx 0 0

θ1 − ρ04

ϕ1 − ρ05

θ2 − ρ04

ϕ2 − ρ05

θ3 − ρ04

ϕ

3 − ρ05

(3.59)

As pictured in figures 3.12 and 3.13 it can be seen that the antisymmetric modesare due to the bending qbA and shearing qsA part. Their value will be obtainedminimizing the dissipation energy of the structure.

36




q

Αα

1/2¡

¢

Αα

q¤

Αγ

1/2 ¥¦ Αγ

1/2 §¨ Αβ

q

Αβ

Figure 3.13: natural antisymmetric shearing straining modes of TRIC element

Azimuthal or drilling modes The drilling straining modes are grouped inthe vector qd and are calculated from the nodal rotations perpendicular to theelement plane (see fig. 3.14).

qdα

qdβ

qdγ

Figure 3.14: natural drilling straining modes of TRIC element

qdαqdβqdγ

=

ψ1 − ρ06

ψ2 − ρ06

ψ3 − ρ06

(3.60)

3.5.4 Natural strains in the element

As obtained in section 3.2 the energy dissipation can be calculated integratingthe specific dissipation over the element. The specific dissipation D depends onstrains, expressed either in local or natural coordinates. In this section a relationbetween natural modes and natural strains is obtained.Internal coordinates of the element are now introduced: these are useful in thedescription of the position of a point over the element.

ηα =Y αlα

ηβ =Y βlβ

ηγ =Y γlγ

ζ α =Z αhα

ζ β =Z βhβ

ζ γ =Z γhγ

37




¡

¢

η ,α α

£

η ,β β

£

η ,γ γ

£

ζ ,Ζ α α

ζ ,Ζ β β

ζ ,Ζ γ γ

¤

γ

¤

α

¤

β

Figure 3.15: internal coordinate of TRIC element

where the relation ζ α + ζ β + ζ γ = 1 holds. The jacobian matrix of the transfor-mation between ζ and η coordinates is here reported:

∂ζ i∂ηj

=

0 1 −1−1 0 11 −1 0

(3.61)

Strains from axial and bending modes

If a section perpendicular to the mid-plane of the shell in the undeformed stateremains perpendicular after the deformation occurs then their displacements innatural coordinates will be given by:

ui = u0i − z

j

∂w j

∂Y ii, j = α,β,γ (3.62)

where ui are the displacements parallel to the side i and wj the displacements

perpendicular to the element plane due to the symmetric bending mode qSj .Therefore the membranal strains in the natural reference system are expressed bythe equation 3.63:

mi = q0mi

−zj

∂ 2wj

∂Y 2i

i, j = α,β,γ (3.63)

where the subscript mi refers to the membrane strains in the i direction.The displacements in z direction are due to the symmetric and antisymmetricbending modes while the drilling modes impose only in-plane deformations. In

38




the following, symmetric and antisymmetric bending mode are treated separatelyfor the sake of simplicity.

Symmetric bending modes Displacements in z direction due to the sym-metric bending modes qS are described by the following expression:

wS =

1

2lαζ βζ γqSα +

1

2lβζ γζ αqSβ +

1

2lγζ αζ βqSγ (3.64)

The boundary conditions are verified, for example considering only the naturalmode qSα:

wS |ζβ=0 = 0 w

S|ζγ=0 = 0 (3.65)

∂w S

∂Y α|ζα=ζβ=0 = −1

2qSα

∂w S

∂Y α|ζα=ζγ=0 =

1

2qSα (3.66)

Along the sides β = 0 and γ = 0 the rotations are not constant but go to 0 as(1 − ζ α). The displacements in the plane of the element are obtained by meansof the equation (3.62) and a subsequent derivation along Y j leads to the naturalstrains:

uSj = −z∂w

S

∂Y j(3.67)

mSj = −z∂ 2w

S

∂Y 2j

(3.68)

and in explicit form, using the jacobian reported in equation (3.61):

mSα = zqSαlα

(3.69)

mSβ = zqSβlβ

(3.70)

mSγ = zqSγlγ

(3.71)

Antisymmetric bending modes As introduced in subsection 3.5.3 the ben-ding antisymmetric modes are a part of the whole antisymmetric mode. In orderto identify the shear and the bending part a bending parameter χ is introducedso that:

qbA = χqAqsA = (1

−χ)qA

Therefore the displacements in z direction due to the antisymmetric bendingmodes qbA are described by the following expression:

wA =

1

2lαζ βζ γ(ζ β − ζ γ)qbAα +

1

2lβζ γζ α(ζ γ − ζ α)qbAβ +

1

2lγζ αζ β(ζ α− ζ β)qbAγ (3.72)

39




The boundary conditions are verified, for example considering only qAα as in thesymmetric case:

wA|ζβ=0 = 0 w

A|ζγ=0 = 0 (3.73)

∂w A

∂Y α|ζα=ζβ=0 =

∂w A

∂Y α|ζα=ζγ=0 =

1

2qbAα (3.74)

Along the sides β = 0 and γ = 0 the rotations are not constant but go to 0 as(1 −ζ α)2. As in the symmetric case the displacements in the plane of the elementare obtained by means of the equation (3.62) and a subsequent derivation alongY j leads to the natural strains:

mAj = −z∂ 2w

A

∂Y 2j

(3.75)

If computations are performed, the following expressions for the strains due tothe antisymmetric bending modes are obtained:

mAα = −z1

l2α

3lα(ζ γ − ζ β)qbAα + lβζ αqbAβ − lγζ αqbAγ

(3.76)

mAβ = −z1

l2β

−lαζ βqbAα + 3lβ(ζ α − ζ γ)qbAβ + lγζ βqbAγ

(3.77)

mAγ = −z1

l2γ

lαζ γqbAα − lβζ γqbAβ + 3lγ(ζ β − ζ α)qbAγ

(3.78)

Strains from antisymmetric shearing modes

The antisymmetric shearing modes involve only natural transverse strains. Fromfigure 3.13 it can be observed that the variation of strains over the elements is alinear one and it is expressed by the following relation:

γ si =qAi

2

s

(1 − ζ i) (3.79)

Strains from azimuthal modes

In the assembly of flat elements the azimuthal degrees of freedom are often ne-glected and substituted with fictitious couple M z. Many works deal with thisargument and the reader can refer to some publications listed in the bibliogra-phy [4,5,23,65,84,112,130]. In the present formulation of the TRIC element the

drilling modes effect has been completely considered, differently from the originalformulation where fictitious terms were employed in order to eliminate the singu-larity in the stiffness matrix [18].The azimuthal modes give rise to in-plane deformations like the one pictured infigure 3.16. It is clear that only natural membrane strains are involved. In order

40




qda

v ( )ga gh

y’

1

2

3

R r

x’

Figure 3.16: strains due to the drilling mode qdα

to calculate these strains the displacements over a linear segment ρ, linking a ver-tex to a point R lying on the opposite side, are assumed similar to those imposedat the sides. If we refer to the drilling mode qdα the displacements on the segmentρ are:

vρα(ηρ) =

ηρ − 2η2ρ + η3

ρ

lρ1qdα (3.80)

where lρ1 is the segment length and 0 < ηρ < 1 the dimensionless coordinate.The displacements direction is perpendicular to the segment considered and somefurther developments lead to explicit relations. Let P be a point on the segmentρ with internal coordinate (ζ 1, ζ 2, ζ 3). ei is the unit vector in the direction of sidei and (xj , yj) the coordinate of point j in the local reference system. Thereforethe following relations hold:

eα = x3 − x

2

lαi + y

3 − y

2

lα j = cαx i + sαx j (3.81)

eβ =x1 − x3

lβi +

y1 − y3lβ

j = cβxi + sβx j (3.82)

eγ =x2 − x1

lγi +

y2 − y1lγ

j = cγx i + sγx j (3.83)

OP = O1 + 1P = x1i + y1 j + ζ βlγeγ − ζ γlβeβ (3.84)

and the coordinates of P in the local reference system are:

xP = x1(1 − ζ β − ζ γ) + ζ βx2 + ζ γx3yP = y1(1

−ζ β

−ζ γ) + ζ βy2 + ζ γy3

The unit vector in the direction of the segment ρ, from vertex 1 to point R, is:

eρ1 =xP − x1

|1P | i +yP − y1

|1P | j = xP 1i + yP 1 j

41




It is now possible to obtain the unit vector normal to the segment ρ and directedas the displacements:

nρj = k ∧ eρj = −yPji + xPj j j = 1, 2, 3 (3.85)

In order to find the displacement of point P it is now necessary to know thedimensionless coordinate ηρ and this is possible if the length of segment ρ isknown. It can be observed that the point R has internal coordinates (0, ζ Rβ , ζ Rγ )

where ζ Rβ and ζ Rγ are related by the following relation of proportionality (see figure3.17:

1P =ζ P γ lγ

sin χ=

ζ P β lβ

sin τ ⇒ ζ P β

ζ P γ=

ζ Rβζ Rγ

It is now possible to calculate ζ Rβ and ζ Rγ :

1

2

3

R P

c

t

lg

lb

Figure 3.17: scheme used to find the position of the point R

ζ Rβ + ζ Rγ = 1

ζ Rβ = ζ RγζP βζP γ

→ζ Rβ =

ζP βζP β

+ζP γ

ζ Rγ =ζP γ

ζP β

+ζP γ

Therefore the segment length and the dimensionless coordinate are:

lρ1 =

ζ Rβ x2 + ζ Rγ x3 − x1

2

+

ζ Rβ y2 + ζ Rγ y3 − y1

(3.86)

ηρ1 =|1P |lρ1

=

xP − x1ζ Rβ x2 + ζ Rγ x3 − x1

= ζ P β + ζ P γ = 1 − ζ α (3.87)

and the displacement vector for a point P on the segment 1 R is:vρα = (ηρ − 2η2

ρ + η3ρ)lρ1qdαnρ = (ζ 2α − ζ 3α)lρ1qdα(−yP 1i + xP 1 j) (3.88)

vρα = ζ 2αqdα(−(y1(ζ α − 1) + ζ βy2 + ζ γy3)i + (x1(ζ α − 1) + ζ βx2 + ζ γx3) j)(3.89)

42




In order to calculate the strains in the natural system the displacement vector isprojected on the element sides.

vραi = vρα • ei i = α,β,γ (3.90)

The strains are calculated with the following formula:

mi =j

∂vρji∂Y j

i, j = α,β,γ (3.91)

where the index j refers to the drilling mode and the index i to the strain direction.Omitting the calculus (which can be found in appendix A) the contribute of drilling modes to natural strains is:

mα = 4Ω

l2α

ζ α(ζ βqdβ − ζ γqdγ) (3.92)

mβ = 4

Ω

l2β ζ β(ζ γqdγ − ζ αqdα) (3.93)

mγ = 4Ω

l2γ

ζ γ(ζ αqdα − ζ βqdβ) (3.94)

Total natural strains

It is now possible to write the natural strains as a function of natural modesρN by means of two matrixes bm and bs. The vector ρN contains the so calledextended natural modes where the antisymmetric modes are split in the bendingand shearing parts.

=

bm 0

0 bs

ρN (3.95)

where:

ρN 15×1

=q0m qS qbA qd qsA

T (3.96)

bm =

1 0 0 z

lα0 0

−3z(ζγ−ζβ)

lα

−zlβ

l2αζα

zlγ

l2αζα 0 4 Ω

l2αζαζβ −4 Ω

l2αζαζγ

0 1 0 0 zlβ

0 zlα

l2β

ζβ−3z(ζα−ζγ )

lβ

−zlγ

l2β

ζβ −4 Ω

l2β

ζαζβ 0 4 Ω

l2β

ζβζγ

0 0 1 0 0 zlγ

−zlαl2γ

ζγzlβ

l2γζγ

−3z(ζβ−ζα)

lγ4 Ωl2γζαζγ −4 Ω

l2γζβζγ

(3.97)

bs =

1−ζα2 0 0

01−ζβ2 0

0 01−ζγ2

(3.98)

3.5.5 The element dissipation

The rigid-plastic problem requires the definition of the dissipation. This is donein terms of natural straining modes. As seen in subsection 2.2.3 the dissipationcan be expressed in terms of strains:

43




D = e De = e Ωe

σ0λdx (3.99)

where:λ =

T Θ (3.100)

The contribution of an element to the dissipation is split in two parts: the con-tribution of membrane strains and the contribution of transversal shear strains.This can be done because there is no coupling between membrane strains and thetransverse natural modes (see eq. 3.95). Equation (3.101) becomes:

D =e

Ωe

σ0

λ2m + λ2

sdx (3.101)

where

λ2m = T mΘmm (3.102a)

λ2s = T s Θss (3.102b)

Matrix Θm and Θs are obtained by equation (2.30) keeping in mind the incom-pressible relation (2.32c): zz = −xx − yy .

Θm =1

3

4 2 02 4 00 0 1

(3.103)

Θs =1

3

1 00 1

(3.104)

Relations (3.102) have to be expressed in terms of natural strains. This can bedone immediately for the membrane part while for the contribution of transverse

modes a short manipulation is necessary.

λ2m = ˙T mT

T mΘmT m ˙m (3.105)

or in a more compact way:

λ2m =

T

mΓmm (3.106)

Regarding the contribution of transverse strains, from eq. (3.54) it can be seenthat the two cartesian transversal strains in local coordinates have to be expressedin terms of the three natural transversal strains. This can be done by solving onlytwo equations at time and this leads to three different solutions:

γ xzγ yz

=

1

2Ω

lαyβ −lβyα

−lαxβ lβxα

γ sαγ sβ

(3.107)

γ xzγ yz

= 12Ω

lβyγ −lγyβ−lβxγ lγxβ

γ sβγ sγ

(3.108)γ xzγ yz

=

1

2Ω

lγyα −lαyγ

−lγxα lαxγ

γ sγγ sα

(3.109)

44




where Ω is the area of the element, lµ the side µ length and xµ, yµ the projections

of side µ on local reference axes. Now three different values of λ2s can be obtained

and their mean value is used in order to calculate the dissipation due to the shearstrains:

λ2s =

1

36Ω2

γ sαγ sβγ sγ

T l2α(l2β+l2γ) −lαlβ(yαy

β+xαx

β) −lγ lα(yγy

α+xγx

α)−lαlβ(yαy

β+xαx

β) l2β(l2γ+l2α) −lβlγ(yβy

γ+xβx

γ)−lγ lα(yγy

α+xγx

α) −lβlγ(yβy

γ+xβx

γ) l2γ(l2α+l2β)

γ sαγ sβγ sγ

(3.110)

or in a more compact way:

λ2s =

T

s Γss (3.111)

Plastic flow, and subsequently the dissipation power, can now be expressed interms of natural strains:

λ2 = T

mΓmm + T

s Γss (3.112)

The symmetric matrixes Γm and Γs are functions of the element attitude onlywhile the natural strains depend on the position where they are calculated.The procedure adopted in order to calculate the energy dissipation of the elementis reported in appendix B.

3.5.6 Nodal equivalent loads

When an element is subject to a pressure load, normal to its surface, the effectshave to be expressed by means of appropriate nodal forces and moments appliedto the nodes. Let us consider a uniform pressure p in the direction of z axis (seefig. 3.18) and the displacements of the element surface:

w (ζ α, ζ β, ζ γ) = ρ03 + ρ04y − ρ05x + wS + w

A (3.113)

where single terms of equation (3.113) have been defined in the previous subsec-tions. The work equality is now imposed:

Ae pw (ζ α, ζ β, ζ γ) dA = pT e u

e (3.114)

where the vector p collects the nodal equivalent loads expressed in the localreference system.Without loss of generality the local axis x can be imposed parallel to the side α.

45




1

2

3

G

x'

y'z'

Figure 3.18: the pressure load

Therefore, if integrations are performed one obtains:

p =

00Ω3

Ω24

(lγsγx − lβsβx)Ω24

(lβcβx − lγcγβx)000Ω3

− Ω24

lγsγxΩ24

(lγcγx − lα)0

00Ω3

Ω24

lβsβxΩ24

(lα − lβcβx)0

(3.115)

Between displacements in local and global reference system relation 3.45 holds,therefore:

pT e = pT

e Te (3.116)

3.5.7 The minimization procedure

In section 3.2 the solution procedure has been described in a general way and themethod proposed can be applied to any element without notable modifications.Actually, for the AX2P element presented in section 3.4 the procedure appliesinalterate and only the matrix Λ0e, typical of the element, has to be calculated.

46




On the contrary, the natural modes of the TRIC element depends on the di-splacements ue and on the bending coefficients χe. Therefore both have to be

considered by the minimization procedure. In particular the bending coefficientis assumed constant in the respective element and its effects are restricted to theelement it self. In the following, the necessary modifications will be described,deferring to the appendixes for the details of the procedure.In the original work [18], the antisymmetric modes were split in the bendingand shear part by considering their stiffnesses: a closed form was obtained andin a subsequent publication the benefits of the procedure were explained [14].In particular, the introduction of shear modes permits to avoid the locking phe-nomenon, without having recourse to different expedients needed when using otherelements [31, 32,39, 77,114].In the present work the TRIC element has been modified in a suitable way toadapt it to the limit analysis. As a consequence the bending coefficients χe cannotbe obtained in a closed form but they have to be considered in the minimization

procedure. This permits to obtain the vector χ of bending coefficients that as-sure the lower dissipation. In this way the shear effects are accounted for, thusavoiding locking phenomena.Let us consider the expanded natural modes grouped in the vector ρN defined inequation (3.96). Being χe constant over the element e the shear and bending partof the antysimmetric modes are obtained in a simple way:

qbA = χeqA (3.117a)

qsA = (1 − χe) qA (3.117b)

where qA is the vector of the antisymmetric modes obtained as explained inequation (3.59).The minimization problem (3.13) becomes:

s = minu,χ

D (u,χ)

= minu,χ

nee=1

Ωe

σ0

uT LT

e TT e CT

e ΛT e (χ) ΠeΛe (χ) CeTeLeudx

(3.118a)

subject to pT u = 1 (3.118b)

where:

Πe =

Πm 0

0 Πs

=

bm 0

0 bs

T Γm 0

0 Γs

bm 0

0 bs

(3.119)

Λe (χ)15×12

= I

I

χeI

I

(1 − χe) I

(3.120)

and

47




• u is the vector of nodal variables of the whole structure in the global re-ference system;

• Le is the connectivity matrix of the element e;

• Te is the coordinate transformation matrix defined in (3.45);

• χ is the vector of bending coefficient χe;

• Λe (χ) is the matrix relating the natural modes and the extended natural

modes.

The problem can be solved minimizing the following Lagrangean function:

L (u,χ, α) = D (u,χ) − α

pT u − 1

(3.121)

At solution the following equalities holds:

∂D

∂ u− αp = 0 (3.122a)

∂D

∂ χ= 0 (3.122b)

Stationarity respect to the displacements leads to the following non-linear expres-sion:

e

Ωe

σ0Z

λedx u − αp = 0 ⇒ Ku − αp = 0 (3.123)

whereZ = LT

e TT e CT

e ΛT e (χ) ΠeΛe (χ) CeTeLe (3.124)

while the bending coefficient of the element e that leads to a minimum of thelagrangean function can be obtained as follows:

χe = −

ΩeρT N

B

λeρN dx

ΩeqT A

A

λeqAdx

(3.125)

A more detailed dealing of the argument can be found in appendix C. Solutionof the non-linear equation (3.123) is provided by means of an iterative proceduresimilar to that reported in equation (3.21b). If some elements are found to dis-sipate less than a given tollerance value (see equation (3.23)) they are set rigid.Proper constraints are imposed to the solution procedure by means of the matrixG0 introduced in equation (3.14).

48



Chapter 4

Test example: simply

supported plate

In this chapter some simply supported plates subject to transversal pressure arestudied in order to validate the approach presented in chapter 2 and the imple-mentation by means of the finite element method developed in chapter 3.Plates and stiffened plates are typically found in ship hulls, platform decks, civilbuildings and many other structures where they have to withstand normally staticloads and in some cases dynamic forces and explosions, with possible impact of dropped objects. In this case test experience shows that the use of static plasticdisplacement models, combined with momentum and energy considerations, couldbe useful. In the formulation of the present approach these aspects have not been

considered, but according to the author they could be included with little effort.The analyses here presented are all concerned with a simply supported plate subject to a static transversal pressure load. The limit load furnished by the methodproposed in this work, will be verified by comparison with some upper bound va-lues available in the literature. Moreover some incremental analyses, performedby means of the commercial code ABAQUS, will be used in order to validate thecollapse behaviour foreseen by the sequential limit analysis. The investigation isnot restricted to the thin plate only, but thick plates will also be considered inorder to check the capabilities of the proposed element.The geometry and the mechanical characteristics of the studied plates are re-ported in section 4.1. Different plates are considered, with slenderness β = 2a/hchanging from β = 1 up to β = 100. An estimate of the limit load, by means of a

mechanism model, is presented in section 4.2 for the two extreme cases. Finallyin sections 4.3 and 4.4 the limit load and the post-collapse behaviour, obtainedby means of the sequential limit analysis, will be discussed and compared withthe results available in literature and with the elasto-plastic analyses performedby the commercial code ABAQUS.

49



Chapter 4. Test example: simply supported plate

4.1 Geometry and mechanical characteristics

All the studied plates are of square shape and simply supported at the four sides.Their geometric and mechanical characteristics are reported in table 4.1 while asketch is pictured in figure 4.1. The slenderness β is defined as:

¡

¡

¢ £ ¥ § ¨ £

Figure 4.1: simply supported plate

Side 2a (mm) 20

Thickness h (mm) 0.2

÷20

Slenderness β 100÷1

Yield stress σ0 (MPa) 200

Young modulus E (GPa) 200

Table 4.1: Geometric and mechanics characteristics of plates

β =2a

h

Simply supported plates subject to transversal pressure represent one of the moststudied case. In literature many results are available and they are obtained refer-ring to a dead pressure p0 defined as:

p0 =M 0a2

(4.1)

50



4.2. Mechanism model

where M 0 is the limit bending moment

M 0 = σ0

h2

4 (4.2)

and h the plate thickness. If equation (4.2) is subsituted in equation (4.1) thedead pressure can be expressed as a function of the slenderness:

p0 =σ0

β 2(4.3)

Therefore the dead pressure p0 considered in the analyses is not constant butdepends on the slenderness of the plate considered. Thus the limit pressure canbe expressed as p = αp0 where α is the plastic multiplier obtained by the limitanalysis.

4.2 Mechanism model

In this section the limit load of two different plates will be obtained by applying asuitable mechanism collapse. In particular, a thin plate with β = 100 and a verythick plate with β = 1 will be considered.

4.2.1 Thin plate

Let us consider a thin plate, with slenderness β = 100, subject to a transversalpressure p. Being the plate thin, a mechanism like that pictured in figure 4.2(a) issupposed to well describes the collapse of the plate. Kirchhoff’s hypotheses are as-

sumed. The dotted diagonals represent the hinge lines: these have to be regardedas the limit case of narrow strips exhibiting cylindrical curvature. Hence defor-mations vanish everywhere except in the hinge lines where plastic deformationsconcentrate. Due to the presence of the uniform pressure, deformations occur in arather large zone, but, in order to simplify the computations, the aforementionedhypothesis is considered. If a virtual displacement δ, normal to the plate surface,is applied in the point E, the rotations of the four rigid parts are:

• part 1: θx = δa

θy = 0

• part 2: θx = 0 θy = − δa

• part 3: θx = − δa

θy = 0

• part 4: θx = 0 θy = δa

Along the hinge lines the relative rotation is:

θr =√

2δ

a

51




q3

q2

q1

q4

Yield lines

E

A B

CD

x

y

4

3

2

1

(a) bending collapse mechanism

Yield lines

A B

CD

vertical displacementconstant over the plate

x

y

(b) punch collapse mechanism

Figure 4.2: two different collapse mechanism for a square plate

The limit load is obtained by equating the internal and the external work:

Le =

A

pwdA =4

3 pa2δ

Li = 42√

3M 0

√2

δ

a

√2a =

16√3

M 0δ

Le = Li ⇒ p = 6.92 M 0a2

Thus, if p0 represents the dead pressure, the plastic multiplier α is constant:

p = 6.92M 0a2

= 6.92 p0 ⇒ α = 6.92 (4.4)

This simple mechanism, adopted to introduce the calculation of the limit load of athin plate, is the same mechanism adopted by Baldacci et al. [20], but many othercollapse mechanisms have been presented in the past, each one leading to differentmultipliers. For sake of comparison some of these plastic multipliers are reportedin table 4.2. In the same table, where available, the lower bound multipliers

obtained by equilibrated approaches are also indicated. In the following sections,the more accurate plastic multiplier obtained by Iliouchine [66] will be used asthe analytical reference value for thin plates.

(1)corners are allowed to lift up.

52



4.2. Mechanism model

dead pressure, p0 = M 0a2

Reference Lower bound Upper bound Method

Baldacci et al. [20] 5.15 6.92 Analytical

Capsoni, Corradi [33] — 6.256 Numerical

Del Rio Cabrera [28] (1) — 6.348 Analytical

Hodge [57] 5.16 6.98 Analytical

Hodge, Belytschko [58] 6.216 6.42 Numerical

Iliouchine [66] — 6.6 Analytical

Table 4.2: Upper and lower bounds for a simply supported plate subject to

transversal pressure load

4.2.2 Thick plate

Let us now consider a thick plate, with slenderness β = 1, subject to a transversalpressure p. From the geometrical point of view a slenderness β = 1 corresponds toa cube, therefore the theory of plates cannot be applied. Actually, the proposedfinite element is not suitable for analyses on such geometry, but some characte-ristics can be highlighted with this example.The collapse is supposed to happen by means of a punch mechanism such as theone pictured in figure 4.2(b) where plastic strains develop near the support. If avirtual displacement δ normal to the plate surface is applied, the collapse load can

be calculated by equating the internal and the external work in the hypothesisthat only shear strains are involved:

Le =

A

pwdA = 4 pa2δ

Li = 4τ 02ahδ =8√

3ahσ0δ =

16a2

√3β

σ0δ

Le = Li ⇒ p =4√

3

σ0

β

Thus, if p0 represents the dead pressure, the plastic multiplier α become:

p =

4

√3

σ0

β =

4

√3 βp0 ⇒ α =

4

√3 β (4.5)

The plastic multipliers obtained both for the thin and the thick plates are reportedin figure 4.3 where, for a better illustration, the logarithmic scale has been usedfor the β -axis.

53




4.3 Finite element limit analysis

In this section the approach developed in the present work will be used in order toobtain the limit load of simply supported plates subject to transversal pressure.Mesh accuracy and thickness variation will be considered. The latter investigationwill bear out the good behaviour of the shell element also for medium thick plates.Being the problem symmetric, only a quarter of plate has been considered in theanalyses.As done in section 4.2 the plates are considered subject to a dead pressure p0 thatdepends on the plate slenderness β :

p0 =M 0a2

=σ0

β 2

4.3.1 Mesh variation

In order to study the accuracy and effectiveness of the element developed in section3.5, some analyses have been performed on a thin plate, with slenderness β = 100,using different meshes. The models characteristics are indicated in table 4.3. Theplastic multiplier α and the bending parameter χ, introduced in subsection 3.5.4,are also reported.Being the TRIC element non-conforming, the convergence to the exact solutioncan be approached from lower values despite the kinematic approach used. Thishappens for the plate under consideration as it is evident from figure 4.3. Thisis not a drawback, as demonstrated by [] where a mixed formulation was usedto achieve better results. For sake of comparison the upper bound multiplier,

Nodes Elements Degrees of freedom Collapse multiplier χ

9 8 24 5.93 0.997

15 16 49 6.11 0.988

27 36 109 6.16 0.971

49 72 217 6.15 0.932

64 98 295 6.16 0.908

91 148 445 6.22 0.862

Table 4.3: results for a simply supported plate subject to transverse pressure for

β = 100

furnished by Iliouchine [66], and the results obtained in [33,102], using a Kirchhoff finite element, are reported in figure 4.3. Both the numerical analyses clearlyconverge to an upper bound multiplier lower than the analytical one. Moreover,

54



4.3. Finite element limit analysis

being the Kirchhoff element compatible, the value of 6.256 reported in [33] canbe considered as a reference multiplier that is well approximated by the analyses

performed with the TRIC element.Unlike the quadrilateral finite element used in [33], the TRIC element can alsodeal with shear strains by means of the bending parameter χ. Its mean valuereported in table 4.3, seems to be influenced by the element dimension, but theminimization procedure always guarantees the most accurate result. In conclusionthe comparison reveals a good behaviour of the element and of the approach usedwhen applied to thin plates.

0 25 50 75 100

Elements

5

5.5

6

6.5

7

C o l l a p s e m u l t i p l i e r

¢ ¡ £ ¥

§ ¨

£¦ ! ¨ " ¨ $ %

[33,102]

Figure 4.3: collapse multiplier obtained using different meshes

4.3.2 Thickness variation

Among the characteristics of the original TRIC element [18], one of the mostnotable is the ability to deal with medium thick shells. The analysis of mediumthick plates requires the use of elements based on Mindlin’s theory, which canaccount for the shear effect over the thickness. Unfortunately, locking phenom-ena occur when these elements are used for the discretization of thin plates, andsome tricks have to be used in order to solve this problem [33,107]. On the otherhand the TRIC element introduces the antisymmetric shear modes and, in its

original formulation, the locking problem is avoided in a very simple but efficientway [18]. In the present formulation the antisymmetric shear modes are obtainedby introducing a parameter χ and minimizing the dissipation with respect to it(see section 3.5.7). This approach allows the study of thin and medium thickshells avoiding locking problems occuring with classic Mindlin’s formulation.

55




In order to check the ability of the element in the study of thin and thick struc-tures some analysis have been performed on plates of slenderness varying between

β = 1 and β = 100. The 98 elements mesh has been chosen for the analyses. The

1 10 100

β

0

2

4

6

8


¡ £ ¥

§ ¨ ¨

" $ ( ¨ 0 "

£ ¨ 3 ¨ $

[33,107]

Figure 4.4: Collapse multiplier for different slendernesses

two plastic mechanism presented in section 4.2 give an idea of what happens inthe extreme cases: the folding mechanism reported in figure 4.2(a) well describesthe collapse phenomena associated to very thin plates, while the punch mecha-nism (see figure 4.2(b)) can be used with very stocky plates (in this case β = 1

refers to a cube). The simple calculations performed in section 4.2 permit to drawthe two curves reported in figure 4.4, which identify an approximate upper boundenvelope of the plastic multiplier for plates with varying slenderness. The resultsobtained by finite element limit analysis are also reported and compared with theanalytical solutions proposed.From the values obtained it can be stated that the TRIC element performs verywell in the limit analysis of thin and medium thick plates, where results are com-parable with the ones obtained by Capsoni and Corradi using Mindlin’s elementswith mixed formulation; nevertheless, for very stocky plates the TRIC elementfurnishes the lowest values but these can not be considered a better approxima-tion of the exact plastic multiplier because of the non-conforming formulation of the element.

The good behaviour of the procedure, introduced in order to deal with shearstrains, is confirmed by the graph of figure 4.5, where the values assumed by thebending parameter χ are plotted vs. the slenderness parameter β . As expectedwhen the slenderness decreases the bending parameter reduce it self and this cor-responds to a greater influence of shear strains. The velocity fields related to

56



4.3. Finite element limit analysis

1 10 100

β

0

0.2

0.4

0.6

0.8

1

χ

Figure 4.5: the values assumed by the bending coefficient χ for different slender-nesses

the plastic load of the plates considered are reported in figures 4.6 and 4.7; thelikeness with the simple collapse mechanisms reported in figure 4.2 is manifest:in particular figure 4.6 shows how the energy is dissipated in a major way by theelements over the diagonal, while in the punch case (figure 4.7) the whole dissi-pation is concentrated on the elements lying on the boundaries. The examples just reported show that the TRIC element is suitable both for limit analysis of thin and medium thick structures subject to bending and shear strains. Duringthe analyses, all the element showed in plane rigidity and the dissipation was dueto the bending natural modes only. As explained in appendix C, the axial modeswere set to zero, and the relevant constraints were taken in account during thesolution process. This avoid the infinity that occurs in the computation of thematrix K when only bending modes are present.The following examples will demonstrate the good behaviour of the element whenalso in plane strains are involved.

57




(a) dissipation density

(b) collapse mode

Figure 4.6: simply supported plate with slenderness β = 100

(a) dissipation density

(b) collapse mode

Figure 4.7: simply supported plate with slenderness β = 1

58



4.4. Post collapse behaviour

4.4 Post collapse behaviour

The use of the sequential limit analysis permits to study the post collapse be-haviour of a shell structure assumed to be rigid-plastic. In this section someresults concerning the collapse evolution of a plate subject to a transverse pres-sure will be presented, either with or without in plane constraints. In order toverify the accuracy of the approach used, some comparisons with elasto-plasticanalyses performed by means of the code ABAQUS, will be shown.

4.4.1 Simply supported plate

The first example refers to a simply supported plate with slenderness β = 100and thickness h = 0.2 mm (see figure 4.1). Its geometric and mechanical char-acteristics are reported in table 4.4 The analyses have been executed on three

a

(mm)

h

(mm)β

E

(GPa)

σ0

(MPa)ν

10 0.2 100 200 200 0.3

Table 4.4: geometric and mechanical characteristics of the considered plate

different models whose most important characteristics are reported in table 4.5.The meshes M1 and M2 are pictured in figure 4.8. Although the post-collapsebehaviour is expected to be stable, the modified Riks algorithm, implemented inthe code ABAQUS, has been used for the incremental analyses. The automatic

step increment, used by default, allows to deal with any nonlinearity. On the con-trary the sequential limit analysis requires to specify a maximum displacementbetween the steps: it has been imposed to be umax = 0.3 mm, an acceptablevalue if compared with the plate thickness. The parameters used to predict thecollapse behaviour of the plate under consideration are summarized in table 4.6.

Model MeshElement

typeNodes Elements

Degrees of

freedom

A M1 TRIC 64 98 295

B M1 S3R 64 98 384

C M2 S9R5 289 64 1734

Table 4.5: characteristics of the models used in the analyses

59




¡

(a) mesh M1 (b) mesh M2

Figure 4.8: meshes used for the post collapse analysis of simply supported plateswith slenderness β = 100

The A and B model are suitable for a correct comparison of the results because of

model umax kd keinitial arc

length

maximum

arc length

A 0.3 mm 0.001 0.001 — —B-C — — — 1 1.5

Table 4.6: parameters used in the sequential limit analysis

the same mesh used. Actually the CPU time required by the method proposed inthis work is shorter than that needed by the incremental analyses (see table 4.7).This is mainly due to the major stability of the present approach that, in order toachieve the same solution, requires fewer step than those necessary to the incre-mental analisys. A significant difference between the post collapse behaviour isclear observing the curves reported in figure 4.9(a), moreover it has to be pointed

out that, the second order effect being stiffening, this example does not involveparticular difficulties to the incremental analyses and the accuracy depends onthe element used. Therefore a second incremental analysis has been performedadopting the C model where a more accurate element and a different mesh havebeen used. The results obtained are reported in figure 4.9(b) and compared with

60




0 1 2 3 4 5

Displacement (mm)

0

40

80

120

160


¢ ¡ £ ¥ § ©

¢ ¡ £ ¥ §

β

(a) model A and B

0 1 2 3 4 5

Displacement (mm)

0

40

80

120

160


¢ ¡ £ ¥ § ©

¢ ¡ £ ¥ §

β

(b) model A and C

Figure 4.9: comparison between the results obtained by the sequential limit anal-ysis and the incremental elasto-plastic approach

the load-displacement curve furnished by the sequential limit analysis: in thiscase there are only minor discrepancies due to the elastic part of the solution notconsidered by the limit analysis.The deformed configurations corresponding to a maximum displacement of 4.2mm are reported in figure 4.10(a) for the sequential limit analysis, and in figure

4.10(b) for the elasto-plastic incremental analysis. The two collapse modes arevery similar and exhibit a major energy dissipation near the corner. It seemsthat the TRIC element has a good accuracy while the classic triangular shellelement performs in a more stiffer way. Comparable results between the sequen-

Model MethodNumber of

iterationsCPU time

A S.L.A. 15 23”

B Incremental 47 29”

C Incremental 58 1’24”

Table 4.7: comparison of solution times for a displacement of 4 mm. The analyseshave been performed on a PC-Athlon 1200 MHz

61




(a) Sequential limit analysis (b) Incremental elasto-plastic analysis

Figure 4.10: deformed state at a maximum displacement of 4.2mm

tial limit analysis and the elasto-plastic incremental approach can be achieved bymeans of a more accurated element like the S9R5. In this case the solution timegrows but the stability of the problem does not involve particular problems to theconvergence of the incremental analysis.

4.4.2 Simply supported plate with in plane renstraints

In practical situations, plate elements are often not simply supported, but in planeconstraints whose effect is to increase the stiffness of the structure when subjectto large displacements, can exist at the boundaries. In order to evaluate this effect

some analyses have been performed on the same plate with slenderness β = 100 just presented introducing in plane constraints at the boundaries. The sequentiallimit analysis and the incremental approach have been applied to models A andC respectively.As for the case of simply supported plate, the present approach well describes thepost-collapse behaviour of such structures, even though the load-displacementcurve is not so smooth (see figure 4.11(a)). The deformed state obtained usingthe limit analysis for a maximum displacement of 2.5 mm is pictured in figure4.11(b).

The stiffness increment is clearly shown in figure 4.4.2 where the curves ob-tained both with and without in plane constraints are reported.

62




0 0.5 1 1.5 2 2.5

Displacement (mm)

0

40

80

120

160


¡ £ ¥ § ©

¡ £ ¥ §

β

(a) Pressure-displacement comparison

(b) Deformed state

Figure 4.11: Results obtained for a simply supported plate with in plane restraints.Slenderness β = 100

0 1 2 3 4 5

Displacement (mm)

0

40

80

120

160


¡ £ ¥ § ©

β

!

# $

% ' ) ' %

Figure 4.12: Comparison between simply supported plates with or without inplane constraints.

63




64



Chapter 5

Test examples: some shells

In chapter 4 some examples related to the post-collapse behaviour of simply sup-ported plates have been presented. It has been shown that the TRIC element,modified in a suitable way, performs better than classical triangular element andit can deal with medium thick shells too. Even though plate structures are verycommon in civil, naval and other relevant engineering fields, some applicationsrequire the use of parts typologically different, capable of developing relevantstrains at constant or decreasing applied force. Example of such parts can befound in crash problems, where the energy absorption capabilities of structuralcomponents are of prominent interest for the assessment of safety in a wide rangeof applications, such as design of vehicles and, in general, components that haveto withstand collisions. The research in this field has experienced a considerablegrowth in recent years, made possible by the ever increasing computer power and

stimulated by the introduction of codes more and more concerned with safety inevery aspect. In several industrial applications, the design involves the definitionof suitable zones at planned deformation, able to absorb significant impact [75].Such zones are usually rather localized and deform by developing large strains, sothat the consequent deceleration is acceptable. As an example, such a solutionhas been recently adopted in the design of rail vehicles to enhance safety [76].Thick or medium thick shells are suitable as energy absorbers, since they bucklein the plastic range, permitting considerable dissipation and significant displace-ments at decreasing force. Axisymmetric shells, like cylinders or cones, are oftenemployed to this purpose because of high effectiveness and low cost. A first, yetmeaningful indication on their crashworthiness is provided by static analyses as-sessing the behavior of the structure after collapse and measuring the energy that

it can dissipate [72,74].Different methods may be employed to predict the post-collapse curve. The mostcomplete simulations, although computationally demanding, are provided by largedisplacement, incremental elastic plastic analyses. This approach furnishes a lotof information sometime unnecessary to the designer and some cases can be solved

65



Chapter 5. Test examples: some shells

only using advanced solver not always implemented. Even simple structures likecylinder parts require the use of these solver, being their post-collapse behaviour

unstable. Simpler and faster alternatives, such as mechanism analysis, which fol-lows the post-collapse evolution of elementary mechanisms consisting of circum-ferential hinges where bending dissipation is concentrated, while the regions inbetween experience membrane flow only, are also available [3,43,69,89]. However,such procedures often consider only axisymmetric collapse modes and axisymme-tric geometric imperfections. Furthermore, the change in shape of the mechanismduring the post-collapse evolution can hardly be followed to great accuracy.A good balance between computational efficiency and accuracy of results is pro-vided by sequential limit analysis adopted in the formulation of the present ap-proach. In the last decade this method has received an incresing interest andhas been applied to different problems where plastic deformations are predomi-nant [46, 62, 63, 110]. Its effectiveness and stability have been proved in manysituations, even where incremental methods exhibit difficulties to follow the post-

collapse path.The advantages of the approach presented in this work and its ability to describealso unstable post-collapse behaviour will be shown studying the collapse of threedifferent shells often used as energy absorber and here listed:

• a thick cylindrical shell subject either to axial load or external pressure;

• a medium thick frusta subject both to axial load and external pressure;

• a square cross section tube subject to axial load.

5.1 Cylinder S1

Despite their simplicity, cylindrical structures are often used as a part of energyabsorber due to their chashworthiness and low cost, therefore many studies havebeen conducted in the past in order to predict the collapse behaviour and theabsorbed energy of such elements. Alexander [3] proposed a simple collapse me-chanism, which became universally known as concertina mode, that was adoptedand modified by many researchers in order to predict the mean collapse load of cylindrical and conical structures [43,88–90]. This method can be applied only toaxisymmetric structures that collapse in axisymmetric way and if imperfectionshave to be considered their shape can only be axisymmetric.The concertina mode is not the only collapse mechanism that a cylindrical shellcan exhibit, but for elements axially compressed with R/t > 40 ÷ 50 [120] the

collapse entails the formation of axial and circumferential waves to make a dia-mond mode also known as the Yoshimura pattern [129]. This more complicatedcollapse mechanism has been widely investigated both theoretically [69, 94] andexperimentally [86, 87] and the interest in the method is shown by recent publi-cations on the topic [52].

66



5.1. Cylinder S1

The limitations of the concertina and diamond mode approaches can be overcomeby means of finite element analysis, which permits to consider any imperfection

shape and to detect any collapse mode. In particular the approach proposed inthis work and based on the sequential limit analysis is able to supply in an efficientand stable way the post-collapse behaviour of any shell part, proposing itself asan alternative tool to the study of such structures.In this section a cylindrical shell labelled S1 subject either to axial compression orexternal pressure will be considered. The geometric and mechanics characteristicsof the shell are reported in table 5.1, while a sketch is pictured in figure 5.1. Dueto geometric characteristics the collapse is expected to be axisymmetric.The values of the squash load P 0 and of the elastic buckling load P E for theaxially compressed cylinder are also indicated. The dimensionless parameterλ =

P 0/P E gives an indication on the slenderness of the shell and the value

of 0.18 states that it is very stocky. Cylinder S1 was previously studied by

R

(mm)

th

(mm)

H

(mm)

σ0

(MPa)

E

(GPa)

P 0

(kN)

P E

(kN)λ

19.5 1.0 25.0 200 200 24.5 760 0.18

Table 5.1: geometrical and mechanical characteristics of cylinder S1

R

Hth

Figure 5.1: the S1 cylinder

Seitzberger and Rammerstofer, who furnished also experimental data that were

used for validation of the proposed method. In [110] the limit analysis problemwas formulated on the basis of Ilyushin yield surface [27, 66, 104] describing theyield criterion in terms of generalized stresses. With respect to the work above,the approach proposed exhibits two main differences. First, the definition of theshell yield surface is avoided and the dissipation power is directly defined by closed

67




form integration over the thickness, which allows for considering more general ma-terials, such as those governed by Hill or Tsai-Wu criterion [34,44,119]. Secondly,

the limit analysis procedure proposed in [30], which has the distinctive featureof detecting and eliminating from the problem the finite elements that are notinvolved in the collapse mechanism, is employed with significant computationalsaving when plastic flow affects limited zones of the structure only, as typicallyoccurs for energy absorbers.

5.1.1 Axisymmetric mechanism approach

In order to obtain a first estimate of the collapse behaviour, the approach de-veloped by Andronicou and Walker [6] has been applied to the S1 cylinder. Themechanism shape is assumed to be in the form of a concertina having three hingeswith straight segments between them as shown in figure 5.2(a). A rigid-plastic

idealisation of the material behaviour is assumed, in which elastic deformationsand strain hardening are ignored. The work done in bending at three hingesduring a virtual change of deformation is:

dW 1 = 4M π (D + h sin θ) dθ (5.1)

where the limit moment M depends on the axial stress:

M =σ0t2

4

1 −

P

σ0t

2

(5.2)

The material between the hinges stretches longitudinally and the related work is:

dW 2 = 2πσ2th2 cos θdθ (5.3)

where the circumferential stress σ2 is assumed constant over the mechanism. Itsvalue can be obtained if a yield criterion is assumed, for example for the von Misescriterion:

σ21 − σ1σ2 + σ2

2 = σ0 ⇒ σ2 =1

2

σ1 −

4σ2

0 − 3σ21

(5.4)

The last contribution to the internal work is due to the longitudinal stretch inthe mechanism that, being the whole mechanism plasticized, can be found byimposing the flow rule:

dW 3 = πDP 2σ1 − σ2

2σ2 − σ1

h2

R

cos θdθ (5.5)

Finally the external work is:

dW e = 2πDPh

sin θ +

2σ1 − σ2

2σ2 − σ1

h

2Rcos θ

(5.6)

68



5.1. Cylinder S1

Equating the external and internal virtual work terms the equilibrium equationis given by:

2√3

t 1 − P 2 (D + h sin θ) − h2 cos θ P − 4 − 3P 2− 2P hD sin θ = 0 (5.7)

where P = P σ0t

Andronicou and Walker considered the length of the mechanism

D

t

θ

h

θ

P

P

(a) axisymmetric mechanism

0 2 4 6 8 10

Shortening (mm)

0

5

10

15

20

25

F o r c e ( k N )

¡¤ £ ¥ ¦ § ¨ ¥ " £ £ $ " §

'

( ¨ ¥ 0 ¨ ¦ 3 " £

4

$ § ¨ £

4 ¨ 5 6 " ¦ ¥ 7 0

(b) the load shortening curve

Figure 5.2: axisymmetric mechanism approach

variable with the deformation θ and the load P in such a manner as to minimisethe required load for each deformed state. The minimisation of equation (5.7)with respect to h gives:

h =

√3DP + tP 2 − t

√3cot θ

√4 − 3P 2 − P

(5.8)

Substituting equation (5.8) in (5.7) and solving it by an appropriate method(for example Newton-Raphson) the collapse load P can be found as a functionof the evolution variable θ. The obtained load-shortening curve is reported in

figure 5.2(b) where it is compared with experimental data and axisymmetric finiteelement limit analysis. The shortening may be obtained by:

∆ = 2h (1 − cos θ) +2σ1 − σ2

2σ2 − σ1

h2

Rsin θ (5.9)

69




5.1.2 The present approach

In order to get a deeper knowledge of the collapse evolution the proposed ap-

proach was used to analyze the cylinder S1 subject either to axial load or externalpressure. In this subsection the results obtained will be presented and comparedwith the elastic-plastic counterpart.Different models (see table 5.2 and figures 5.3) were used: the first two modelslabelled A and B are of axisymmetric type and they are described by elementsAX2P and SAX1 for limit and incremental analysis respectively; models C, Dand E describe a quarter of the cylinder, while models F and G describe the wholecylinder surface.

Model MeshElement

type

Nodes ElementsDegrees of

freedom

Notes

A M1 AX2P 51 50 148 Axisym.

B M1 SAX1 51 50 148 Axisym.

C M2 TRIC 159 272 751 Quarter

D M2 S3R 159 272 960 Quarter

E M3 S9R5 525 120 3156 Quarter

F M4 TRIC 220 400 1081 Whole cyl.

G M5 S9R5 841 200 5046 Whole cyl.


Axial load

The post-collapse behaviour of a cylindrical shell subject to axial compressionload is expected to be unstable because of the second order geometric effects. Inthese situations a step to step approach can deal with convergency problems andthe time needed to obtain the required solution can be greater than using directmethods. Some recent studies showed that direct methods can be more efficient

than traditional ones [62] in the post-collapse analysis of structures that exhibitan unstable behaviour.For the case under consideration experimental results are available and they areused as validation of the procedure adopted. The post-collapse curves obtainedby sequential limit analysis are reported in figure 5.4(a) and compared with the

70



5.1. Cylinder S1

18 19

Radius (mm)

0

5

10

15

20

25

H e i g h t ( m m )

node 51

node 1

node 21

node 11

node 31

node 41

(a) mesh M1

(b) mesh M2 (c) mesh M3

(d) mesh M4

(e) mesh M5

Figure 5.3: the different meshes used for the analysis of S1 cylinder

experimental results described in [110] and referring to a considerably longer spe-cimen. Because of this fact, comparatively large displacements are present priorthe collapse, but the process zone is very nearly the same and, as far as the

post-collapse response is concerned, comparison is meaningful. In figure 5.4(a)the post-critical experimental curve was shifted toward left, so as to depart fromthe collapse load. Results of incremental analyses are reported in figure 5.4(b):comparison shows that present results are in good agreement with all alternativesolutions, while the element S3R seems to be too stiffer as pointed out in sec-

71




Model MethodP max

(MPa)

Number of

iterationsCPU Time

Exper. 24.50 — —

A S.L.A 24.67 29 15”

B Incremental 24.50 1098 4’02”

C S.L.A 24.75 18 54”

D Incremental 24.50 1529 23’12”

E Incremental 24.45 1475 25’16”

F S.L.A 24.85 18 1’31”

G Incremental 24.43 1477 40’09”

Table 5.3: results obtained for the S1 cylinder subject to axial load. The CPUtimes refer to a shortening of 8 mm. The analyses have been performed on aPC-Athlon 1200 MHz

tion 4.4.As experimentally obtained, the collapse of the cylinder shell is axisymmetricand the evolution of the mechanism up to contact is shown in figure 5.5(a) whilein figure 5.5(b) is reported a deformed configuration corresponding to an end-shortening of 4.2 mm.

For such analysis where axisymmetric collapse occurs the AX2P element give thebest result in shorter time (see table 5.3).

72



5.1. Cylinder S1

0 2 4 6 8 10

Shortening (mm)

0

5

10

15

20

25

F o r c e ( k N )

¡¤ £ ¥ ¦ § ¨ ¥ " £ £ $ " §

%

& ¨ ¥ 0 ¨ ¦ 2 " £

4

$ § ¨ £

4

$ § ¨ £

4

$ § ¨ £ 5

(a) comparison with experimental data

0 2 4 6 8 10

Shortening (mm)

0

5

10

15

20

25

F o r c e ( k N )

¡¤ £ ¥ ¦ § © ! ¥ # £ £ % # §

' % § © £

' % § © £ )

'

% § © £ 0

'

% § © £ 1

'

% § © £ 2

(b) comparison with incremental analyses

Figure 5.4: load-shortening curves for the S1 cylinder subject to axial load

18 22 26 ¡ ¢ £ ¡ ¤ ¥ ¦ ¦ § ¢ ¨ ©

0

5

10

15

20

25

!

" #

$ %

%

&

S1

(a) Mesh M1

(b) Mesh M2

Figure 5.5: evolution of the collapse obtained by sequential limit analysis for theS1 cylinder subject to axial load

73




Pressure load

Although no experimental tests were available for comparison, the investigation

was extended to the case of external pressure. The resulting collapse modes showa particular aspect: while the axisymmetric analysis imposes an axisymmetriccollapse mechanism, the general shell analysis permits to recognize more complexcollapses. Indeed, this happens both for limit and incremental analyses where,after a beginning axisymmetric collapse, subsequent deformations concentrate infew narrow zones of the shell (figure 5.6) and the radial displacement becomesmeaningless. For this reason the curves reported in figure 5.7(a) refer to theaxisymmetric meshes only, while the other analyses are compared in terms of thetop displacements as reported in figure 5.7(b). This can be done because thenodes on the top edge are constrained to have the same vertical displacement.

Model Method P lim (kN)

Number of

iterations CPU Time

A S.L.A 10.86 8 4”

B Incremental — 83 15”

C S.L.A 11.43 8 13”

D Incremental — 190 1’49”

E Incremental — 130 1’55”

F S.L.A 11.52 19 56”

G Incremental —130

3’30”

Table 5.4: results obtained for the S1 cylinder subject to external pressure. TheCPU times refer to a top displacement of 0.2 mm. The analyses have been per-formed on a PC-Athlon 1200 MHz

The axisymmetric curve reported in figure 5.7(a) is in good qualitative agreementwith that produced by incremental analysis, but transverse displacements aresomewhat larger. This may be caused by the fact that the mechanism involvesthe entire cylinder (see figure 5.8) and, the material being assumed as rigid-plastic,only the membrane stresses in the deformed configuration contrast the increasingpressure. In this situation the role played by the elastic stiffness, even if not

dramatic, appears of engineering significance.The curves reported in figure 5.7(b) are due to an initial axisymmetric behaviour,even in the models from C to G where shell elements have been used: indeed, thecurves are similar and the vertical shift is merely due to the coarser mesh used inthe F model. After this initial behaviour something happens and the deformations

74



5.1. Cylinder S1

-

Figure 5.6: deformed mesh obtained by the model C analysis

0 2 4 6 8

Transversal displacement (mm)

0

4

8

12

16

P r e s s u r e ( M P a )

¢ ¡¤ £ ¥ ¦ § © ! " $ & © ) ) 2 ©

4 5

§ © £ 8

4 5 § © £ @

(a) pressure - radial displacement

0 0.2 0.4 0.6 0.8

Top edge displacement (mm)

8

10

12

14

16

P r e s

s u r e ( M P a )

¡¤ £ ¥ ¦ § ¨

! " $ %

& ) 0 2 4

6 9 @ 0 6

(b) pressure - top base displacement

Figure 5.7: results obtained for the S1 cylinder subject to external pressure

develop in few small zones detectable in figure 5.6: the limit pressure drops anda local collapse occurs.

75




10 15 20 ¡ ¢ £ ¡ ¤ ¥ ¦ ¦ § ¢ ¨ ©

0

5

10

15

20

25

30

!

" #

$ %

%

&

S1

' ( 0 2 4 6

¡

Figure 5.8: evolution of the collapse obtained by the A model analysis

76



5.2. Conical shells

5.2 Conical shells

Conical shells are structural elements often used in civil and marine applications.In the former case they may be parts of tanks and silos while in marine andoffshore structures they appear as parts of underwater housings or as transitionelements between cylindrical parts. Loading conditions are often complex andmay include bending and torsion, but axial compression and external pressureare considered the main actions to be taken into consideration in determining thestability and strength of these structures.Studies on elastic buckling of perfect conical shells were developed both analyti-cally [79, 108], and experimentally [49, 50], and originated mainly from aeronau-tical research. Other studies dealt with the effects of pre-buckling nonlinearitiesand boundary conditions, or were oriented to producing imperfection sensitivitycurves by means of numerical models [98]. In most current codes dealing with thisproblem [48], the critical buckling load of conical shells is reduced to the study of

an equivalent cylinder, while different approaches are proposed for elastic-plasticcases [54].When slenderness of conical shells is not high, yield stress is usually achievedin a section of the shell and the buckling phenomenon becomes influenced bythe plastic effects. The analysis of the plastic buckling process is usually tack-led by means of finite element models [42], that may become burdensome anduneconomic in the design phase. Mechanism models such as that presented insubsection 5.1.1 have been developed and validated [43,96] for conical shells too,but with the same limitations of the cylindrical counterpart.Being based on a finite element discretization the method proposed in this workovercome the mechanism models’ limitations proposing itself as an efficient toolfor the prediction of the collapse load and the post-collapse behaviour of such

structures. The capabilities of the sequential limit analysis are investigated bysimulating an experimental test made by Ross et al. whose particulars can befound in [105]. Some medium thick conical shells labelled TICC4, TICC5 andTICC6 were subject to external pressure and to axial load due to the pressureover the small cap. In all the tested specimen the collapse mode was asymmetricwith the formation of some circumferential waves and subsequent rupture near thebottom edge. Circumferential deformations were recorded during the experimentsby means of ten strain gauges applied to the bottom internal part of the shell.Unfortunately shortening data are not available and the comparison between theanalyses and experimental test will be done in terms of the maximum achievedload and the corrispondent collapse mode.

5.2.1 The TICC5 cone

The cone labelled TICC5 is here analyzed. Its geometric properties are reportedin table 5.5 while a sketch of the shell can be found in figure 5.9. Due to the asym-

77




metric collapse experimentally obtained by Ross et al. the mechanism model isnot applied to the case under consideration, therefore the sequential limit anal-

ysis and the incremental elasto-plastic approach will be presented using generalshell elements. The mechanical properties adopted in the analyses are indi-

Ri

(mm)

Rs

(mm)

th

(mm)

H

(mm)α

52.75 14.75 3.5 100 20.8

Table 5.5: geometric characteristics of cone T5

Hth

R

R ¡

α

Figure 5.9: the TICC5 cone

cated in table 5.6. In the same table are also reported the squash load and theelastic buckling load due to the axial compression. The dimensionless parameterλ =

P 0/P E gives an indication on the slenderness of the cone and the value of

0.17 states that it is very stocky. Specimen TICC5 was machined from a solidbillet of aluminium alloy in a very precisely way so that the maximum initialimperfection was of 0.0096 mm. Thus, imperfections were considered negligible.Furthermore, in order to avoid secondary bending stresses, during the test the

model was hung vertically from its bigger base while the pressure was applied onthe lateral surface and on the small cap. Therefore the cone was subject bothto external pressure and axial load. The strain gauges applied to the internalsurface recorded a linear behaviour until a pressure of about 220 bar, when somecircumferential waves formed, while the rupture occurred for an external pressure

78



5.2. Conical shells

σ0

(MPa)

E

(GPa)ν

P 0

(kN)

P E

(kN)λ

250 61.9 0.3 75.8 2516 0.17

Table 5.6: mechanical characteristics of cone T5

of 275.9 bar. Obviously this value was influenced by the hardening of the ma-terial not considered in the numerical analyses. It is important to observe thata descending solution path can not be followed by the experimental equipmentand only the maximum load can be used as comparison. However, from the engi-neering point of view, the descending path represents an important informationin order to understand the dangerousness of every structure.

From the analytical point of view the post-collapse behaviour of cone TICC5 wasstudied both by the sequential limit analysis and by elastic-plastic incrementalapproach. Two models labelled H and I have been used and their characteristicscan be found in table 5.7. In figure 5.10 the meshes M6 and M7 used for the dis-cretization are reported. The post-collapse behaviour of the cone was obtained

Model MeshElement

typeNodes Elements

Degree of

freedomNotes

H M6 TRIC 390 720 1981 Whole cone

I M7 S9R5 882 210 5298 Whole cone


by the sequential limit analysis in 30 step with a maximum displacement of 2.0mm. On the contrary the elastic-plastic incremental analysis was performed usingthe modified Riks method implemented in the commercial code ABAQUS withautomatic evaluation of the step amplitude. This procedure is strongly recom-mended when the analysis is expected to be highly non-linear as in this case. Theload-shortening curve reported in figure 5.11(a) is not meaningful because, for theparticular load case under consideration, the area under the curve does not rep-resent the dissipated energy. Moreover the top displacements are very smalls andthe elastic strains can become the relevant part of the whole shortening. Indeed

the elastic effects, present prior the collapse, can be recognized from the dashedcurve pictured in figure 5.11(a) where it is manifest that their amplitude is notnegligible. Therefore a comparison in terms of the dissipated energy, which is analways growing quantity, is also reported in figure 5.11(b). Although the curvesare not coincident their shape is similar: both exhibit a drop in the pressure

79




(a) mesh M6 (b) mesh M7

Figure 5.10: the different meshes used for the analysis of the TICC5 cone

followed by an horizontal path where the shell proceeds to dissipate at constantpressure. The horizontal gap between the two curves is probably due to the elasticdisplacements that influence the collapse mechanisms detected by the methods.As reported in figures 5.12 and 5.13, the post-collapse behaviour, foreseen by thenumerical analyses, is rather different: the incremental analysis identify threewaves (see figure 5.13) as experimentally obtained [105], while the limit methodproposed in this work enhance the deformations present in one wave only (figure5.12). The maximum loads obtained are reported in table 5.8 and compared withthe experimental collapse value. The latter is sensibly higher than the valuesnumerically obtained and this is probably due to the hardening in the material.Being impossible to compare the two analyses for the same top displacement, theCPU solution times, reported in table 5.8, refer to a dissipated energy of 2 MJ.In spite of the fact that the same dissipation can be obtained by the sequentiallimit analysis in fewer step than those required by the incremental solution, theneeded CPU time is five times higher. This is in clearly disagreement with theresults obtained for the plate and the cylindrical shell where the sequential limitanalysis has always been faster than the incremental one.

80



5.2. Conical shells

0 0.05 0.1 0.15 0.2 0.25Shortening (mm)

0

5

10

15

20

25

30


¢ ¡ £ ¥ § ©

¡ ¥

¡ ¥ ©

Experimentalrupture

(a) load-shortening curve

0 1000000 2000000

Dissipation (J)

0

5

10

15

20

25

30


¡ £ ¥ § ©

¡ ¥

¡ ¥ ©

(b) load-dissipation curve

Figure 5.11: shortening and dissipation of the cone TICC5. The dots refer to thedeformated states reported in figures 5.12 and 5.13

Model Method P max (MPa)Number of

iterationsCPU Time

Exper. 27.59 — —

H S.L.A 24.00 30 12’05”

I Incremental 22.00 59 2’40”

Table 5.8: results obtained for the TICC5 cone subject to axial load and external

pressure. The CPU times refer to a dissipation of 2 MJ. The analyses have beenperformed on a PC-Athlon 1200 MHz

81




(a) dissipation = 703.6 kJ

(b) dissipation = 1286.3 kJ

| .

|

(c) dissipation = 1605.2 kJ

(d) dissipation = 1874.5 kJ

Figure 5.12: some steps of the collapse foreseen by the sequential limit analysis

82



5.2. Conical shells

-

- - -- - -

(a) dissipation = 74.8 kJ

-

- -------------

- - -- - -

(b) dissipation = 235.5 kJ

-

- - -- - -

(c) dissipation = 770.6 kJ

-

- -------------

- - -- - -

(d) dissipation = 1267.0 kJ

Figure 5.13: some steps of the collapse foreseen by the elasto-plastic incrementalanalysis

83




5.3 Square tube

In previous sections the method proposed has been applied to the study of thepost-collapse behaviour of cylindrical shells and frusta. An accurate investigationof Wirsching and Slater [124] permitted to underline that circular tubes provideone of the best energy-absorption capability. Nevertheless, in many applicationsuch as cars, railways coaches and ships, tubular frames are rarely used andoccasional impacts are absorbed by sheet metal or plastic structures. In orderto understand the post-collapse behaviour of such structures many experimentswere conducted over square tubes [2,69,113,121–123] and small model coaches [82].

Dynamic effects were also investigated [56,71] and the crumpling load was pointedout to be generally higher than the corrisponding static value.Referring to the static analyses some advances in the mechanics of crumplingof square tubes have been made, working out some theoretical models whichinclude the plastic work done by travelling plastic hinges. The latter concepthas been introduced [73,97,125] for experimental indication of the rolling radiusand [69,123] for theoretical predictions, but has also been applied with success tothe analysis of cylindrical shells subject to axial load. Obviously these models arenot as general as possible and they can be applied only in particular situations:for example initial imperfections or not classic boundary conditions are difficultto deal with. Recent advances in numerical methods applied to limit analysissuggest to use the finite element method for these problem too.Huh et al. [62] developed a degenerated four-node shell element suitable for limit

analysis and checked their approach by simulating some square tubes. In parti-cular the capacity of dealing with strain-hardening materials was introduced intothe formulation of the limit problem, simply by tracking the effective plastic straincalculated from successive iterations. Nevertheless this is not a challenge problemand this capability could be easily implemented in the present work. The maindifference between this method and the work published by Huh et al. is due tothe element used. While for the four-node element employed in [62] the reducedintegration technique was necessary to avoid locking phenomena and the zero-energy modes were eliminated by using phisical stabilisation, the TRIC element,here modified in a suitable way, do not need particular care and the lockingproblem is avoided by its natural formulation and by minimizing respect to thebending parameter χ.

In order to check the ability of the TRIC element to describe also the post-collapse behaviour of square tubes, an analysis is performed on the same specimenstudied by Huh et al. [62], by means of the approach present in this work. Sincethe possibility to consider strain-hardening materials was not implemented theresults are expected to be lower than that obtained in the original work.

84



5.3. Square tube

5.3.1 Axial load

The considered specimen is a square tube whose dimensions are indicated in

table 5.9. The symbols used are evident from figure 5.14(a) where a sketch of thetube is reported. An indication of the specimen slenderness may be useful to

a

(mm)

h

(mm)

t

(mm)

50 50 1.4

Table 5.9: geometrics characteristics of the tube considered

h

a a

t

(a) the specimen geometry

¢ ¡

(b) the initial imperfection

Figure 5.14: specimen geometry and imperfections assumed for the analyses

understand which type of collapse can occur. During the tests the bottom andtop edges were welded to a plate, therefore they can be considered clamped andthe elastic buckling load can be found as the buckling load of four square plateswith two sides simply supported and the others clamped. From [26] the buckling

load is:P E = 4th

π2D

h26.7

where the coefficient 6.7, depending on the boundary conditions, can be found ingraphs or tables from the literature [26]. The squash load is calculated neglecting

85




the boundary conditions effects with the following expression:

P 0

= 4thσ0

The mechanical properties, the squash and the buckling loads are reported in table5.10. A value of 0.45 for the slenderness parameter λ states that the tube is stocky,therefore the collapse can be studied by means of the limit analysis. For sake of

σ0

(MPa)

E

(GPa)ν

P 0

(kN)

P E

(kN)λ

270 205 0.3 75.6 381.5 0.45

Table 5.10: mechanical characteristics of the tube considered

comparison two analyses have been performed by the limit method proposed inthis work and by the classic incremental approach. The models adopted, whosecharacteristics are reported in table 5.11, have been labelled J and K, being thefirst used for the sequential limit analysis while the second has been used for theelasto-plastic method. The corresponding meshes M8 and M9 are pictured infigure 5.3.1: in particular the mesh M8 consists of 441 nodes and 800 elementsfor a total of 2281 degrees of freedom while the M9 consists of 1089 nodes and256 elements for a total of 6534 degrees of freedom. Being the tube symmetriconly a quarter has been considered and proper boundary conditions are imposedat the vertical edges. Moreove, clamped boundary conditions are applied at thetop and bottom edges as imposed by the test equipment. The post-collapse of

Model MeshElement

typeNodes Elements

Degrees of

freedomNotes

J M8 TRIC 441 800 2281 Quarter

K M9 S9R5 1089 256 6534 Quarter


the square tube was obtained by the sequential limit analysis in 14 steps: thefirst five with a maximum displacement of 0.3 mm and the following steps with

a maximum displacement of 2 mm. This permits to better follow the collapse inthe beginning, reducing the approximation errors of the subsequent steps. On thecontrary, the modified Riks algorithm has been used in the incremental analysiswhere the step amplitude was automatically setted by the solver. As reportedin table 5.12 the incremental simulation need much more steps than the method

86



5.3. Square tube

¡

(a) M8 mesh

(b) M9 mesh

Figure 5.15: the mesh used in the analyses

proposed to achieve a vertical displacement of 12 mm after which the contactoccurs. This leads to a notable difference in solution times as can be seen fromthe values reported. The load-shortening curves just obtained are reported in

figure 5.16(a) and compared with experimental data obtained by Huh et al. Forsake of comparison the experimental curve was traslated towards left in orderto do not consider the elastic effects. The limit load and the related collapsemode detected by the present approach agree well with experimental results andincremental analysis. On the contrary the post collapse behaviour is influencedby strain hardening effects which are not considered in the present work: this isthe reason because experimental results are higher than the values here obtained.Anyway the sequential limit analysis is a stable and very efficient tool to predictthe behaviour of structures that undergo plastic flow during the collapse. Inparticular this effectiveness is much more manifest when dealing with decreasingcollapse evolution.The collapse mode analytically obtained, whose evolution is reported in figure

5.17, is similar with the experimental deformation reported in figure 5.16.

87




Model MethodP max

(MPa)

Number of

iterationsCPU Time

Exper. 80.0 — —

J S.L.A 76.57 14 7’56”

K Incremental 24.50 970 37’30”

Table 5.12: results obtained for the square tube subject to axial load. The CPUtimes refer to a shortening of 12 mm. The analyses have been performed on aPC-Athlon 1200 MHz

0 2 4 6 8 1 0 12

Shortening (mm)

0

20

40

60

80

100

F o r c e

( k N )

¡ £ ¥ § © £ ©

© § " © $ ¥ %

' ( 0

© % 2

' ( 0

© % 3

(a) load-shortening curves (b) during the experiment [62]

Figure 5.16: square tube subject to axial load

88



5.3. Square tube

(a) shortening = 0.6 mm

(b) shortening = 1.2 mm

¡

(c) shortening = 3.1 mm

¡

(d) shortening = 9.7 mm

Figure 5.17: some steps of the collapse

89




90



Chapter 6

Conclusions

6.1 A critical survey

A method to follow the post-collapse behaviour of shells that buckle in the plasticrange has been proposed. The method is set on the theoretycal basis of limitanalysis. In particular the kinematic approach is formulated, in conjunction withfinite element modelling, so as to reduce the problem to the search of the essen-tially free minimum of a convex function. Such function is not differentiable whereplastic strain rates vanish, namely in the rigid parts of the body, therefore, in thesolution procedure the elements detected as rigid are eliminated and appropriate

constraints equations are set. Two shell elements, formulated accordingly to thenatural approach, have been proposed:

• an axisymmetric one, based on Kirchhoff’s hypotheses, suitable for the ana-lysis of thin shell structures;

• a triangular shell element whose natural modes depends on nodal displace-ments and on bending parameter χe. This element is suitable both for theanalysis of thin and medium thick shells.

Regarding the triangular shell element, the solution procedure has been modifiedto account for the effects of the bending parameter, thus avoiding the lockingphenomenon typical of Mindlin’s elements.The method proposed has been applied to study the post-collapse behaviour of

some shell structures; the results obtained permit to draw the following conclu-sions:

• the method is much more stable than the incremental approach and biggersteps can be used in order to obtain the desidered solution;

91



Chapter 6. Conclusions

• the analyses performed with the approach proposed are generally faster thanthe incremental counterparts;

• the triangular element proposed is more accurate than classic triangularelement and it performs well with thick shells too.

In conclusion, the method proposed yields some important benefits to thecollapse analysis, proposing itself as an efficient tool for the prediction of thecollapse load and the post-collapse behaviour of structures that buckle in theplastic range.

6.2 Future developments

Some work has to be done in order to increase the capabilities of the method. First

of all, the procedure implemented to set as rigid the elements that do not exhibitplastic strains has to be modified. Actually the effectiveness of the whole methodcan be negatively affected when the related constraints are applied, furthermore,improvement in this topic could be exploited by a future contact algorithm.In order to deal with crash problems, the method has to be modified to accountfor inertial forces rising from high acceleration, as well as a contact algorithm hasto be implemented. Although only consistency conditions have to be satisfied,this is always a challenge problem, especially when dealing with contact and self-contact situations.Another interesting characteristic we are dealing with, is the ability to considertwo different sets of load. A set composed of constant load, and a second setwhere loads, instead, are affected by a load multiplier.

92



Appendix A

Strains from drilling modes

As seen in section 3.5.4 the displacement vector of a point P on a generic elementsubject to a drilling mode qdα is:

vρα = ζ 2αqdα(−(y1(ζ α − 1) + ζ βy2 + ζ γy3)i) + (x1(ζ α − 1) + ζ βx2 + ζ γx3) j) (A.1)

Natural strains due to the drilling modes are calculated from the displacementsin P . They are found to be:

mi =j

∂vρji∂Y i

i, j = α,β,γ (A.2)

where vρji is the displacement in P due to the drilling mode j projected onthe side i.In the following the calculations necessary to expand the relations A.2 are re-ported.

vραα = vρα • eα = ζ 2αqdα( − ((ζ α − 1)y1 + ζ βy2 + ζ γy3)i+

(x1(ζ α − 1) + ζ βx2 + ζ γx3) j) • (cαxisαx j)(A.3)

vρβα = vρβ • eα = ζ 2βqdβ( − (ζ αy1 + (ζ β − 1)y2 + ζ γy3)i+

(ζ αx1 + (ζ β − 1)x2 + ζ γx3) j) • (cαx isαx j)(A.4)

vργα = vργ•

eα = ζ 2γqdγ(−

(ζ αy1 + ζ βy2 + (ζ γ−

1)y3)i+

(ζ αx1 + ζ βx2 + (ζ γ − 1)x3) j) • (cαxisαx j) (A.5)

If the projections of side j on local axes x, y are called xj , yj the equationA.5 can be written in a more simplified way:

93



Appendix A. Strains from drilling modes

vραα

= ζ 2α

qdα

(−

(ζ β

yγ −

ζ γ

yβ

)cαx

+ (ζ β

xγ −

ζ γ

xβ

)sαx

) (A.6)

vρβα = ζ 2βqdβ((ζ αyγ − ζ γyα)cαx − (ζ αxγ − ζ γxα)sαx) (A.7)

vργα = ζ 2γψδ(−(ζ αyβ − ζ βyα)cαx + (ζ αxβ − ζ βxα)sαx) (A.8)

The calculus of the components parallel to sides β and γ can be done changingthe direction cosines. The strains are derived from the definition:

mα =dvραdY α

=1

lα

dvραdηα

=1

lα j

∂vρjα∂ζ α

∂ζ α∂ηα

+∂vρjα

∂ζ β

∂ζ β∂ηα

+∂vρjα

∂ζ γ

∂ζ γ∂ηα

(A.9)

mβ =dvρβdY β

=1

lβ

dvρβdηβ

=1

lβ

j

∂vρjβ

∂ζ α

∂ζ α∂ηβ

+∂vρjβ

∂ζ β

∂ζ β∂ηβ

+∂vρjβ

∂ζ γ

∂ζ γ∂ηβ

(A.10)

mγ =dvργdY γ

=1

lγ

dvργdηγ

=1

lγ

j

∂vρjγ

∂ζ α

∂ζ α∂ηγ

+∂vρjγ

∂ζ β

∂ζ β∂ηγ

+∂vρjγ

∂ζ γ

∂ζ γ∂ηγ

(A.11)

From the derivatives of ζ respect to η (equation 3.61):

mα =1

lα

j

−∂vρjα

∂ζ β+

∂vρjα∂ζ γ

(A.12a)

mβ =1

lβ

j

∂vρjβ

∂ζ α− ∂vρjβ

∂ζ γ

(A.12b)

mγ =1

lγ

j

−∂vρjγ

∂ζ α+

∂vρjγ∂ζ β

(A.12c)

(A.12d)

Partial derivatives of v respect to the internal coordinate ζ are now calculated:

∂vραα∂ζ α

= 2ζ αqdα((−ζ βyγ + ζ γyβ)cαx + (ζ βxγ − ζ γxβ)sαx)

= 2ζ αqdα(ζ β(−yγcαx + xγsαx) + ζ γ(yβcαx − xβsαx))

(A.13)

94



It can be observed that the terms in the internal parentheses are the projec-tions of sides β and γ on the direction perpendicular to side α, therefore they are

the height hα of the triangle. Introducing the element area Ω the derivatives canbe written in a more simplified way:

∂vραα∂ζ α

= 2ζ αqdα2Ω

lα(ζ β + ζ γ) = 4

Ω

lαζ α(1 − ζ α)qdα

∂vραα∂ζ β

= ζ 2αqdα(−cαxyγ + sαxx

γ) = 2

Ω

lαζ 2αqdα

∂vραα∂ζ γ

= ζ 2αqdα(cαxyβ − sαxx

β) = 2

Ω

lαζ 2αqdα

∂vρβα∂ζ α

= ζ 2βqdβ(cαxyγ − sαxx

γ) = −2

Ω

lαζ 2αqdα

∂vρβα∂ζ β

= 2ζ βqdβ((ζ αyγ − ζ γyα)cαx + (−ζ αxγ + ζ γxα)sαx)

= 2ζ βqdβ(ζ α(−cαxyγ + sαxx

γ) − ζ γ(cαxy

α − sαxx

α))

= −4Ω

lαζ αζ βqdβ

∂vρβα∂ζ γ

= ζ 2βqdβ(−cαxyα + sαxx

α) = 0

∂vργα∂ζ α

= ζ 2γqdγ(−cαxyβ + sαxx

β) = −2

Ω

lαζ 2γqdγ

∂vργα∂ζ β

= ζ 2γqdγ(cαxyα − sαxx

α) = 0

∂vργα∂ζ γ

= 2ζ γqdγ((−ζ αyβ + ζ βyα)cαx + (ζ αxβ − ζ βxα)sαx)

= 2ζ γqdγ(−ζ α(cαxyβ − sαxx

β) + ζ β(cαxy

α − sαxx

α))

= −4Ω

lαζ αζ γqdγ

∂vραβ∂ζ α

= 2ζ αqdα(ζ β(−cβxyγ + sβxx

γ) + ζ γ(cβxy

β − sβxx

β))

= −4Ω

lβζ αζ βqdα

∂vραβ∂ζ β

= ζ 2αqdα(−cβxyγ + sβxx

γ) = −2

Ω

lβζ 2αqdα

∂vραβ∂ζ γ

= ζ 2αqdα(cβxyβ − sβxx

β) = 0

∂vρββ∂ζ α

= ζ 2βqdβ(cβxyγ − sβxx

γ) = 2

Ω

lβζ 2βqdβ

95




∂vρββ∂ζ β

= 2ζ βqdβ(ζ α(cβxyγ − sβxx

γ) − ζ γ(cβxy

α − sβxx

α))

= 4 Ωlβ

ζ β(1 − ζ β)qdβ

∂vρββ∂ζ γ

= ζ 2βqdβ(−cβxyα + sβxx

α) = 2

Ω

lβζ 2βqdβ

∂vργβ∂ζ α

= ζ 2γqdγ(−cβxyβ + sβxx

β) = 0

∂vργβ∂ζ β

= ζ 2γqdγ(cβxyα − sβxx

α) = −2

Ω

lβζ 2γqdγ

∂vργβ∂ζ γ

= 2ζ γqdγ(−ζ α(cβxyβ − sβxx

β) + ζ β(cβxy

α − sβxx

α))

=−

4Ω

lβζ βζ γqdγ

∂vραγ∂ζ α

= 2ζ αqdα(ζ β(−cγxyγ + sγxx

γ) + ζ γ(cγxy

β − sγxx

β))

= −4Ω

lγζ αζ γqdα

∂vραγ∂ζ β

= ζ 2αqdα(−cγxyγ + sγxx

γ) = 0

∂vραγ∂ζ γ

= ζ 2αqdα(cγxyβ − sγxx

β) = −2

Ω

lγζ 2αqdα

∂vρβγ∂ζ α

= ζ 2βqdβ(cγxyγ − sγxx

γ) = 0

∂vρβγ∂ζ β

= 2ζ βqdβ(ζ α(cγxyγ − sγxx

γ) − ζ γ(cγxy

α − sγxx

α)) = −4

Ω

lγζ βζ γqdβ

∂vρβγ∂ζ γ

= ζ 2βqdβ(−cγxyα + sγxx

α) = −2

Ω

lγζ 2βqdβ

∂vργγ∂ζ α

= ζ 2γqdγ(−cγxyβ + sγxx

β) = 2

Ω

lγζ 2γqdγ

∂vργγ∂ζ β

= ζ 2γqdγ(cγxyα − sγxx

α) = 2

Ω

lγζ 2γqdγ

∂vργγ∂ζ γ

= 2ζ γqdγ(−ζ α(cγxyβ − sγxx

β) + ζ β(cγxy

α − sγxx

α))

= 4Ω

lγ ζ γ(1 − ζ γ)qdγ

From equationa A.12a, A.12b and A.12c the strains due to the drilling modesare now calculated:

96



α

mα =

1

lα −2

Ω

lα ζ 2αqdα + 2

Ω

lα ζ 2αqdα = 0

βmα =1

lα

4

Ω

lαζ αζ βqdβ

γmα =

1

lα

−4

Ω

lαζ αζ γqdγ

mα = 4Ω

l2α

ζ α(ζ βqdβ − ζ γqdγ) (A.14)

αmβ =1

lβ

−4

Ω

lβζ αζ βqdα

βmβ =

1

lβ

2

Ω

lβζ 2βqdβ + 2

Ω

lβζ 2βqdβ

= 0

γmβ = 1lβ4 Ω

lβζ βζ γqdγ

mβ = 4Ω

l2β

ζ β(ζ γqdγ − ζ αqdα) (A.15)

αmγ =1

lγ

4

Ω

lγζ αζ γqdα

βmγ =

1

lγ

−4

Ω

lγζ βζ γqdβ

γmγ =

1

lγ

2

Ω

lγζ 2γqdγ + 2

Ω

lγζ 2γqdγ

= 0

mγ = 4

Ω

l2γ

ζ γ(ζ αqdα − ζ βqdβ) (A.16)

and the equations 3.92, 3.93 and 3.94 are demonstrated.

97




98



Appendix B

Details on the compilation

of the energy dissipation

As seen in section 3.5.4 the strains in the element can be formulated in termsof natural modes. The expressions are here reported and in the following somecalculations will be done in order to obtain a more simple and easily implementableform.

mα =q0mα + z

qSαlα

− z1

l2α

3lα(ζ γ − ζ β)qbAα + lβζ αqbAβ − lγζ αqbAγ

+ 4

Ω

l2α

ζ α(ζ βqβ − ζ γqγ)

(B.1a)

mβ =q0mβ + z qSβ

lβ− z 1

l2β

−lαζ βqbAα + 3lβ(ζ α − ζ γ)qbAβ + lγζ βqbAγ

+ 4Ω

l2β

ζ β(ζ γqγ − ζ αqα)

(B.1b)

mγ =q0mγ + z

qSγlγ

− z1

l2γ

lαζ γqbAα − lβζ γqbAβ + 3lγ(ζ β − ζ α)qbAγ

+ 4

Ω

l2γ

ζ γ(ζ αqα − ζ βqβ)

(B.1c)

sα = qsAα(1 − ζ α)

2(B.1d)

sβ = qsAβ(1 − ζ β)

2(B.1e)

sγ = qsAγ(1 − ζ γ)

2(B.1f)

99



Appendix B. Details on the compilation of the energy dissipation

where qbA + qsA = qA.The dissipated energy in each element can be calculated by means of the following

equation:

De =

Ωe

σ0

ρT mΠm

˙ρm + ˙ρT s Πs˙ρsdx (B.2)

where for convenience the transversal shear modes are separated from theother ones:

ρm12×1

=q0m qS qbA qd

T (B.3)

ρs3×1

=qsA

(B.4)

The terms of symmetric matrixes Πm12×12 and Πs3×3, defined in equation 3.119, arenow evaluated

Πm(1, 1) = Γm11

Πm(1, 2) = Γm12

Πm(1, 3) = Γm13

Πm(1, 4) =z

lαΓm11

Πm(1, 5) =

z

lβ Γm12

Πm(1, 6) =z

lγΓm13

Πm(1, 7) = 3ζ β − ζ γ

lαzΓm11 +

lαζ βl2β

zΓm12 − lαζ γl2γ

zΓm13

Πm(1, 8) = − lβζ αl2α

zΓm11 − 3ζ α − ζ γ

lβzΓm12 +

lβζ γl2γ

zΓm13

Πm(1, 9) =lγζ α

l2α

zΓm11 − lγζ βl2β

zΓm12 + 3(ζ α − ζ β)

lγzΓm13

Πm(1, 10) = −4Ωl2β

ζ αζ βΓm12 + 4Ωl2γ

ζ αζ γΓm13

Πm(1, 11) =4Ω

l2α

ζ αζ βΓm11 − 4Ω

l2γ

ζ βζ γΓm13

100



Πm(1, 12) = −4Ω

l2α

ζ αζ γΓm11 +4Ω

l2β

ζ βζ γΓm12

Πm(2, 2) = Γm22

Πm(2, 3) = Γm23

Πm(2, 4) =z

lαΓm21

Πm(2, 5) =z

lβΓm22

Πm(2, 6) =z

lγΓm23

Πm(2, 7) = 3

(ζ β

−ζ γ)

lα z

Γm21 +

lαζ β

l2β z

Γm22 −lαζ γ

l2γ z

Γm23

Πm(2, 8) = − lβζ αl2α

zΓm21 − 3(ζ α − ζ γ)

lβzΓm22 +

lβζ γl2γ

zΓm23

Πm(2, 9) =lγζ α

l2α


zΓm22 + 3(ζ α − ζ β)

lγzΓm23

Πm(2, 10) = −4Ω

l2β

ζ αζ βΓm22 +4Ω

l2γ

ζ αζ γΓm23

Πm(2, 11) =4Ω

l2α


l2γ

ζ βζ γΓm23

Πm(2, 12) = −4Ω

l2α


l2β

ζ βζ γΓm22

Πm(3, 3) = Γm33

Πm(3, 4) =z

lαΓm31

Πm(3, 5) =z

lβΓm32

Πm(3, 6) =z

lγΓm33

Πm(3, 7) = 3(ζ β − ζ γ)

lαzΓm31 +

lαζ βl2β

zΓm32 − lαζ γl2γ

zΓm33

101




Πm(3, 8) = − lβζ αl2α

zΓm31 − 3(ζ α − ζ γ)

lβzΓm32 +

lβζ γl2γ

zΓm33

Πm(3, 9) = lγζ αl2α


zΓm32 + 3 (ζ α − ζ β)lγ

zΓm33

Πm(3, 10) = −4Ω

l2β

ζ αζ βΓm32 +4Ω

l2γ

ζ αζ γΓm33

Πm(3, 11) =4Ω

l2α


l2γ

ζ βζ γΓm33

Πm(3, 12) = −4Ω

l2α


l2β

ζ βζ γΓm32

Πm(4, 4) =z2

l2α

Γm11

Πm(4, 5) =z2

lαlβΓm12

Πm(4, 6) =z2

lαlγΓm13

Πm(4, 7) = 3(ζ β − ζ γ)

l2α

z2Γm11 +ζ βl2β

z2Γm12 − ζ γl2γ

z2Γ13

Πm(4, 8) = − lβζ αl3α

z2Γm11 − 3(ζ α − ζ γ)

lαlβz2Γm12 +

lβζ γlαl2

γ

z2Γm13

Πm(4, 9) = lγζ αl3α

z2Γm11 − lγζ βlαl2

β

z2Γm12 + 3 ζ α − ζ βlαlγ

z2Γm13

Πm(4, 10) = − 4Ω

lαl2β

zζ αζ βΓm12 +4Ω

lαl2γ

zζ αζ γΓm13

Πm(4, 11) =4Ω

l3α

zζ αζ βΓm11 − 4Ω

lαl2γ

zζ βζ γΓm13

Πm(4, 12) = −4Ω

l3α

zζ αζ γΓm11 +4Ω

lαl2β

zζ βζ γΓm12

Πm(5, 5) =z2

l2β

Γm22

Πm(5, 6) =z2

lβlγΓm23

102



Πm(5, 7) = 3(ζ β − ζ γ)

lαlβz2Γm21 +

lαζ βl3β

z2Γm22 − lαζ γlβl2

γ

z2Γm23

Πm(5, 8) = −ζ αl2α

z2Γm21 − 3(ζ α − ζ γ)

l2β

z2Γm22 +ζ γl2γ

z2Γm23

Πm(5, 9) =lγζ αl2αlβ

z2Γm21 − lγζ βl3β

z2Γm22 + 3(ζ α − ζ β)

lβlγz2Γm23

Πm(5, 10) = −4Ω

l3β


lβl2γ

zζ αζ γΓm23

Πm(5, 11) =4Ω

l2αlβ


lβl2γ

zζ βζ γΓm23

Πm(5, 12) =

−4Ω

l2αlβzζ αζ γΓm21 +

4Ω

l3β

zζ βζ γΓm22

Πm(6, 6) =z2

l2γ

Γm33

Πm(6, 7) = 3(ζ β − ζ γ)

lαlγz2Γm31 +

lαζ βl2βlγ

z2Γm32 − lαζ γl3γ

z2Γm33

Πm(6, 8) = − lβζ αl2αlγ

z2Γm31 − 3(ζ α − ζ γ)

lβlγz2Γm32 +

lβζ γl3γ

z2Γm33

Πm(6, 9) =ζ αl2α

z2Γm31 − ζ βl2β

z2Γm32 + 3(ζ α − ζ β)

l2γ

z2Γm33

Πm(6, 10) = − 4Ω

l2βlγ


l3γ

zζ αζ γΓm33

Πm(6, 11) =4Ω

l2αlγ


l3γ

zζ βζ γΓm33

Πm(6, 12) = − 4Ω

l2αlγ

zζ αζ γΓm31 +4Ω

l2βlγ

zζ βζ γΓm32

Πm(7, 7) = 9(ζ β − ζ γ)2

l2α

z2Γm11 + 3ζ β(ζ β − ζ γ)

l2β

z2(Γm21 + Γm12) +l2αζ 2βl4β

z2Γm22

− 3

ζ γ(ζ β

−ζ γ)

l2γ z

2

(Γm31 + Γm13) −l2αζ βζ γ

l2βl2γ z

2

(Γm32 + Γm23) +

l2αζ 2γ

l4γ z

2

Γm33

103




Πm(7, 8) = −3lβζ α(ζ β − ζ γ)

l3α

z2Γm11 + 3lαζ γ(ζ α − ζ γ)

lβl2γ

z2Γm32 +lβζ αζ γ

lαl2γ

z2Γm31

− 9 (ζ α − ζ γ)(ζ β − ζ γ)lαlβ

z2Γm12 + lαζ βζ γlβl2

γ

z2Γm23 − ζ αζ βlαlβ

z2Γm21

+ 3lβ(ζ β − ζ γ)ζ γ

lαl2γ

z2Γm13 − 3lαζ β(ζ α − ζ γ)

l3β

z2Γm22 − lαlβζ 2γl4γ

z2Γm33

Πm(7, 9) = 3lγζ α(ζ β − ζ γ)

l3α

z2Γm11 +lγζ αζ β

lαl2β

z2Γm21 − 3lα(ζ α − ζ β)ζ γ

l3γ

z2Γm33

− 3lγζ β(ζ β − ζ γ)

lαl2β

z2Γm12 − lαlγζ 2βl4β

z2Γm22 + 3lα(ζ α − ζ β)ζ β

l2βlγ

z2Γm23

+ 9(ζ α − ζ β)(ζ β − ζ γ)ζ γ

lαlγz2Γm13 +

lαζ βζ γl2βlγ

z2Γm32 − ζ αζ γlαlγ

z2Γm31

Πm(7, 10) = −12Ωζ αζ β(ζ β − ζ γ)

lαl2β

zΓm12 − 4Ωlαζ αζ 2βl4β

zΓm22 +4Ωlαζ αζ βζ γ

l2βl2γ

zΓm32

+12Ωζ αζ γ(ζ β − ζ γ)

lαl2γ

zΓm13 +4Ωlαζ αζ βζ γ

l2βl2γ

zΓ23 − 4Ωlαζ αζ 2γl4γ

zΓm33

Πm(7, 11) =12Ωζ αζ β(ζ β − ζ γ)

l3α

zΓm11 +4Ωζ αζ 2β

lαl2β

zΓm21 − 4Ωζ αζ βζ γlαl2

γ

zΓm31

− 12Ωζ βζ γ(ζ β − ζ γ)

lαl2γ

zΓm13 − 4Ωlαζ 2βζ γ

l2βl2γ

zΓm23 +4Ωlαζ βζ 2γ

l4γ

zΓm33

Πm(7, 12) = −12Ωζ αζ γ(ζ β

−ζ γ)

l3α

zΓm11 −4Ωζ αζ βζ γ

lαl2β

zΓm21 +4Ωζ αζ 2γ

lαl2γ

zΓm31

+12Ωζ βζ γ(ζ β − ζ γ)

lαl2β

zΓm12 +4Ωlαζ 2βζ γ

l4β

zΓm22 − 4Ωlαζ βζ 2γl2βl2γ

zΓm32

Πm(8, 8) =l2βζ 2αl4α

z2Γm11 + 3ζ α(ζ α − ζ γ)

l2α

z2(Γm21 + Γm12)

− l2βζ αζ γ

l2αl2γ

z2(Γm31 + Γm13) + 9(ζ α − ζ γ)2

l2β

z2Γm22

− 3(ζ α − ζ γ)ζ γ

l2γ

z2(Γ32 + Γ23) +l2βζ 2γl4γ

zΓm33

104



Πm(8, 9) = − lβlγζ 2αl4α

z2Γm11 − 3lγζ α(ζ α − ζ γ)

l2αlβ

z2Γm21 + 3lβ(ζ α − ζ β)ζ γ

l3γ

z2Γm33

+ lγζ αζ βl2αlβ

z2Γm12 + 3 lγζ β(ζ α − ζ γ)l3β

z2Γm22 − 3 lβζ α(ζ α − ζ β)l2αlγ

z2Γm13

− ζ βζ γl2βlγ

z2Γm32 − 9(ζ α − ζ β)(ζ α − ζ γ)

lβlγz2Γ23 +

lβζ αζ γl2αlγ

z2Γm31

Πm(8, 10) = 4Ωζ 2αζ βl2αlβ

zΓm12 + 12Ωζ αζ β(ζ α − ζ γ)

l3β

zΓm22 − 4Ωζ αζ βζ γ

lβl2γ

zΓm32

− 4Ωlβζ 2αζ γ

l2αl2γ

zΓm13 − 12Ωζ αζ γ(ζ α − ζ γ)

lβl2γ

zΓm23 + 4Ωlβζ αζ 2γ

l4γ

zΓm33

Πm(8, 11) = −4Ωlβζ 2αζ β

l4α

zΓm11 − 12Ωζ αζ β(ζ α − ζ γ)

l2αlβ

zΓm21 + 4Ωlβζ αζ βζ γ

l2αl2γ

zΓm31

+ 4Ωlβζ αζ βζ γ

l2αl2γ

zΓm13 + 12Ωζ βζ γ(ζ α − ζ γ)

lβl2γ

zΓm23 − 4Ωlβζ βζ 2γ

l4γ

zΓm33

Πm(8, 12) = 4Ωlβζ 2αζ γ

l4α

zΓm11 + 12Ωζ αζ γ(ζ α − ζ γ)

l2αlβ

zΓm21 − 4Ωlβζ αζ 2γ

l2αl2γ

zΓm31

− 4Ωζ αζ βζ γ

l2αlβ

zΓm12 − 12Ωζ βζ γ(ζ α − ζ γ)

l3β

zΓm22 + 4Ωζ βζ 2γlβl2

γ

zΓm32

Πm(9, 9) =l2γζ 2αl4α

z2Γm11 − l2γζ αζ β

l2αl2β

z2(Γm21 + Γm12) + 3ζ α(ζ α − ζ β)

l2α

z2Γm31

+

l2γζ 2βl4β

z2Γm22 − 3ζ β(ζ α

−ζ β)

l2β

z2(Γm32 + Γ23)

+ 3ζ α(ζ α − ζ β)

l2α

z2(Γm31 + Γm13) + 9(ζ α − ζ β)2

l2γ

z2Γm33

Πm(9, 10) = −4Ωlγζ 2αζ βl2αlβ2

zΓm12 + 4Ωlγζ αζ 2β

l4β

zΓ22 − 12Ωζ αζ β(ζ α − ζ β)

l2βlγ

zΓm32

+ 4Ωζ 2αζ γl2αlγ

zΓm13 − 4Ωζ αζ βζ γ

l2βlγ

zΓm23 + 12Ωζ αζ γ(ζ α − ζ β)

l3γ

zΓm33

Πm(9, 11) = 4Ωlγζ 2αζ β

l4α

zΓm11 − 4Ωlγζ αζ 2β

l2αl2β

zΓm21 + 12Ωζ αζ β(ζ α − ζ β)

l2αlγ

zΓm31

− 4Ωζ αζ βζ γ

l2αlγ

zΓm13 + 4Ωζ 2βζ γ

l2βlγ

zΓm23 − 12Ωζ βζ γ(ζ α − ζ β)

l3γ

zΓm33

105




Πm(9, 12) = −4Ωlγζ 2αζ γ

l4α

zΓm11 + 4Ωlγζ αζ βζ γ

l2αl2β

zΓm21 − 12Ωζ αζ γ(ζ α − ζ β)

l2αlγ

zΓm31

+ 4Ω lγζ αζ βζ γl2αl2β

zΓm12 − 4Ω lγζ 2βζ γ

l4β

zΓm22 + 12Ω ζ βζ γ(ζ α − ζ β)l2βlγ

zΓm32

Πm(10, 10) = 16Ω2ζ 2αζ 2β

l4β

Γm22 − 16Ω2 ζ 2αζ βζ γl2βl2γ

(Γm32 + Γm23) + 16Ω2ζ 2αζ 2γ

l4γ

Γm33

Πm(10, 11) = −16Ω2ζ 2αζ 2βl2αl2β

Γm21 + 16Ω2 ζ 2αζ βζ γl2αl2γ

Γm31 + 16Ω2ζ αζ 2βζ γ

l2βl2γ

Γm23

− 16Ω2ζ αζ βζ 2γ

l4γ

Γm33

Πm(10, 12) = 16Ω2 ζ 2αζ βζ γ

l2αl2β Γm21 − 16Ω

2ζ 2αζ 2γl2αl2γ Γm31 − 16Ω

2ζ αζ 2βζ γ

l4β Γm22

+ 16Ω2ζ αζ βζ 2γ

l2βl2γ

Γm32

Πm(11, 11) = 16Ω2ζ 2αζ 2β

l4α

Γm11 − 16Ω2ζ αζ 2βζ γ

l2αl2γ

(Γm31 + Γ13) + 16Ω2ζ 2βζ 2γ

l4γ

Γm33

Πm(11, 12) = −16Ω2 ζ 2αζ βζ γl4α

Γm11 + 16Ω2ζ αζ βζ 2γ

l2αl2γ

Γm31 + 16Ω2ζ αζ 2βζ γ

l2αl2β

Γm12

− 16Ω2ζ 2βζ 2γl2βl2γ

Γm32

Πm(12, 12) = 16Ω2ζ 2αζ 2γ

l4α

Γm11 − 16Ω2ζ αζ βζ 2γ

l2αl2β

(Γm12 + Γm21) + 16Ω2ζ 2βζ 2γ

l4β

Γm22

Πs(1, 1) =(1 − ζ α)2

4Γs11

Πs(1, 2) =(1 − ζ α)(1 − ζ β)

4Γs12

Πs(1, 3) =(1 − ζ α)(1 − ζ γ)

4Γs13

Πs(2, 2) =

(1

−ζ β)2

4 Γs22

Πs(2, 3) =(1 − ζ β)(1 − ζ γ)

4Γs23

106



Πs(3, 3) =(1 − ζ γ)2

4Γs33

The integral (B.2) can be evaluated analitically respect to the thickness whilethe integration over the mid-surface of the element has to be performed numeri-cally. In order to solve the integration over the thickness the dependence of Πm

and Πs on z is exploited:

Πm =

Πmm zΠmS zΠb

mA Πmd

zΠSm z2ΠSS z2ΠbSA zΠSd

zΠbAm z2Πb

AS z2ΠbAA zΠb

Ad

Πdm zΠdS zΠbdA zΠdd

Πs = Πs

AAwhere all the sub-matrixes are of shape 3x3.The integral (B.2) can be threfore written in the following way:

De =

Se

h2

− h2

σ0

am + as + bz + cz2dzdS (B.5)

where h is the thickness of the element and:

am = qT mΠmmqm + 2qT mΠmdqd + qT d Πddqd (B.6a)

as = qsAT

ΠsAAqsA (B.6b)

b = 2qT mΠmsqS + 2qT mΠbmAqbA + 2qT SΠSdqd + 2qbA

T ΠbAdqd (B.6c)

c = qT SΠSSqS + 2qT SΠbSAqbA + qbA

T ΠbAAqbA (B.6d)

It is pointed out that b eing Πm and Πs positive definite in E P therefore b2−4ac ≤0.The following cases have been examinated separately:

• c = 0 and b = 0 (corresponding to strains due to axial, shear and drillingmodes) h

2

− h2

am + as + bz + cz2dz =

√am + ash (B.7)

• am = 0 and b = 0 (corresponding to strains due to bending and shear modesonly)Let us indicate with r1 the following square root:

r1 =

4as + ch2

107




therefore:

h2

−h2 am + as + bz + cz2dz =

h

4 r1 +as

2√c lnr1 +

√ch

r1 − √ch (B.8)

• am = 0, as = 0 and b = 0 (corresponding to strains due to bending modesonly) h

2

−h2

am + as + bz + cz2dz =

√c

h2

4(B.9)

• b2 − 4ac = 0

h2

− h

2 a + bz + cz2dz =

2bh + ch2

8√

csign (b + ch) +

2bh − ch2

8√

csign (b − ch)

(B.10)

• b2 − 4ac < 0Let us indicate with r2 and r3 the following square roots:

r2 =

4a + 2bh + ch2

r3 =

4a − 2bh + ch2

Therefore: h2

− h2

a + bz + cz2dz =

1

8c32

√

c [(b + ch) r2 + (ch − b) r3] +b2 − 4ac

ln

b√

c− √

ch + r3

− ln

b√

c+

√ch + r2

(B.11)

The integration over the mid-plane can be performed by mean of a classicalquadrature procedure, in particular for the present element three gauss pointshave been considered at the following coordinate:

•G1:ζ

α= 1

6ζ β

= 2

3ζ γ

= 1

6• G2:

ζ α = 1

6ζ β = 1

6ζ γ = 2

3

• G3:

ζ α = 2

3 ζ β = 16 ζ γ = 1

6

108



Appendix C

Details on the minimization

procedure

The minimization procedure implemented for the limit problems to be solvedwith the general shell triangular element has been presented in subsection 3.5.7skipping some important aspects for the sake of simplicity. In this appendixboth the analytical integration over the thickness of the matrix K introducedin equation (3.123) and the theoretical derivation of equation (3.125) will bepresented.

C.1 Analytical integration to obtain K

The integration of the matrix K over the thickness presents much more problemsthan the integration of the dissipation due to the presence of the denominator.Using the notation introduced in appendix B and equation (3.124) the matrix K

can be written as:

K =e

LT e TT

e CT e

Se

h2

−h2

σ0Πz (χ)√

am + as + bz + cz2dzdSCeTeLe (C.1)

where

Πz = ΛT

e (χ) ΠeΛe (χ) = Πmm zΠmS zχeΠ

bmA Πmd

zΠSm z2ΠSS z2χeΠbSA zΠSd

zχeΠbAm z2χ2eΠ

bAS z2χ2eΠ

bAA+(1−χe)2Πs

AA zχeΠbAd

Πdm zΠdS zχeΠbdA Πdd

(C.2)

As for the integration of the power dissipation the values of the coefficient am,as, b and c lead to the following cases:

109



Appendix C. Details on the minimization procedure

• c = 0, b = 0This case happens only when no bending deformations occur.

h2−h

2

1√a

dz =h√

a(C.3a)

h2

−h2

z√a

dz = 0 (C.3b)

h2

−h2

z2

√a

dz =h3

12√

a(C.3c)

• am = 0, as = 0, b = 0This case can occur only when the axial (qm) and drilling (qd) modes arenulls and have to be applied as constraints. The element is therefore mem-

branally rigid and this allows to avoid the infinity of the first integral. h2

−h2

1√cz2

dz = ∞ (C.4a)

h2

−h2

z√cz2

dz = 0 (C.4b)

h2

−h2

z2

√cz2

dz =h2

4√

c(C.4c)

• am = 0, b = 0Also in this case the axial (qm) and drilling (qd) modes are nulls and have

to be applied as constraints. The positive term as allows to integrate thefollowing terms:

h2

− h2

1√as + cz2

dz =1√

cln

r1 +√

ch

r1 − √ch

(C.5a)

h2

− h2

z√as + cz2

dz = 0 (C.5b)

h2

− h2

z2

√as + cz2

dz =h

4cr1 − as

2c√

cln

r1 +√

ch

r1 − √ch

(C.5c)

• b2 − 4ac = 0From the physics point of view there exist a point on the section whereλ2 = 0, therefore in that point no dissipation occurs.This problem has tobe specifically dealed imposing some constraints rising from the variation

110



C.1. Analytical integration to obtain K

of the coefficient a. In particular if bch

≤ 1 the point is into the sectionotherwise it is external and no problem occurs.

Let us recalling the definition of the power dissipation:

D =e

Ωe

σ0λdx (C.6)

The matrix K is obtained as a consequence of the derivation process:

∂D

∂ u=e

Ωe

σ0∂ λ

∂ udx = Ku (C.7)

but

λ =

a (u) + b (u) z + c (u) z2 (C.8)

and therefore:

Ku =e

Ωe

σ0

2

∂a∂ u

+ ∂b∂ u

z + ∂c∂ u

z2

λdx (C.9)

But being a = b2

4cits variation depends on the variation of b and c:

∂a

∂ u=

2bc ∂b∂ u

− b2 ∂c∂ u

4c2(C.10)

while the plastic flow can be expressed as follows being c > 0:

λ = b2

4c + bz + cz2 =1

2√c |b + 2cz| (C.11)

The variation of the power dissipation become:

Ku =e

Ωe

σ0

√c∂a∂ u

+ ∂b∂ u

z + ∂c∂ u

z2

|b + 2cz| dx

=e

Ωe

σ0

√c

12c

(b + 2cz) ∂b∂ u

− 14c2

b2 − 4c2z2

∂c∂ u

|b + 2cz| dx

(C.12)

Without loss of generality σ0 can be supposed constant over the elementtherefore the expression of the variation become:

Ku =e

σ0

√c

1

2c

Ωe

b + 2cz

|b + 2cz|∂b

∂ udx−

1

4c2

Ωe

(b − 2cz) (b + 2cz)

|b + 2cz|∂c

∂ udx

(C.13)

111




The variation of a, b and c can be easily found if the coefficient are expressedin the following way:

a = ˙ρT N Πa ˙ρN = uT LT

e TT e CT

e ΠaCeTeLeu (C.14a)

b = ˙ρT N Πb ˙ρN = uT LT

e TT e CT

e ΠbCeTeLeu (C.14b)

c = ˙ρT N Πc ˙ρN = uT LT

e TT e CT

e ΠcCeTeLeu (C.14c)

where Le is the connectivity matrix, Te the coordinate transformation ma-trix, Ce the matrix of the natural modes while the Πi matrix are herereported:

Πa

12×12=ΛT

e Πa

Λ =

Πmm · · Πmd

· · · ·· · (1 − χe)

2ΠsAA ·

Πdm · · Πdd

(C.15a)

Πb

12×12=ΛT

e Πb

Λ =

· ΠmS χeΠb

mA ·ΠSm · · ΠSd

χeΠbAm · · χeΠb

Ad

· ΠdS χeΠbdA ·

(C.15b)

Πc

12×12=ΛT

e Πc

Λ =

· · · ·· ΠSS χeΠb

SA ·· χeΠb

AS χ2eΠb

AA ·· · · ·

(C.15c)

Therefore the variations of b and c are:

∂b

∂ u = 2LT e T

T e C

T e Π

b

CeTeLeu (C.16a)∂c

∂ u= 2LT

e TT e CT

e ΠcCeTeLeu (C.16b)

and do not depend on z. Now the integrations over the thickness can beperformed:

– bch

≤ 1

h2

− h2

b + 2cz

|b + 2cz|dz =

− b2c

−h2

−dz +

h2

− b2c

dz =b

c(C.17a)

h2−h

2

(b − 2cz) (b + 2cz)|b + 2cz| dz = − b2c

− h2

− (b − 2cz) dz+ h2

− b2c

(b − 2cz) dz =3b2

2c− ch2

2

(C.17b)

112



C.1. Analytical integration to obtain K

Therefore:

K = e σ0

√cLT

eTT

eCT

e Se b

c2Πb

− 3b2

4c3 −h2

4cΠc dS CeTeLe

(C.18)

– bch

> 1

h2

−h2

b + 2cz

|b + 2cz|dz = sign

b

ch

h (C.19a)

h2

−h2

(b − 2cz) (b + 2cz)

|b + 2cz| dz = sign

b

ch

bh (C.19b)

Therefore:

K = e

σ0√cL

T

e T

T

e C

T

e sign b

ch Se h

c Π

b

−bh

2c2 Π

c dS CeTeLe

(C.20)In order to avoid possibly indefiniteness of the matrix K the constraintsdue to the equation (C.10) have to be imposed. From equations (C.14)and (C.10) one obtains the following relation:

LT e TT

e CT e

Πa − bc

2Πb +

b2

4c2Πc

CeTeLeu = 0 (C.21)

The constraints (C.21) can be imposed using the same procedure adoptedfor the rigid elements.

• b2 − 4ac < 0

h2−h

2

1√a + bz + cz2

dz = 1√c

ln x + 1 + r3

x − 1 + r4(C.22a)

h2

−h2

z√a + bz + cz2

dz =h

2√

c

−x ln

x + 1 + r3

x − 1 + r4+ r3 − r4

(C.22b)

h2

−h2

z2√a + bz + cz2

dz =h2

8√

c

(3x + 1) r4 − (3x − 1) r3+3x2 − 4y

ln

x + 1 + r3

x − 1 + r4

(C.22c)

where

x =b

ch(C.23a)

y = 4ach2

(C.23b)

r3 =

y + 2x + 1 (C.23c)

r4 =

y − 2x + 1 (C.23d)

113




C.2 Minimization respect to χ

At solution the equation (3.122b) must be satisfied and this condition allow toobtain the correct bending factor.Let us suppose the bending factor χe constant over a whole element without anycorrelation with the bending factor of adjacent elements. Subject to this hypo-thesis the minimization can be performed separately on any element. Pointingout the dependance on χe one obtains:

De =

Ωe

σ0

ρT ΛT

e (χe) ΠeΛeρdx =

Ωe

σ0

λ2 (χe)dx (C.24)

The variation respect to χe is:

∂De

∂χe

= Ωe

σ0

∂ λ2

∂χ

2λ

dx (C.25)

If the calculations are performed:

λ2 (χe) =qT mΠmmqm + 2zqT mΠmSqS + 2zχeqT mΠbmAqA+

2qT mΠmdqd + z2qT SΠSS qS + 2z2χeqT SΠbSAqA+

2zqT SΠSdqd + qT A

z2χ2

eΠbAA + (1 − χe)2

ΠsAA

qA+

2zχeqT AΠbAdqd + qT d Πddqd

(C.26)

and

∂ λ2 (χe)

∂χe = 2zqT

mΠ

b

mAqA + 2z2

qT

SΠ

b

SAqA + 2χeqT

A z2

Π

b

AA + Π

s

AA qA

− 2qT AΠsAAqA + 2zqT AΠb

Adqd

(C.27)

Expression (C.27) can be written in a more simple way if two matrix, A and B,are introduced:

∂ λ2 (χe)

∂χe

= qT AAqAχe + ρT N BρN (C.28)

Substituting equation (C.28) in (C.25) and using the condition (3.122b) lead to:

χe = −

Ωe

ρT NBρNλ

dx

Ωe

qT AAqAλ

dx(C.29)

and the equation (3.125) is demonstrated.The analytical evaluation of the integrals over the thickness can be done as forthe matrix K (see section C.1).

114



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. . . and to my brothers

post critical behaviour of shells: a sequential limit analysis approach

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