3. UNIVERSITÀ MEDITERRANEA DI REGGIO CALABRIA
DOTTORATO DI RICERCA IN
INGEGNERIA MARITTIMA, DEI MATERIALI E DELLE STRUTTURE
(XXVI CICLO)
S.S.D. ICAR/08
STRUCTURAL ANALYSIS AT ULTIMATE LIMIT
STATES OF REINFORCED CONCRETE ELEMENTS:
A NUMERICAL APPROACH FOR THE EVALUATION
OF PEAK LOAD AND COLLAPSE MECHANISM
By
Dario De Domenico
Supervisors:
Professor P. Fuschi
Professor A.A. Pisano
Coordinator:
Professor F. Arena
REGGIO CALABRIA, JANUARY 2014

5. The grand aim of all science is to cover
the greatest number of empirical facts
by logical deduction from the smallest
possible number of hypotheses or axioms
ALBERT EINSTEIN (1879-1955)

7. TABLE OF CONTENTS
ACKNOWLEDGEMENTS I
ABSTRACT III
SOMMARIO V
LIST OF FIGURES VII
LIST OF TABLES XVII
1 GENERAL INTRODUCTION 1
1.1 Aims and scope of the thesis ...........................................................................................1
1.2 Outline of the thesis.........................................................................................................4
1.3 Related publications and original contributions of the thesis ..........................................5
2 FUNDAMENTALS: CONSTITUTIVE ASSUMPTIONS AND LIMIT ANALYSIS APPROACH 9
2.1 Introduction .....................................................................................................................9
2.2 Plasticity: some fundamental concepts..........................................................................10
2.2.1 The plasticity laws..............................................................................................14
2.2.2 Drucker’s stability postulate...............................................................................16
2.2.2.1 Principle of maximum plastic dissipation.............................................18
2.2.2.2 Normality rule and convexity of yield surface......................................20
2.2.3 Variational principles.........................................................................................22
2.3 Limit analysis ................................................................................................................27
2.3.1 Introduction........................................................................................................27
2.3.2 Standard limit analysis .......................................................................................27
2.3.2.1 Lower bound theorem (static theorem).................................................29
2.3.2.2 Upper bound theorem (kinematic theorem)..........................................30
2.3.3 Nonstandard limit analysis.................................................................................31
2.3.3.1 Radenkovic’s theorems.........................................................................33
3 LIMIT ANALYSIS PROCEDURES 35
3.1 Introduction ...................................................................................................................35
3.2 Hand-based procedures .................................................................................................36
3.2.1 Strut-and-tie model.............................................................................................36
3.2.2 Stringer-panel method........................................................................................41
3.2.3 Yield-line theory ................................................................................................44
3.2.4 Hillerborg’s strip method ...................................................................................50

8. 3.3 Computer-based procedures..........................................................................................54
3.3.1 Mathematical programming...............................................................................55
3.3.1.1 Linear programming.............................................................................58
3.3.1.2 Nonlinear programming .......................................................................62
3.3.2 Modulus variation procedures............................................................................76
3.3.2.1 Introductory remarks............................................................................76
3.3.2.2 Reduced modulus method ....................................................................77
3.3.2.3 GLOSS R-node method........................................................................78
3.3.2.4 The mα -method ....................................................................................79
3.3.2.5 Elastic compensation method (ECM)...................................................83
3.3.2.6 Linear matching method (LMM)..........................................................95
4 CONSTITUTIVE MODELS FOR CONCRETE 111
4.1 Introduction.................................................................................................................111
4.2 Main features of concrete behaviour...........................................................................113
4.2.1 Observations from experimental tests..............................................................113
4.2.2 Literature review of concrete constitutive models ...........................................119
4.3 Failure criteria for concrete and concrete-like materials.............................................137
4.3.1 Stress representations and stress invariants......................................................138
4.3.2 Strain representations and strain invariants......................................................147
4.3.3 Characteristics of concrete failure surface .......................................................149
4.3.4 Maximum-tensile-stress criterion (Rankine)....................................................153
4.3.5 Mohr–Coulomb criterion (with tension cutoff)................................................154
4.3.6 Drucker–Prager criterion .................................................................................158
4.3.7 Bresler–Pister criterion ....................................................................................160
4.3.8 Ottosen criterion ..............................................................................................161
4.3.9 Hsieh–Ting–Chen criterion..............................................................................162
4.3.10 Willam–Warnke criterion ................................................................................163
4.3.11 Menétrey–Willam criterion..............................................................................171
4.3.12 Generalized failure criterion ............................................................................179
4.3.13 Bigoni–Piccolroaz criterion .............................................................................182
4.4 Concrete models oriented to computational analysis ..................................................184
4.4.1 Lubliner-type models.......................................................................................184
4.4.1.1 Barcelona model.................................................................................185
4.4.1.2 Model by Lee and Fenves ..................................................................193
4.4.2 Menétrey–Willam-type model for concrete.....................................................195

9. 4.4.2.1 Menétrey–Willam-type yield surface .................................................200
5 THE PROPOSED FE-BASED LIMIT ANALYSIS APPROACH 205
5.1 Introductory remarks ...................................................................................................205
5.2 Linear matching method applied to the M–W-type yield surface ...............................207
5.2.1 Step #1: location of the matching point on the M–W-type yield surface.........216
5.2.2 Step #2: complementary energy density surface at matching ..........................223
5.3 Elastic compensation method applied to the M–W-type yield surface........................234
6 NUMERICAL PREDICTIONS AGAINST EXPERIMENTAL FINDINGS ON REAL PROTOTYPES 241
6.1 Introduction .................................................................................................................241
6.2 Standard tests on simple concrete specimens ..............................................................244
6.2.1 Compression test on a cubic plain concrete specimen .....................................244
6.2.2 Splitting tensile test on a cubic plain concrete specimen .................................247
6.2.3 Splitting tensile test on cylindrical specimens..................................................250
6.3 Steel-reinforced concrete elements..............................................................................255
6.3.1 Beams...............................................................................................................256
6.3.2 Walls ................................................................................................................263
6.3.3 Slabs.................................................................................................................268
6.4 FRP-reinforced concrete elements...............................................................................275
6.4.1 Beams...............................................................................................................277
6.4.2 Slabs.................................................................................................................279
6.4.3 Results and comments for the FRP-RC prototypes..........................................281
7 CONCLUSIONS AND FUTURE WORK 285
APPENDIX 289
A.1 The rate of energy dissipation per unit volume for a material obeying the von Mises
yield condition.............................................................................................................289
A.2 Proof of convergence of the linear matching method for a material obeying the von
Mises yield condition ..................................................................................................291
A.3 Proof of convergence of the linear matching method for a general class of yield
conditions ....................................................................................................................294
A.4 Numerical procedure for locating the point having a given normal on the deviatoric
plane of the M–W-type yield surface ..........................................................................296
REFERENCES 299

11. ACKNOWLEDGEMENTS
I would like to thank the University Mediterranea of Reggio Calabria, which awarded me a
scholarship to undertake this PhD programme, and the PAU Department, which has offered me
excellent working conditions for these 3 years. All staff and colleagues I met during this period
were warm and kind, helping me integrate into a familiar environment.
This thesis would never have been written without the support and constant encouragement
of my supervisors, Professor Paolo Fuschi and Professor Aurora Pisano. During the course of the
PhD programme, their invaluable ideas, guidance and trust helped me overcome a number of
difficulties arising in this research. They have taught me how to do my best and always be self-
critical to improve my work further, how to face scientific problems in a rigorous, well-founded
manner, and how not to give up in the face of difficulties that research work inevitably involves.
What I have appreciated most is the professionalism and expertise in what they do and the love
and enthusiasm they always have in their work, which has really motivated me during this PhD
course.
I wish to thank my parents who have been the source of love and education for their
inspiration and encouragement throughout the course of my studies including this Doctorate. It
has been a joy to feel their deep affection and support during the course of the PhD. They have
always done their best to create a serene, peaceful atmosphere. I believe that to be a good
student you have to be a responsible son, and I hope with all my heart not to ever disappoint
them.
Last, but definitely not least, I am extremely grateful to Valentina for her love, support,
patience and understanding shown since she has met me. She has been really supportive in
thousands of occasions during this period. She so often and so cheerfully sacrificed many
weekends and holiday times so that I could do my work peacefully and finish this thesis in time.
My love and gratitude for her cannot be expressed in words.
i

13. ABSTRACT
A numerical methodology, based on limit analysis theory, is developed to predict the peak
load, collapse mechanism and critical zones of reinforced concrete elements. The dilatancy of
concrete implies the lack of associativity and underlies the adoption of a nonstandard limit
analysis approach. Therefore, the proposed methodology makes use of two different numerical
methods of limit analysis in order to search for both an upper and a lower bound to the actual
peak load separately, namely: the Linear Matching Method and the Elastic Compensation
Method. Both methods, originally conceived for von Mises type materials, are here extended to
deal with a 3D plasticity model for concrete. This model is derived from the triaxial Menétrey–
Willam failure criterion equipped with a cap in compression to limit the concrete strength in the
high hydrostatic compression regime. It is formulated in terms of stress invariants known as the
Haigh–Westergaard cylindrical coordinates.
Basically, the expounded methods allow to simulate the limit state (non-linear) solution by
carrying out, within an iterative procedure, linear elastic analyses based on conventional finite
elements (FEs). They produce a limit-type distribution of stresses or strain and displacement
rates to be applied within the static and kinematic theorem of nonstandard limit analysis. Such
distributions are obtained by systematically adjusting the elastic moduli of the elements within
the discrete FE model. As conventional linear elastic FE analyses are required, the proposed
methodology can easily be applied using any commercial FE-code and can address even
problems with a large number of degrees of freedom.
The efficiency and reliability of the promoted methodology is illustrated with several
applications to reinforced concrete structural elements of engineering interest. Large-scale
prototypes of beams, walls and slabs, tested up to collapse, are numerically analyzed. The
comparison between experimental findings and the corresponding numerical results proved quite
satisfactory and establishes the proposed methodology as a simple design-tool, of practical
connotation, for reinforced concrete structures.
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15. SOMMARIO
In questa tesi viene sviluppata una metodologia numerica, basata sulla teoria dell’analisi
limite, al fine di valutare il carico di picco, il meccanismo di collasso e le zone critiche di
elementi in calcestruzzo armato. Il fenomeno della dilatanza del calcestruzzo implica una legge
di flusso di tipo non associato e l’adozione di un approaccio di analisi limite non standard.
Pertanto, la metodologia proposta si avvale di due distinti metodi numerici di analisi limite per la
ricerca di un limite superiore ed inferiore rispetto all’effettivo carico di picco separatamente,
ovvero il Linear Matching Method e l’Elastic Compensation Method. Entrambi i metodi,
originariamente concepiti per materiali alla von Mises, sono qui estesi per gestire un modello
alla plasticità 3D per il calcestruzzo. Questo modello è ricavato dal criterio di rottura triassiale di
Menétrey e Willam, dotato di un cap in compressione per limitare la resistenza del calcestruzzo
nel regime delle alte compressioni idrostatiche. Esso è formulato in termini di invarianti di
tensione noti come coordinate cilindriche di Haigh–Westergaard.
Sostanzialmente, i metodi esposti consentono di simulare la soluzione allo stato limite (non-
lineare) effettuando, all’interno di una procedura iterativa, analisi elastico-lineari agli elementi
finiti. Essi producono una distribuzione limite di campi di tensione e di incrementi di
deformazione e spostamento da applicare all’interno dei teoremi statico e cinematico dell’analisi
limite non standard. Tali distribuzioni sono ottenute aggiornando sistematicamente i moduli
elastici dei vari elementi all’interno del modello discreto agli elementi finiti. Poiché sono
richieste analisi elastico-lineari convenzionali agli elementi finiti, la metodologia proposta può
essere facilmente applicata per mezzo di un qualunque codice agli elementi finiti e può
rivolgersi anche a problemi aventi molti gradi di libertà.
L’efficacia e l’affidabilità della metodologia proposta è illustrata con numerose applicazioni
a elementi strutturali in calcestruzzo armato di interesse ingegneristico. Sono stati
numericamente analizzati prototipi reali (a larga scala) di travi, muri e piastre testati in
laboratorio fino al collasso. Il confronto tra dati sperimentali e i corrispondenti risultati numerici
si è rivelato abbastanza accurato e pone la metodologia proposta come un semplice strumento
pratico di progetto per strutture in calcestruzzo armato.
v

17. LIST OF FIGURES
Figure 2.1 Stress-strain diagrams: a) ductile metal (low-carbon steel), simple tension and compression, with
yield-point phenomenon; b) typical concrete or rock, simple compression and tension (Lubliner [250])
...............................................................................................................................................................12
Figure 2.2 Compression tests on concrete or rock: stress against longitudinal strain and volume strain
(Lubliner [250]) .....................................................................................................................................14
Figure 2.3 Material models: a) elastic-perfectly plastic idealization; b) rigid-perfectly plastic idealization 15
Figure 2.4 External agency and Drucker’s stability postulate: a) existing system; b) existing system and
external agency (Chen and Han [93]).....................................................................................................17
Figure 2.5 Stable and unstable stress-strain curves in the Drucker’s sense: a) stable materials with σ ε > 0 ;
b) and c) unstable materials with σ ε < 0 .............................................................................................18
Figure 2.6 Drucker’s stability postulate: illustration in stress space.............................................................19
Figure 2.7 Principle of maximum plastic dissipation: illustration in the uniaxial stress-strain plane (Lubliner
[250]) .....................................................................................................................................................19
Figure 2.8 Consequences of principle of maximum plastic dissipation: a) normality rule; b) convexity of
the yield surface violated if inequality (2.17) is not satisfied.................................................................21
Figure 2.9 Principle of maximum plastic dissipation (Kelly [208]) .............................................................21
Figure 2.10 Structural model considered for expressing the fundamental variational principles (Le [233]) 22
Figure 2.11 Plastic strain increment vectors in the Rendulic plane: a) for the associated flow rule the yield
surface coincides with the plastic potential contour; b) for the nonassociated flow rule plastic potential
and yield surface do not coincide...........................................................................................................32
Figure 2.12 Radenkovic’s theorems for nonstandard limit analysis: a) first theorem for computing an upper
bound; b) second theorem for computing a lower bound.......................................................................33
Figure 3.1 Equilibrium of the single strut (Nielsen and Hoang [302]) .........................................................38
Figure 3.2 Illustration of the strut action in a corbel (Nielsen and Hoang [302]).........................................39
Figure 3.3 Strut and tie model for a plate with one concentrated load (Nielsen and Hoang [302])..............40
Figure 3.4 Stringers and panels as building blocks of a reinforced concrete wall model (Blaauwendraad and
Hoogenboom [55]).................................................................................................................................42
Figure 3.5 Model of a square wall with hole at its centre using the stringer-panel method: a) real
(continuous) model of the wall; b) the stringer-panel (discrete) model .................................................43
Figure 3.6 Simply supported beam model using the stringer-panel method: a) real (continuous) model of
the beam; b) an exploded view of the stringer-panel (discrete) model showing that all elements are in
equilibrium.............................................................................................................................................43
Figure 3.7 A stringer-panel model can be assembled from simple components...........................................44
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18. Figure 3.8 Moment-curvature relationship for a reinforced concrete slab section: idealization for the
application of the yield-line method.......................................................................................................46
Figure 3.9 Convention for drawing yield lines and representing support conditions in the yield-line method
...............................................................................................................................................................47
Figure 3.10 Collapse mechanism for a rectangular simply supported slab...................................................47
Figure 3.11 Sketch of an isotropically reinforced slabs................................................................................48
Figure 3.12 Geometrical sketch for deriving the energy dissipation along a yield line ................................49
Figure 3.13 Equilibrium of a small element of a slab of sides dx and dy......................................................51
Figure 3.14 Distribution of the coefficient a for different support conditions............................................52
Figure 3.15 Graph of a convex function.......................................................................................................57
Figure 3.16 Linearization of a yield surface for RC plates: a) original, non-linear (conic) yield surface by
Nielsen [298]; b) linearized yield surface suitable for linear programming (Poulsen and Damkilde
[340])......................................................................................................................................................58
Figure 3.17 A polyhedron  (shown shaded) represented as intersection of five halfspaces, with outward
normal vectors ,...,a a51 (Boyd and Vandenberghe [63]).......................................................................59
Figure 3.18 Geometric interpretation of a linear program, with the feasible set  , which is a polyhedron,
shaded (Boyd and Vandenberghe [63])..................................................................................................60
Figure 3.19 Illustration of a linear programming problem solved by the simplex method (Larsen [229]) ...61
Figure 3.20 Illustration of a linear programming problem solved by the interior-point method (Larsen
[229])......................................................................................................................................................62
Figure 3.21 Geometric interpretation of the optimality condition (3.32) for a general nonlinear program: the
feasible set  is shown shaded; some level curves of f0 are shown as dashed lines; the point *x is
optimal since *( )xf−∇ 0 defines a supporting hyperplane (shown as a solid line) to  at *x (Boyd
and Vandenberghe [63]).........................................................................................................................63
Figure 3.22 The pie slice shows all points of the form 1 1 2 2x xθ θ+ , where 1 2, 0θ θ ≥ (Boyd and
Vandenberghe [63])................................................................................................................................64
Figure 3.23 Boundary of second-order cone in
3
R , that is 2 2 1/2
1 2 1 2{( , , ) | ( ) }x x t x x t+ ≤ (Boyd and
Vandenberghe [63])................................................................................................................................65
Figure 3.24 Boundary of positive semidefinite cone in 2
S (Boyd and Vandenberghe [63]).......................67
Figure 3.25 Backtracking line search (Boyd and Vandenberghe [63]).........................................................70
Figure 3.26 Newton’s method interpreted as minimizer of second-order Taylor approximation (Boyd and
Vandenberghe [63])................................................................................................................................71
Figure 3.27 Newton’s method interpreted as steepest descent direction in Hessian norm (Boyd and
Vandenberghe [63])................................................................................................................................72
Figure 3.28 Determination of reference volume (after Seshadri and Mangalaramanan [366]).....................80
Figure 3.29 Graphical representation of method for calculating q as proposed in [5] ..................................82
viii

19. Figure 3.30 Four different strategies for calculating modulus-adjustment index q: a) fixed strain method; b)
method based on strain energy equilibrium principle; c) circular arc method; d) method based on strain
energy conservation law (Yu and Yang [415]) ......................................................................................83
Figure 3.31 Stress resultants, i.e. forces and moments, of shell element (Yu and Yang [415])....................87
Figure 3.32 Stress resultants, i.e. forces and moments, of beam element (Yu and Yang [416])...................87
Figure 3.33 3D geometrical sketch, in the stress space , ,σ σ σ1 2 6 for lower bound evaluation by the ECM
applied to orthotropic materials with Tsai–Wu type yield surface (Pisano and Fuschi [321]) ...............92
Figure 3.34 Variation in elastic strain energy U and energy dissipation D with applied load, used in the
calculation of the upper bound limit load (Mackenzie et al. [259]) .......................................................94
Figure 3.35 The iterative process of the LMM (Ponter and Carter [331])....................................................99
Figure 3.36 The sufficient condition for convergence requires that the surface of constant W which is
tangent to the yield surface at the matching point, otherwise lies outside the yield surface (Ponter et al.
[334]) ...................................................................................................................................................104
Figure 3.37 Matching procedure at the generic Gauss point for ( )
= const.W W=
0
homothetic to the Tsai–
Wu-type yield surface: geometrical sketch in the dimensionless Z = 0 plane (Pisano and Fuschi [320])
.............................................................................................................................................................108
Figure 4.1 Typical uniaxial compressive and tensile stress-strain curve for concrete ................................115
Figure 4.2 Stress-strain relationships of concrete under biaxial-compression test (Chen [91])..................115
Figure 4.3 Triaxial stress-strain relationship for concrete (Chen [91]).......................................................116
Figure 4.4 Uniaxial tensile stress-elongation curve (Chen and Han [93])..................................................117
Figure 4.5 Volumetric strain under uniaxial and biaxial compression (Chen and Han [93])......................117
Figure 4.6 Uniaxial compressive stress-strain curves for concrete of different compressive strengths (Chen
and Han [93]).......................................................................................................................................118
Figure 4.7 Response of concrete to uniaxial cyclic loading (Chen [91])....................................................119
Figure 4.8 Uniaxial stress-strain curve, pre- and postfailure regime (Chen [91]).......................................122
Figure 4.9 Evolution of subsequent loading surfaces during hardening: a) isotropic hardening; b) kinematic
hardening .............................................................................................................................................124
Figure 4.10 Configuration of a cylinder bar subjected to uniaxial tension: a) nominal (damaged)
configuration; b) effective (undamaged) configuration (Voyiadjis and Kattan [401]).........................128
Figure 4.11 Different crack models within the FE framework (Mordini [282]).........................................131
Figure 4.12 Different cracking models: a) discrete fracture model; b) smeared crack model. ...................131
Figure 4.13 Global and local cracking coordinate systems ........................................................................132
Figure 4.14 Representation of the nine stress components with respect to a Cartesian coordinate system in
the positive faces..................................................................................................................................138
Figure 4.15 Representation of the three principal stresses..........................................................................139
Figure 4.16 Representation of the octahedral plane....................................................................................141
Figure 4.17 Cartesian coordinate system with axes parallel and perpendicular to the octahedral plane.....141
ix

20. Figure 4.18 Decomposition of the traction vector
)( an
t acting on the octahedral plane into the octahedral
normal stress octσ and the octahedral shear stress octτ ........................................................................142
Figure 4.19 Three-dimensional principal stress-space: π−plane and decomposition of the stress vector into
hydrostatic and deviatoric stresses .......................................................................................................143
Figure 4.20 Axonometric view of the π−plane...........................................................................................143
Figure 4.21 Representation of four different stress states: a) three-dimensional principal stress space; b)
projected view of the π−plane ..............................................................................................................144
Figure 4.22 The stress state expressed by the Haigh–Westergaard coordinates: a) three-dimensional
principal stress space; b) projected view on the corresponding deviatoric plane const.ξ = ................145
Figure 4.23 The von Mises and Tresca yield surfaces: a) three-dimensional principal stress space; b) π -
plane.....................................................................................................................................................146
Figure 4.24 General shape of the failure surface for an isotropic, pressure-dependent material like concrete:
a) compressive and tensile meridians; b) a deviatoric section..............................................................151
Figure 4.25 Deviatoric sections of failure concrete surface for different hydrostatic pressure values........152
Figure 4.26 Rankine’s maximum-principal-stress criterion expressed with the Haigh–Westergaard
coordinates: a) tensile and compressive meridians; b) cross-sectional shape on the π−plane ..............153
Figure 4.27 Mohr–Coulomb failure criterion in the ( )n
σ τ− stress space .................................................154
Figure 4.28 The Mohr–Coulomb criterion: a) meridian plane; b) π−plane.................................................155
Figure 4.29 The Mohr–Coulomb criterion: axonometric view in principal stress space.............................156
Figure 4.30 The Mohr–Coulomb criterion with tension cutoff in ( )n
σ τ− coordinate system..................157
Figure 4.31 Mohr’s circles corresponding to stress states which involve different failure modes: a) sliding
failure; b) separation failure .................................................................................................................157
Figure 4.32 The Drucker–Prager criterion: axonometric view in principal stress space.............................158
Figure 4.33 Drucker–Prager criterion: a) meridian plane; b) π−plane........................................................159
Figure 4.34 The Mohr–Coulomb criterion, solid line, and Drucker–Prager criterion, dashed line, matched in
the π−plane...........................................................................................................................................159
Figure 4.35 Bresler–Pister criterion: a) axonometric view in principal stress space; b) π−plane (source
Wikipedia)............................................................................................................................................160
Figure 4.36 Hsieh–Ting–Chen criterion: a) axonometric view in principal stress space; b) three cross-
sectional shapes in deviatoric plane......................................................................................................162
Figure 4.37 Deviatoric section of the Willam–Warnke failure surface.......................................................163
Figure 4.38 Elliptic trace of the Willam–Warnke failure surface for /θ π≤ ≤0 3 ...................................164
Figure 4.39 Three-parameter Willam–Warnke failure surface in the meridian plane.................................165
Figure 4.40 Three-parameter Willam–Warnke criterion with straight-line meridians and non-circular
deviatoric sections: a) axonometric view in principal stress space; b) π−plane (source Wikipedia) ....167
Figure 4.41 Five-parameter Willam–Warnke failure surface in the meridian plane...................................167
x

21. Figure 4.42 Deviatoric sections of the Willam–Warnke failure surface: a) affinity for the three-parameter
model; b) non-affinity for the five-parameter models..........................................................................169
Figure 4.43 Five-parameter Willam–Warnke failure surface with curved meridians and non-circular
deviatoric sections: axonometric view in principal stress space...........................................................169
Figure 4.44 Six-parameter failure surface with elliptical cap as proposed by Argyris et al. [17]...............171
Figure 4.45 Hoek and Brown criterion with curved meridians and noncircular deviatoric sections: a)
axonometric view in principal stress space; b) deviatoric sections at various levels of hydrostatic
pressure................................................................................................................................................173
Figure 4.46 Influence of eccentricity e on the deviatoric trace of the failure surface: a) triangular shape for
.e = 0 5 ; b) elliptical (smooth) shape for . e< <0 5 1 ; c) circular shape for e = 1 ..............................174
Figure 4.47 Polar plot of the inverse of the elliptic function ( , )r eθ given by Eq. (4.104) for values of the
eccentricity parameter ranging from .e = 0 5 to e = 1 .........................................................................175
Figure 4.48 Extended Leon model with curved meridians and affine, non-circular deviatoric sections: a)
axonometric view in principal stress space; b) deviatoric sections at various levels of hydrostatic
pressure................................................................................................................................................176
Figure 4.49 Menétrey–Willam failure surface: a) meridian sections at θ = 0 and /θ π= 3 ; b) deviatoric
sections at three generic values of hydrostatic pressure .......................................................................177
Figure 4.50 Menétrey–Willam failure surface: axonometric view in principal stress space.......................177
Figure 4.51 Influence of the eccentricity parameter e on the deviatoric plane: a) .e = 0 5 ; b) .e = 0 6 ...178
Figure 4.52 Comparison, along the meridian plane / 3θ π= , of six well-known failure conditions
obtainable through a suitable setting of the coefficients entering the generalized failure criterion, Eq.
(4.113)..................................................................................................................................................181
Figure 4.53 Bigoni–Piccolroaz criterion: a) axonometric view in principal stress space; b) π−plane (source
Wikipedia) ...........................................................................................................................................184
Figure 4.54 Lubliner et al. yield criterion in plane stress space .................................................................187
Figure 4.55 Deviatoric trace of the Lubliner et al. yield criterion in the deviatoric plane corresponding to
two different values of ..................................................................................................................190
Figure 4.56 Compressive and tensile meridians of the Lubliner et al. yield criterion ................................191
Figure 4.57 Evolution of the yield surface during hardening (after Chen and Han [93]) ...........................195
Figure 4.58 Loading function for cap model..............................................................................................197
Figure 4.59 Three-surface loading function with the two-invariant elliptical cap model proposed by
DiMaggio and Sandler [121] and a tension cutoff limit plane..............................................................198
Figure 4.60 Adopted Menétrey–Willam-type yield surface with cap in compression: a) compressive and
tensile meridians in the Rendulic plane; b) 3D sketch in principal stress space...................................202
Figure 5.1 Stress point at yield Y
 : a) 3D sketch of stress and strain components in the Haigh–
Westergaard representation, ( , )x y denoting the orthogonal Cartesian system associated to the
cylindrical coordinates ( , , )ξ ρ θ at Y
ξ ξ= ; b) deviatoric section at Y
ξ ξ= ; c) Rendulic section at
Y
θ θ= ................................................................................................................................................209
cK
xi

22. Figure 5.2 Geometrical sketch, in the principal stress space, at the current GP within the current element:
stress point ( , , )x yL ξ ρ ρ  
 with pertinent strain rates ( , , )yxd dvε ε ε    
lying on the complementary
energy equipotential surface ( ) ( ) ( ) ( ) ( )
( , , , , , ) const.
k k k k k
W K Gξ ρ ξ ρ− − − − −
=
 1 1 1 1 1
and representing the
computed fictitious solution .................................................................................................................211
Figure 5.3 Geometrical sketch, in the principal stress space, at the current GP within the current element:
stress point M of coordinates ( , , )Y Y Y
yx MyM Mx
ρ ρ ρ ρξ ξ= = = lying on the M–W-type yield surface at
which the outward normal has given components, namely ( , , )yxd dvε ε ε    
representing the “solution at
yield” to be matched.............................................................................................................................213
Figure 5.4 Geometrical sketch, in the principal stress space, of the matching procedure fulfilled at the
generic Gauss point within the current element at the generic step from iteration ( 1)k# − to ( )k# :
( )
( ) k
⋅ denotes a modified value of the fictitious quantity ( )⋅ which allows to achieve the matching;
L = stress point pertaining to the fictitious linear strain rate solution ( )k −
 1
ε 
; M = stress point “at
yield” of given normal assumed as strain rate at collapse ( )kc −
 1
ε ...........................................214
Figure 5.5 Location of matching point M on the M–W-type yield surface guided by the assigned fictitious
linear strain rate solution ε 
: a) meridian plane at Mθ θ= ; b) deviatoric plane at Mξ ξ= ..............217
Figure 5.6 Analytical procedure to search for the point iP having a given normal ε 
on the meridian plane
of the M–W-type yield surface.............................................................................................................218
Figure 5.7 Analytical procedure to search for the point iP having a given normal dε 
on the deviatoric
plane of the M–W-type yield surface ...................................................................................................221
Figure 5.8 Iterative procedure to search for the matching point analysing both the meridian planes and the
deviatoric sections of the M–W-type yield surface and taking into account their reciprocal influence: a)
meridian planes; b) deviatoric planes ...................................................................................................222
Figure 5.9 Construction of the prolate spheroid
( )
const.
k
W = at matching: deviatoric section, at
ˆ
M
ξ ξ ξ= = , of the M–W-type yield surface and of the spheroid
( )
const.
k
W = ...................................226
Figure 5.10 Deviatoric section, at ˆ
M
ξ ξ ξ= = , of the M–W-type yield surface and of the spheroid
( )
const.
k
W = at matching for the case 0M
θ = ......................................................................................227
Figure 5.11 Deviatoric section, at ˆ
M
ξ ξ ξ= = , of the M–W-type yield surface and of the spheroid
( )
const.k
W = at matching for the case / 3M
θ π= ................................................................................228
Figure 5.12 Construction of the prolate spheroid ( )
const.k
W = at matching: section on the plane belonging
to the sheaf of axis ˆξ and passing through M with location of the reference system ˆ ˆ ˆ( , , )O ρξ the
generatrix ellipse is referred to: a) matching point M on the Menétrey–Willam surface; b) matching
point M on the cap surface ................................................................................................................230
Figure 5.13 Geometrical sketch, in the principal stress space, of the ECM fulfilled within the generic
element in the FE discrete model at iteration ( 1)k# − within the current sequence:
e
e# is the stress
point representing the elastic (averaged) solution at the the − element, while Y
e# is the corresponding
stress point at yield...............................................................................................................................235
Figure 5.14 Location of the stress point at yield Y
e# as intersection between the meridian section of the M–
W-type yield surface for eY
θ θ= and the straight line passing through the origin O and whose slope is
given by the direction | |/
e e
e eO O# #
 
  , where
e
e# is the stress point representing the elastic (averaged)
solution at the the − element..............................................................................................................237
( )k −
 1
ε 
xii

23. Figure 5.15 Location of the “maximum stress” in the whole mesh: among the stress points at elements
1, 2,..., ,...,e n# # # # , e
Re# ≡  represents that farthest away from the M–W-type yield surface .......238
Figure 6.1 3D-solid elements employed to model concrete: a) 27-node elements employed for plain
concrete specimens; b) 8-node elements employed for reinforced concrete structural elements (ADINA
[7]) .......................................................................................................................................................243
Figure 6.2 Generation of re-bars in the FE-model as truss elements embedded within 3D-solid elements
(ADINA [7]) ........................................................................................................................................243
Figure 6.3 Compressive test on cubic specimen: a) geometry, loading and boundary conditions; b) adopted
FE mesh of 343 3D-solid 27-node elements ........................................................................................245
Figure 6.4 Compression test on cubic plain concrete specimen. Values of the upper ( )UBP and lower
( )LBP bounds to the peak load multiplier versus iteration number: LMM prediction, solid lines with
square markers; ECM prediction, solid lines with triangular markers; expected value of the peak load
multiplier ( )EXPP for the given reference load and material properties, dashed lines...........................246
Figure 6.5 Splitting test on cubic specimen: a) geometry, loading and boundary conditions; b) adopted FE
mesh of 343 3D-solid 27-node elements..............................................................................................248
Figure 6.6 Splitting tensile test on cubic plain concrete specimen. Values of the upper ( )UBP and lower
( )LBP bounds to the peak load multiplier versus iteration number: LMM prediction, solid lines with
square markers; ECM prediction, solid lines with triangular markers; expected value of the peak load
multiplier ( )EXPP for the given reference load and material properties, dashed lines...........................249
Figure 6.7 Predicted failure mechanism, in terms of Cartesian strain rates xx
c
ε at collapse (i.e. constructed
by the LMM at the last converged solution), for a cubic specimen subjected to splitting tensile test..249
Figure 6.8 Splitting test on cylindrical specimens: a) geometry, loading and boundary conditions; b)
adopted FE mesh of 720 3D-solid 27-node elements...........................................................................251
Figure 6.9 Splitting tensile tests on cylindrical specimens of short-fiber reinforced concrete. Values of the
upper ( )UBP and lower ( )LBP bounds to the peak load multiplier versus iteration number: LMM
prediction, solid lines with square markers; ECM prediction, solid lines with triangular markers;
experimental value of the peak load multiplier ( )EXPP for the given reference load and material
properties after Denneman et al. [120], dashed lines. a) specimen #1; b) specimen #2; c) specimen #3;
d) specimen #4.....................................................................................................................................253
Figure 6.10 Splitting tensile tests on cylindrical specimens of plain concrete. Values of the upper ( )UBP
and lower ( )LBP bounds to the peak load multiplier versus iteration number: LMM prediction, solid
lines with square markers; ECM prediction, solid lines with triangular markers; experimental value of
the peak load multiplier ( )EXPP for the given reference load and material properties, dashed lines. a)
specimen #5 after Denneman [119]; b) and c) specimen #6 and #7, respectively, after Carmona and
Aguado [74] .........................................................................................................................................254
Figure 6.11 Predicted failure mechanism, in terms of Cartesian strain rates xx
c
ε at collapse (i.e. constructed
by the LMM at the last converged solution), for cylindrical specimens subjected to splitting tensile test:
a) specimen #1 (fiber reinforced concrete); b) specimen #5 (plain concrete).......................................255
Figure 6.12 Bending test on steel-reinforced concrete beams: a) mechanical model of the half analyzed
symmetric specimen prototype showing geometry, loading and boundary conditions; b) typical FE
xiii

24. mesh adopted with 3D solid FEs modelling concrete and embedded 1D truss FEs modelling re-bars and
stirrups..................................................................................................................................................257
Figure 6.13 Bending test on steel-reinforced concrete beams: cross-sections geometry (dimensions in mm)
with re-bars and stirrups arrangement for the nine analyzed specimens (after Vecchio and Shim [397])
.............................................................................................................................................................259
Figure 6.14 Bending test on steel-reinforced concrete beams. Values of the upper ( )UBP and lower ( )LBP
bounds to the peak load multiplier versus iteration number: LMM prediction, solid lines with square
markers; ECM prediction, solid lines with triangular markers; experimental value of the peak load
multiplier ( )EXPP for the given reference load and material properties after Vecchio and Shim [397],
dashed lines. a) beam #A2; beam #B2; c) beam #C3 ...........................................................................260
Figure 6.15 Band plots of the Cartesian strain rate components xx
c
ε in the deformed configuration at the
ultimate value of the acting loads for beams #A2, #B2 and #C3. a), c) and e): results at last converged
solution of the LMM giving the predicted collapse/failure mechanisms; b), d) and f): results obtained
by an elastic analysis of the beams.......................................................................................................262
Figure 6.16 Steel-reinforced concrete walls: a) mechanical model, geometry, boundary and loading
conditions; b) typical FE-mesh of 3D-solid elements for concrete and 1D-embedded-truss elements for
re-bars; c) reinforcement arrangement for walls type 1, having a square-shaped profile; d)
reinforcement arrangement for walls type 2, having a rectangular profile ...........................................263
Figure 6.17 Steel-reinforced concrete walls: values of the upper ( )UBP and lower ( )LBP bounds to the
peak load multiplier against the experimentally detected one ( )EXPP after Lefas et al. [237]..............265
Figure 6.18 Steel-reinforced concrete walls. Values of the upper ( )UBP and lower ( )LBP bounds to the
peak load multiplier versus iteration number: LMM prediction, solid lines with square markers; ECM
prediction, solid lines with triangular markers; experimental value of the peak load multiplier ( )EXPP
for the given reference load and material properties after Lefas et al. [237], dashed lines. a) specimen
#SW16; b) specimen #SW26................................................................................................................266
Figure 6.19 Steel-reinforced concrete walls. Band plots of the principal (compressive) strain rates 3
c
ε in the
deformed configurations at the ultimate value of the horizontal load UB HP F for specimens SW16 and
SW26. a) and c): results obtained at last converged solution of the LMM on the fictitious structure for
specimens SW16 and SW26, respectively, localizing the plastic zone and/or the collapse mechanism; b)
and d): results pertaining to an elastic solution of the real structure, i.e. with the real elastic parameters,
for specimens SW16 and SW26, respectively......................................................................................267
Figure 6.20 Steel-reinforced concrete simply-supported slab #1 (which is the control specimen of El
Maaddawy and Soudki [134]) and slab #2 (coincident with specimen SS4 of Sakka and Gilbert [354]):
a) mechanical model, geometry, boundary and loading conditions; b) reinforcement arrangement along
x direction; c) reinforcement arrangement at cross-section ................................................................268
Figure 6.21 Steel-reinforced concrete continuous-supported slab #3 (coincident with slab CS5 of Sakka and
Gilbert [354]): a) mechanical model, geometry (all dimensions in mm), boundary and loading
conditions; b) reinforcement arrangement at roller support and at mid-span along longitudinal axis; c)
reinforcement arrangement at roller support and at mid-span at cross-section; d) reinforcement
arrangement at interior support along longitudinal axis (top bars along x have a length of 1800mm
and are centred at midspan); e) reinforcement arrangement at interior support in the cross-section....269
xiv

25. Figure 6.22 Steel-reinforced concrete two-way corner-supported slabs #4, #5, #6 and #7, coincident with
specimens S2S-5, S2S-6, S2R-4 and S2R-5 of Sakka and Gilbert [355]: a) mechanical model, geometry
(all dimensions in mm), boundary and loading conditions; b) reinforcement arrangement along x
direction; c) reinforcement arrangement along y direction.................................................................270
Figure 6.23 Steel-reinforced concrete slabs: values of the upper ( )UBP and lower ( )LBP bounds to the
peak load multiplier against the experimentally detected one ( )EXPP ..................................................273
Figure 6.24 Steel-reinforced concrete simply-supported slab #2. Band plots of the principal (compressive)
strain rates 3
c
ε in the deformed configurations at the ultimate value of the acting loads: a) results
obtained at last converged solution of the LMM on the fictitious structure localizing the plastic zone
and/or the collapse mechanism; b) results pertaining to an elastic solution of the real structure, i.e. with
the real elastic parameters; c) comparison of the deformed shapes given in a) and b) in the plane x z− ;
d) photograph of slab #2 at failure after Sakka and Gilbert [354]........................................................273
Figure 6.25 Steel-reinforced concrete corner-supported slab #7. Band plots of the principal (compressive)
strain rates 3
c
ε in the deformed configurations at the ultimate value of the acting loads: a) results
obtained at last converged solution of the LMM on the fictitious structure localizing the plastic zone
and/or the collapse mechanism; b) results pertaining to an elastic solution of the real structure, i.e. with
the real elastic parameters; c) comparison of the deformed shapes given in a) and b) in the plane y z− ;
d) photograph of slab #7 at failure after Sakka and Gilbert [355]........................................................274
Figure 6.26 Four-point bending test on FRP-reinforced concrete beams: a) mechanical model of the half
analyzed symmetric specimen showing geometry (all dimensions in mm), loading and boundary
conditions; b) typical FE mesh adopted with 3D solid FEs modelling concrete and embedded 1D truss
FEs modelling re-bars and stirrups.......................................................................................................278
Figure 6.27 Four-point bending test on FRP-reinforced concrete slabs: a) mechanical model showing
geometry (all dimensions in mm), loading and boundary conditions; b) typical FE mesh adopted with
3D solid FEs modelling concrete and embedded 1D truss FEs modelling re-bars...............................280
Figure 6.28 Four-point bending test on FRP-reinforced concrete elements: a) beam #BG2; b) slab #SC2.
Values of the upper ( )UBP and lower ( )LBP bounds to the peak load multiplier versus iteration
number: LMM prediction, solid lines with square markers; ECM prediction, solid lines with triangular
markers; experimental value of the peak load multiplier ( )EXPP for the given reference load and
material properties after Al-Sunna et al. [8], dashed lines ...................................................................282
Figure 6.29 Beam #BG2. Band plots of principal (compressive) strain rates 3
c
ε in the deformed
configurations at the ultimate value of the acting loads: a) result at last converged solution of the LMM
giving the predicted collapse/failure mechanism; b) result given by an elastic analysis of the beam ..283
Figure 6.30 Beam #SC2. Band plots of principal (compressive) strain rates 3
c
ε in the deformed
configurations at the ultimate value of the acting loads: a) result at last converged solution of the LMM
giving the predicted collapse/failure mechanism; b) result given by an elastic analysis of the slab.....284
Figure A.1 Procedure to search for the point iP having a given normal dε 
on the deviatoric plane of the
M–W-type yield surface.......................................................................................................................297
xv

27. LIST OF TABLES
Table 4.1 Determination of the six parameters entering Eqs. (4.89) ..........................................................168
Table 4.2 Reduction of generalized failure criterion to specific forms.......................................................182
Table 6.1 Splitting tensile tests on cylindrical specimens: concrete type of the analyzed specimens;
specimen number; elastic properties; compressive and tensile strengths; experimentally detected peak
load multiplier values ( )EXPP against the computed values of the upper ( )UBP and lower ( )LBP bounds
to the peak load multiplier....................................................................................................................252
Table 6.2 Geometry and concrete properties of the analyzed specimens (after Vecchio and Shim [397]).258
Table 6.3 Peak load multipliers for the analyzed specimens: columns labelled EXPP and VSP report values
experimentally detected and numerically predicted by Vecchio and Shim [397], respectively; columns
labelled UBP and LBP give the upper and lower bounds evaluated, through the proposed methodology,
by means of the LMM and the ECM, respectively...............................................................................258
Table 6.4 Steel-reinforced concrete walls: specimen number; compressive and tensile concrete strengths;
concrete Young modulus; constant value of the applied vertical load (after Lefas et al. [237]) ..........264
Table 6.5 Steel-reinforced concrete simply-supported slabs #1 and #2 sketched in Figure 6.20: specimen
number; geometrical data; diameters and spacing of re-bars; value of the applied reference load.......271
Table 6.6 Steel-reinforced concrete corner-supported slabs #4, #5, #6 and #7 sketched in Figure 6.22:
specimen number; geometrical data; diameters and spacing of re-bars; value of the applied reference
load ......................................................................................................................................................271
Table 6.7 Steel-reinforced concrete slabs: specimen number; compressive and tensile strengths; elastic
concrete properties...............................................................................................................................272
Table 6.8 Steel-reinforced concrete slabs: specimen number; number of 3D-solid elements and 1D-
embedded truss elements used for the FE analyses..............................................................................272
Table 6.9 Properties of FRP re-bars used as main flexural reinforcement..................................................277
Table 6.10 Four-point bending test on FRP-reinforced concrete beams: specimen number; concrete
compressive and tensile strengths; concrete Young modulus; reinforcement details...........................279
Table 6.11 Four-point bending test on FRP-reinforced concrete slabs: specimen number; compressive and
tensile strengths; concrete Young modulus; reinforcement details; clear concrete cover to the main
rebars....................................................................................................................................................281
Table 6.12 Peak load multipliers for the analyzed FRP-reinforced elements: values experimentally detected
by Al-Sunna et al. [8] ( )EXPP against the values of the upper ( )UBP and lower ( )LBP bounds to the
peak load multiplier numerically predicted by LMM and ECM, respectively .....................................281
xvii

29. 1 GENERAL INTRODUCTION
1.1 Aims and scope of the thesis
From an engineering design viewpoint the problem of safety assessment with regard to
collapse in structures, structural elements and mechanical components is of utmost importance.
As a result, in-depth description of the mechanical behaviour of structures in ultimate conditions
has been one of the main goals pursued by engineers and researchers for decades. For the
purpose of load-carrying capacity assessment of structures and within plasticity theory, a step-
by-step incremental analysis may be performed, following the structural response in the post-
elastic regime up to collapse, progressively. Alternatively, if primary interest is in the final stage
and the problem can be tackled within plasticity theory, the two fundamental theorems of limit
analysis theory allow the determination of the limit (peak) load at collapse in a simpler and more
direct manner. The primary aim of this thesis is to develop a numerical approach, based on limit
analysis theory, that is able to give satisfactory answer to the above problem with regard to
reinforced concrete (RC) structures.
Although extensive research work carried out in recent times has led to a reasonably accurate
understanding of the constitutive behaviour of concrete (and reinforced concrete) structures
under various loading conditions, an exhaustive description of the actual mechanical behaviour
exhibited experimentally by concrete still remains one of the most difficult challenges in the
field of structural engineering, due to several intricate phenomena involved. When subjected to
failure laboratory tests, in either a monotonically increasing or reversed cyclic load scenario,
every concrete element shows a variety of effects, such as cracking in tension, crushing in
compression, dowel action, aggregate interlock of cracked interfaces, strain softening and
degradation of material stiffness, work hardening, volumetric expansion and bond slip between
concrete and eventual reinforcement bars. Other time-dependent effects such as creep, shrinkage
and temperature change contribute to further complicate the problem of modelling the
mechanical behaviour of concrete. At present, despite many attempts made by researchers to put
together various mathematical models dealing with only a few of the above mentioned
phenomena at a time, no unified formulation capable of realistically including all these effects is
yet available.
Many theories have been proposed in literature, using different models, such as empirical,
linear elastic, nonlinear elastic, plasticity-based models, and models based on endochronic
theory of inelasticity, models based on fracture mechanics and continuum damage mechanics,
combined plastic-damage and plastic-fracture models, and micromechanical models. Among
these, the concrete models that have been widely successful, at least in computational
applications are, undoubtedly, those based on a combination of flow theory of plasticity with
fracture mechanics or damage mechanics. A common feature of such coupled models is that
they are all oriented to implementation in finite element (FE) codes able to describe both the
elastic response of concrete structures, at low levels of loading, and their post-elastic behaviour
1

30. after the so-called elastic limit is attained. Usually, FE-based structural analysis is carried on
using a step-by-step incremental scheme with a time-stepping algorithm to integrate flow laws
and consistent tangent operators to assure rapid convergence and accuracy of the solution. Even
if some of the existing concrete models have been successfully employed in well-known
commercial FE codes, a set of material parameters are always needed to correctly describe the
actual post-elastic behaviour in compression or the cracking pattern in tension, especially when
material degradation occurs. These parameters, the experimental definition of which is not easy
and whose values have a strong influence on the numerical results, are properly evaluated only
by performing extensive laboratory tests, which obviously results in high costs. Many,
occasionally not well defined, parameters are sometimes necessary so that the observed
(macroscopic) material behaviour can be faithfully reflected during the analysis. Close
connections between the proposal of the models and the identification of parameters based on
experiments are not always established in a rigorous and consistent manner. Unless such
parameters are properly defined, which often happens when they are extrapolated by parametric
and/or analytical studies, thus proving to be only partially reliable, a lack of knowledge or a
misunderstanding of the nonlinear behaviour may also result in wrong conclusions. Moreover,
some of the existing methods are only partially reliable, since the obtained solution depends on
the load-history followed up to failure through a set of hardly detectable internal variables.
In many practical cases of engineering interest it is indeed sufficient to know only the load
value, or the value of the load multiplier, at which the structure will collapse or deform
excessively. In this regard, limit analysis theory is often invoked to obtain an estimate of the
collapse load of a structure directly, and therefore it plays a crucial role in safety assessment and
structural design. Limit analysis belongs to the so-called Direct Methods, where the adjective
“Direct” refers to the fact that these methods do not follow the evolution of structural response
along a given history of external actions, but rather focus on critical, ultimate states of this
evolution. By making use of limit analysis theory the limit load can be determined without
resorting to iterative or incremental analysis, i.e. without carrying out a complete progressive-
failure analysis of stress and strains in a structure. It should be emphasized that the results
obtained by using limit analysis theory apply to an idealized structure, as the two fundamental
theorems formulated by Drucker et al. [127] are valid under the assumptions of unlimited
ductility, perfect plasticity and small deformation. According to these hypotheses, the
deformation of the structure can increase without limit while the load is held constant, and
neither work hardening of the material nor significant changes in geometry of the structure are
assumed to occur. Nevertheless, limit analysis, with all its inherent limitations in the treatment
of post-elastic phenomena exhibited by concrete structures (localization, fracturing/damaging
mechanisms, etc.), is proposed in this thesis as a simple tool for determining the load-carrying
capacity of a reinforced concrete structure or structural element. The ductile behaviour, which is
an essential requisite for applying limit analysis theory, is actually guaranteed by the presence of
reinforcement bars which mitigate, or even nullify, many complex post-elastic phenomena
exhibited by plain concrete at incipient failure, such as localization and/or fracturing/damaging
mechanisms. These phenomena, due to rather brittle behaviour, cannot be treated by means of
the present methodology which refers to ductile reinforced concrete structures and, in general, to
2

31. concrete structures whose failure mechanism is mainly dominated by crushing of concrete. The
confining effect of reinforcement bars, and the ductile behaviour generated by their presence,
nevertheless makes a limit analysis approach, such as the one here proposed, both applicable and
effective for many applications of engineering interest.
With regard to the expounded numerical limit analysis methodology, a plasticity model for
concrete, with a pressure-sensitive yield surface which arises from the failure criterion proposed
by Menétrey and Willam [276], is adopted. A cap in compression is also adopted to limit
concrete strength in the high hydrostatic compression regime. Moreover, a realistic modelling of
concrete, viewed as frictional material, requires taking dilatancy into account, i.e. the volumetric
expansion under compression clearly indicated by experimental tests. This circumstance makes
the application of the associated flow rule inappropriate for concrete and implies the lack of a
unique peak/collapse load, motivating the adoption of a nonstandard limit analysis approach
[250] for computing two bounds, an upper and a lower bound of the peak load. The key-idea of
the proposed methodology is the combined use of two numerical procedures for limit analysis,
namely the Linear Matching Method (LMM) [331] and the Elastic Compensation Method
(ECM) [256], in order to search for an upper and lower bound to the actual peak/collapse load,
separately. In particular, the former is related to the kinematic approach of limit analysis, being
able to build a compatible collapse mechanism, and hence provides an upper bound to the peak
load value. The latter is a procedure based on the static approach of limit analysis, producing an
admissible stress field, and therefore gives a lower bound to the peak load value. The application
of the LMM also provides some useful information on the expected failure/collapse mechanism
that the structure exhibits at ultimate limit state. Both numerical procedures, originally
formulated with reference to von Mises type materials, are extended to deal with the 3D
plasticity constitutive model adopted for concrete and expressed in terms of a particular set of
stress invariants, namely the Haigh–Westergaard cylindrical coordinates. The extension of the
LMM and the ECM to the Menétrey–Willam-type yield surface required significant
modifications of the original formulation, and several geometrical rationales were necessary to
implement the entire numerical procedure in the Haigh–Westergaard space.
The two procedures may be considered as an alternative to classical programming
procedures in that they do not solve the optimization problem related to limit analysis in a strict
mathematical way, but rather they allow to simulate the limit state (non-linear) solution by
carrying out linear elastic analyses while using conventional finite element methods as the basis
for iterative procedures. Basically, these FE-based procedures produce a limit-type distribution
of the stresses or strain and displacement rates to be applied within the static and kinematic
theorems of limit analysis, respectively. These distributions are obtained by systematically
adjusting the elastic moduli of the various elements within the FE model. As conventional linear
elastic FE analyses are required, these methods can easily be applied to problems with even a
large number of degrees of freedom where programming techniques may take longer to reach
convergence. Another advantage is that these methods are sufficiently flexible to be applied to a
wide range of non-linear material behaviours. For these reasons, these simplified procedures
may be very useful in preliminary estimations and verification processes: they allow one to gain
a quick insight into mechanical behaviour in terms of peak load, failure mode as well as critical
3

32. zones of the analyzed structures, even those having large dimensions, intricate geometry and/or
complex boundary conditions.
Validation of the proposed methodology through numerical applications to reinforced
concrete structural elements is performed and discussed. Applications have regarded a number
of reinforced concrete structural elements of engineering interest, namely beams, walls and
slabs. The reliability of the obtained numerical results has been verified by comparison with
experimental findings available in literature and regarding laboratory tests carried out up to
collapse on real (large-scale) prototypes. Therefore, reference is made to experimental
campaigns, as well as well-documented benchmarks, in an attempt to cover a broad survey of
specimens and structural elements in terms of reinforcement ratio, type of reinforcement bars,
ultimate mechanical behaviour and collapse mode. The comparison between experimental
findings and numerical results has proven reasonably good and establishes the proposed
methodology as a simple design-tool of practical connotation oriented to reinforced concrete
structures.
1.2 Outline of the thesis
The thesis is divided into seven chapters, which are organized as follows.
After this introductory chapter, in Chapter 2, some theoretical fundamentals of plasticity
theory are presented. Attention is first focused on the main assumptions made for modelling
mechanical behaviour of (reinforced) concrete within a plasticity-based approach; then, the
underlying concepts of limit analysis theory are reviewed. The two fundamental theorems of
limit analysis are also stated, both in standard form (for associative materials) and in the
nonstandard one, according to Radenkovic, for nonassociative materials.
Chapter 3 deals with the most commonly used limit analysis procedures. A great number of
limit analysis procedures have, in fact, been developed. In an attempt to classify such a long list
of methods, in the present thesis a distinction is made between hand-based procedures and
computer-based procedures. The first class includes all those simple and practical analytical
tools, founded on the extremum theorems, that can be used without the aid of any software
package, that is, without resorting to finite elements or optimization algorithms (e.g. yield-line
theory, Hillerborg’s strip method, strut-and-tie model). These simplified procedures are
considered of some interest as they were used extensively until some decades ago and they may
still be important as practical design tools for preliminary estimates of load-carrying capacity.
The second class is by far broader than the first and includes all those numerical tools that make
use of classical mathematical programming techniques used in conjunction with FE-based or
more recent mesh-free discretization strategies. To this aim, some preliminary concepts
regarding mathematical programming and convex optimization are also outlined in this chapter.
Also the so-called modulus variation procedures, discussed in section 3.3.2, may be considered
within the second class of limit analysis procedures, as alternative to the above classical
programming procedures. The latter FE-based procedures produce a limit-type solution by
carrying out linear (elastic) analyses and by systematically adjusting the elastic moduli of the
various elements within the FE model. This class of simplified FE-based methods includes the
two numerical procedures that have been adopted within the proposed limit analysis
4

33. methodology: the ECM and the LMM. Some of the most important methods belonging to this
class are reviewed in this chapter.
Chapter 4 addresses constitutive models for concrete. An insight into the main features and
key aspects of concrete experimental behaviour is provided in section 4.2, together with a review
of the major models for concrete available in literature. Among this very wide ranging list of
approaches, attention has been focused on two of the most widespread. The former is the
plasticity approach which uses a failure model, developed specifically for concrete, as the yield
function by applying some corrections, and integrating it into the flow theory of plasticity to
compute strains and stresses in the yielded material. To this aim, the most significant failure
criteria for concrete, or at least those which form the basis of the most common concrete
constitutive models, are presented in section 4.3. The latter approach, instead, combines
plasticity theory with damage mechanics or fracture mechanics. Among these plasticity-based
approaches, two constitutive models are described in details in this thesis: a) the so-called
Barcelona model originating from the work of Lubliner et al. [251] and later modifications by
Lee and Fenves [236]; b) the so-called Menétrey–Willam-type model [276]. They have been
chosen since they are, undoubtedly, the most specifically oriented toward computational analysis
of concrete structures and have been implemented in several FE-codes or within plasticity-based
numerical formulations which are well-known in literature. In particular, the Menétrey–Willam-
type model represents the constitutive model adopted within the framework of the proposed
limit analysis numerical procedure and is illustrated in section 4.4.2.
In Chapter 5, the proposed FE-based limit analysis methodology for concrete is presented
and discussed. The extension of the LMM and the ECM to the Menétrey–Willam-type yield
surface is illustrated with the aid of some sketches giving geometrical and analytical details of
the entire procedure. The algorithms of the LMM and the ECM are also described in the form of
two flow-charts with the aim to synthesize the basic operative steps of the numerical
implementation.
The limit analysis methodology described is applied to a number of reinforced concrete
structural elements in Chapter 6. The reliability and effectiveness of the methodology is
investigated by comparison with experimental findings available in literature and regarding
laboratory tests carried out up to collapse on real (large-scale) prototypes, namely reinforced
concrete beams, walls and slabs. The obtained numerical results are critically analyzed and
compared with experimental findings.
Finally, in Chapter 7, the main conclusions are drawn while suggestions and perspectives for
future work are outlined.
1.3 Related publications and original contributions of the thesis
Parts of this thesis have already been published in, or submitted to, international journals or
presented at national and international conferences. These papers are:
[1] Pisano A.A., Fuschi P., De Domenico D., Steel-reinforced concrete walls and slabs:
peak load predictions by limit analysis, Construction and Building Materials
(submitted to).
5

34. [2] Pisano A.A., Fuschi P., De Domenico D. (2014), Limit Analysis: A Layered Approach
for Composite Laminates, In: K. Spiliopoulos, D. Weichert (eds.), Direct Methods for
Limit States in Structures and Materials (Springer Science + Business Media B.V.).
[3] De Domenico D., Pisano A.A., Fuschi P., (2014), A FE-based limit analysis approach
for concrete elements reinforced with FRP bars, Composite Structures, 107, 594–603.
[4] De Domenico D., Pisano A.A., Fuschi P., (2013), A preliminary design tool for steel-
reinforced concrete elements, Conference Proceedings In: Atti del XXI Congresso
dell’Associazione Italiana di Meccanica Teorica e Applicata (Edizioni Libreria Cortina
Torino).
[5] Pisano A.A., Fuschi P., De Domenico D. (2013), A FE-based limit analysis approach
for concrete elements reinforced with FRP bars, In: Book of abstracts – ICCS17 17th
International Conference on Composite Structures.
[6] Pisano A.A., Fuschi P., De Domenico D. (2013), Peak loads and failure modes of steel-
reinforced concrete beams: predictions by limit analysis, Engineering Structures, 56,
477–488.
[7] Pisano A.A., Fuschi P., De Domenico D. (2013), A kinematic approach for peak load
evaluation of concrete elements, Computers and Structures, 119, 125–139.
[8] Pisano A.A., Fuschi P., De Domenico D. (2013), Peak load prediction of multi-pin
joints FRP laminates by limit analysis, Composite Structures, 96, 763–772.
[9] Pisano A.A., Fuschi P., De Domenico D. (2013), Failure modes prediction of multi-pin
joints FRP laminates by limit analysis, Composites Part B: Engineering, 46, 197–206.
[10] Pisano A.A., Fuschi P., De Domenico D. (2012), A layered limit analysis of pinned-
joint composite laminates: Numerical versus experimental findings, Composites Part
B: Engineering, 43, 940–952.
[11] Pisano A.A., Fuschi P., De Domenico D. (2012), Peak load computation of 3D
concrete elements by Menétrey–Willam pressure sensitive yield condition, Conference
Proceedings In: Atti del XIX Convegno Italiano di Meccanica Computazionale
GIMC2012.
[12] Pisano A.A., Fuschi P., De Domenico D. (2011), Estimation of the limit load of a
multilayer composite laminate, In: Book of abstracts – ICMM2 2nd International
Conference on Material Modelling (Presses des MinesParisTech).
[13] Pisano A.A., Fuschi P., De Domenico D. (2011), A layer-by-layer limit analysis
approach for composite laminates, Conference Proceedings In: Atti del XX Congresso
AIMETA di Meccanica Teorica e Applicata (Publi & Stampa Edizioni).
In particular, among the above contributions, papers [1], [3]–[7] and [11] address the set-up of
the limit analysis methodology for reinforced concrete elements obeying the Menétrey–Willam-
type yield surface, and therefore represent the core, original contribution of the present thesis,
dealing with the main concepts of the proposed numerical procedures. Some geometrical
considerations and specific analytical information of the entire procedure are given in more
detail in chapter 5 of this thesis. Papers [2], [8]–[10], [12] and [13] are, instead, devoted to the
extension of the same limit analysis methodology adopted in this thesis to composite laminates
handled by a Tsai–Wu criterion in plane stress conditions. The extension to composite laminates
6

35. is actually not an original contribution of these papers (see [320], [321]), but the whole
methodology is refined and improved as it is applied via a multilayer approach to deal with
different fiber ply orientations, taking into account the actual stacking sequence of the laminate.
The latter contributions are of a certain importance also in the framework of this thesis since
they demonstrate the general applicability of the proposed methodology to different materials
and different yield conditions. Furthermore, the extension of the two numerical procedures
(LMM and ECM) in the field of multilayer laminates and implementation at lamina level may be
regarded as a first step toward more general applications which address, for instance, structural
elements with two or more constitutive models through the thickness, in order to take account of
different materials at different layers (see suggestions for future work at the end of chapter 7).
7

37. 2 FUNDAMENTALS: CONSTITUTIVE ASSUMPTIONS AND LIMIT
ANALYSIS APPROACH
2.1 Introduction
In this chapter, some theoretical fundamentals for the developments described in subsequent
chapters are briefly presented. Attention is first focused on plastic analysis together with some
fundamental concepts necessary to state the theorems of limit analysis, both being powerful
tools for modelling structural behaviour at ultimate states and gaining an understanding of
structural safety. The underlying concepts of these theorems are therefore reviewed. After some
remarks on plasticity relations, limit analysis theory is introduced, both in its standard form (for
associative materials) and in its nonstandard version, the latter being necessary in the present
study due to the dilatancy exhibited by concrete (see chapter 4 for more details). The two
fundamental theorems of limit analysis, namely the kinematic and the static theorem, are
therefore invoked in the form given by Radenkovic in the 1960s (Radenkovic [344], [345];
Lubliner [250]) as Radenkovic’s first and second theorem, respectively. Such theorems allow
one to determine an upper and a lower bound of the exact limit load multiplier within a
nonstandard limit analysis framework.
It is worth noting that, although plain concrete is not a ductile material, reinforced concrete
can exhibit considerable ductility. In engineering practice, this can be achieved if concrete’s
material properties are conservatively defined and careful attention is paid to the arrangement of
reinforcement bars. These conditions, if properly verified, make applicable and effective a limit
analysis approach, at least as a preliminary analysis and/or design tool to acquire useful
information on peak load, failure modes and critical zones of reinforced concrete structures. A
limit analysis approach can therefore be oriented to practical reinforced concrete structures
where reinforcement bars have a stabilizing influence on fracture/damage phenomena (see e.g.
Bažant [43]). In these cases the strength and ductility behaviour of the restrained concrete can be
adequately described by a simpler plasticity-based model with all its inherent limitations.
Moreover, the ductile response of reinforced concrete has been demonstrated by decades of
experimental tests on large-scale concrete specimens. The underlying concepts of applications of
the theory of plasticity and limit analysis to reinforced concrete have been widely established
and discussed over the past half century in a number of review papers and books such as: Baker
[24]; Prager and Hodge [341]; Hodge [178]; Baker and Heyman [23]; Heyman [170]; Save and
Massonet [360]; Bræstrup et al. [64]; Nielsen et al. [300], [301]; Chen [91]; Chen and Han [93];
Lubliner [250]; more recently, Jirásek and Bažant [192]; Nielsen and Hoang [302]. Obviously,
the treatment of post-elastic phenomena that might be exhibited by concrete structures such as
localization, fracturing/damaging mechanisms, creep, and interface problems is not allowed by
limit analysis theory. The circumstance that with a plasticity-based approach, as the one here
proposed, it is not possible to describe any crack pattern or brittle failure is certainly a limit on
the applicability of the proposed methodology but such behaviour is mainly exhibited by plain
9

38. concrete or weakly steel-reinforced concrete elements not so usual within the steel-reinforced
concrete framed structures or many other common civil engineering applications. In the
following, concrete is governed by a plasticity criterion grounded on the three dimensional
failure surface by Menétrey and Willam [276]. A perfect bond is assumed between re-bars and
concrete (see more details in chapter 6). Moreover, stirrups and re-bars, both steel- and FRP-
bars, are assumed to have, by hypothesis, an indefinitely elastic behavior. Such assumptions are
somewhat different from conventional ones where steel is a plastic material and (plain) concrete
a mainly brittle one. Limit analysis however does refer to the structure as a whole and, in the
present context, it refers to a standard (commonly used) RC-structure whose ductility is assured
by the presence of a proper arrangement and amount of re-bars and whose behavior at ultimate
state—at incipient collapse—is dominated by crushing of confined concrete, re-bars being far
from yielding.
2.2 Plasticity: some fundamental concepts
The theory of linear elasticity is quite limited and its applicability is restricted for modelling
materials which undergo small deformations and which return to their original configuration
upon removal of load. Elastic deformations are, in fact, termed reversible since the energy
expended in deformation is stored as elastic strain energy and is completely recovered upon load
removal. On the contrary, almost all real materials will undergo some permanent, plastic
deformations, which remain after removal of load. Permanent (plastic) deformations involve the
dissipation of energy; such processes are termed irreversible, in the sense that the original state
can be achieved only by the expenditure of more energy. Significant permanent deformations
will usually occur when the stress reaches some critical value, called the yield stress, a material
property (Kelly [208]). The adjective “plastic” comes from the classical Greek verb πλάσσειν,
meaning “to shape”; it thus describes materials, such as ductile metals, clay, or putty, which
have the property that bodies made from them can have their shape easily changed by the
application of appropriately directed forces, and retain their new shape upon removal of such
forces (Lubliner [250]). In this regard, plasticity theory, in its simplest form, deals with materials
that can deform plastically under constant load when the load has reached a sufficiently high
value. These materials are called perfectly plastic materials, and the theory dealing with the
determination of the load-carrying capacity of structures made of such materials is called limit
analysis (Nielsen and Hoang [302]).
The roots of some elementary ideas of the theory of plasticity can be traced back over three
and half centuries. In Galileo’s (1638) calculation of the collapse load of a cantilever, one may
discern the assumption of a uniform distribution of tensile stresses over the cross section. About
a century later, Giovanni Poleni discussed the safety of Michelangelo’s dome of Saint Peter’s
cathedral in Rome in a manner in which one could detect the ideas of the static approach of limit
analysis. Moreover, in the early nineteenth century in the debates of the stability of masonry
arches, vaults and domes among La Hire, Boscovich, Lamé, Clapeyron, Fourier and Pauker one
could also perceive various elementary ideas of plastic analysis (Jirásek and Bažant [192]). The
classical theory of plasticity originated from the study of metals in the late nineteenth century.
The beginning of plasticity theory is probably to be traced back to 1868, when Tresca [391]
10

39. undertook an experimental program into the extrusion of metals and published his famous yield
criterion. Further advances with yield criteria, plastic flow rules, slip lines and plastic friction
were made in the years which followed by Saint-Venant, Levy, Von Mises, Hencky, Prandtl,
among others. The 1940s-1950s saw the advent and development of the classical theory: Prager
[342]; Hill [171], [172], [173]; Drucker et al. [127] and Koiter [215], [216], among others,
brought together many fundamental aspects of the theory into a single framework.
From a general point a view, plasticity theory is concerned with materials which initially
deform elastically, but which deform plastically upon reaching a yield stress. In metals and other
crystalline materials the occurrence of plastic deformations at the micro-scale level is due to the
motion of dislocations and the migration of grain boundaries on the micro-level. In fact metals in
their usual form are polycrystalline aggregates, that is, they are composed of large numbers of
grains, each of which has the structure of a simple crystal. On the other hand, in soils and other
granular materials such as sands, rocks, mortar and concrete plastic flow is due both to the
irreversible rearrangement of individual particles and to the irreversible crushing of individual
particles. The deformation of microvoids and the development of micro-cracks is also an
important cause of plastic deformations in materials such as rocks and concrete (Kelly [208]).
Except in low-porosity crystalline rocks at high temperature, where the crystal plasticity
mechanism operate, in most other cases the inelastic deformation in rocks is a consequence of
the nucleation and propagation of micro-cracks originated within a rock or concrete sample at
the surface of pores or other parts where lack of homogeneity occurs, such as inclusions and
grain boundaries. The nucleation and growth of micro-cracks under overall compressive loads
can, for example, occur due to tensile stress concentrations at the perimeter of microvoids,
bending of the long beam-like grains, or Hertzian contact stresses between the cement and grains
(Lubarda et al. [249]).
Figure 2.1 shows the results of a simple uniaxial tension and compression test on a specimen
of a ductile metal, namely a low-carbon steel (Figure 2.1a), compared with stress-strain diagram
for a typical concrete or rock (Figure 2.1b). Unlike ductile metals, quasi-brittle solids like
concrete and rocks behave quite differently in tension and compression, the highest attainable
stress in compression being many times that in tension. However, in contrast to classically brittle
solids, such as cast iron or glass that fracture almost immediately after the proportional limit is
attained, quasi-brittle solids, such as concrete and many rocks, produce stress-strain diagrams
that are qualitatively similar to those of many ductile materials. Of course, the strain scale is
quite different: in quasi-brittle materials the largest strains attained rarely exceed %1 . The stress
peak represents the onset of fracture, while the decrease in slope of the stress-strain curve
represents a loss in stiffness due to progressive cracking. In fact, looking at Figure 2.1,
following the attainment of the ultimate strength, concrete and many rocks exhibit strain-
softening, that is, a gradual decrease in strength with additional deformation. However, such a
decreasing post-peak portion of the curve is highly sensitive to test conditions (stiffness of the
testing machine) and specimen dimensions, and therefore it cannot be regarded as a material
property (see chapter 4 for more details). Moreover, it is not only compression per se that causes
fracture, but the accompanying shear stresses and secondary tensile stresses (Lubliner [250]).
Nevertheless, the superficial resemblance between the curves makes it possible to apply some
11

40. concepts of plasticity to these materials, as discussed further in this chapter with reference to
concrete.
Figure 2.1 Stress-strain diagrams: a) ductile metal (low-carbon steel), simple tension and compression,
with yield-point phenomenon; b) typical concrete or rock, simple compression and tension (Lubliner [250])
In recent years the term “geomaterials” has become current as one encompassing soils,
rocks, and concrete. What these materials have in common, and in contrast to metals, is the great
sensitivity of their mechanical behavior to pressure, resulting in very different strengths in
tension and compression. Beyond this common feature, however, the differences between soils
on the one hand and rocks and concrete on the other are significant. Soils can usually undergo
very large shearing deformations, and thus can be regarded as plastic in the usual sense,
although in Geotechnics only cohesive, claylike soils (that can be easily moulded without
crumbling) are usually labelled as “plastic”. Rock and concrete, on the other hand, are brittle,
except under high triaxial compression. Nevertheless, unlike classically brittle solids, which
fracture shortly after the elastic limit is attained, concrete and many rocks can undergo inelastic
deformations that may be significantly greater than the elastic strains, and their stress-strain
curves superficially resemble those of plastic solids (Lubliner [250]).
The essential property of soils is that they are particulate, that is, they are composed of many
small solid particles, ranging in size from less than . mm0 001 (in clays) to a few millimetres (in
coarse sand and gravel). Permanent shearing deformation of a soil mass occurs when particles
slide over one another. Beyond this defining feature, however, there are fundamental differences
among various types of soils, differences that are strongly reflected in their mechanical behavior.
Clays are fine-grained soils whose particles contain a significant proportion of minerals known
as clay minerals. The chemistry of these minerals permits the formation of an adsorbed water
film that is many times thicker than the grain size. This film permits the grains to move past one
another, with no disintegration of the matrix, when stress is applied. It is this property that in
Geotechnics is labelled as plasticity. Claylike soils are also generally known as cohesive soils. In
cohesionless soils, such as gravels, sands, and silts, the movement of grains past one another is
resisted by dry friction, resulting in shear stresses that depend strongly on the compression.
a) b)
12

41. Materials of this type are sometimes called frictional materials. As in ductile metals, failure in
soils occurs primarily in shear. Unlike metals, the shear strength of soils is, in most
circumstances, strongly influenced by the compressive normal stress acting on the shear plane
and therefore by the hydrostatic pressure. The dependence of the shear strength of soils on the
normal stress acting on the shearing plane varies with the type and condition of the soil. It ranges
from the simplest Coulomb law of friction in dry cohesionless soils such as gravels, sands, and
silts, to a more complex expression in terms of the Terzaghi’s effective stress in wet
cohesionless soils. In clays, the stresses in the adsorbed water layers play an important role in
determining strength, and in partially saturated clays this role is predominant. In this case the
shear strength of such clays is given approximately by the sum of frictional term, like in
cohesionless soils, and an additional term depending on the so-called cohesion, a material
constant representing the shear strength under zero normal stress.
Unlike soils, materials such as rock, mortar and concrete are generally not plastic in the sense
of being capable of considerable deformation before failure. Instead, in most tests they fracture
through crack propagation when fairly small strains (on the order of %1 or less) are attained,
and can therefore be regarded as quasi-brittle. Concrete, mortar, and many rocks (such as marble
and sandstone) are also unlike characteristically brittle solids like glass and cast iron, which
fracture shortly after the elastic limit is attained. Instead, they attain their ultimate strength after
developing permanent strains that, while small in absolute terms, are significantly greater than
the elastic strains. The permanent deformation is due to several mechanisms, the foremost of
which is the opening and closing of cracks. An important feature of the triaxial behaviour of
concrete, mortar and rocks (including those which are classically brittle in unconfined tests) is
that, if the confining pressure 3σ is sufficiently great, then crack propagation is prevented, so
that brittle behaviour disappears altogether and is replaced by ductility with work-hardening. It
can be said, in general, that rocks and concrete behave in a ductile manner if all three principal
stresses are compressive and close to one another. If the transverse strain 2 3ε ε= is measured in
uniaxial compression tests of rock and concrete specimens in addition to the axial strain 1ε , then
the volumetric strain Vε equals 1 2 3ε ε ε+ + . If the stress 1σ is plotted against Vε (positive in
compression), it is found that Vε begins to decrease from its elastic value at stresses greater than
about half the ultimate strength, reaches zero at a stress near the ultimate strength, and becomes
negative (signifying an increase in volume) in the strain-softening range. Similar results are
obtained in triaxial tests under low confining pressures. This volume increase, which results
from the formation and growth of cracks parallel to the direction of the greatest compressive
stress, is known as dilatancy. This term is sometimes also applied to the swelling of dense
granular soils, although the mechanism causing it is unrelated (Lubliner [250]).
In the following sections, some of the main ingredients of plasticity theory are recalled
without specific reference to concrete structures. On the basis of the above remarks about
plasticity of soils, rocks and concrete, the concepts that will be described below can apply, with
reasonable approximation, to concrete under sufficient confinement.
13