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Robustness of flat slab structures subjected to a sudden columnfailure scenario

Author(s): Herraiz Gómez, Borja

Publication Date: 2016

Permanent Link: https://doi.org/10.3929/ethz-a-010747984

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Robustness of flat slab structures subjected to a sudden column failure scenario

Borja Herraiz Gómez

IBK Bericht Nr. 368, Oktober 2016

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KEYWORDS: Robustness, modelling, column failure, energy balance, concrete slabs, membrane action,comparative assessment.

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Borja Herraiz Gómez: Robustness of flat slab structures subjected to a sudden column failure scenario

Bericht IBK Nr. 368, Oktober 2016

© 2016 Institut für Baustatik und Konstruktion der ETH Zürich, Zürich

Gedruckt auf säurefreiem PapierPrinted in Switzerland

ROBUSTNESS OF FLAT SLAB STRUCTURES SUBJECTED TO

A SUDDEN COLUMN FAILURE SCENARIO

Borja Herraiz Gómez

Institute of Structural Engineering Swiss Federal Institute of Technology Zurich

Zurich October 2016

PREFACE

Following the events of September 11, 2001, the robustness of structures has regained awareness from researchers and code writers. Although it is generally accepted that the failure of a single structural element should not lead to the collapse of the whole structure, there is still no fundamental consensus on how such a verification should be done. For buildings, however, the failure of a column – by whatever reasons – is an appropriate and commonly accepted scenario to judge the achieved degree of robustness.

Borja Herraiz concentrates on the frequent type of multi-storey concrete buildings with flat slabs and regular column patterns. He shows that – especially for internal columns – a compression ring forms in the slab that enables membrane action in the region of the failed column. In this way, the damage remains locally limited, provided that the slab can bear the load by membrane action even if the spans have doubled due to the failed column. Since the sudden removal of a column causes dynamic forces, their influence has to be considered as well.

As an alternative to calculations in the time domain, dynamic influences can be estimated by balancing potential, dynamic and dissipated energies. Herraiz defined and assessed the various assumptions of this method and compared the results with the few documented tests available. He could show that this ap-proach provides reasonable results on the safe side.

By considering geometric non linearity due to large deflections, structural resistances can be mobilised that go far beyond those of linearised yield-line theory. Finally, Herraiz proposes a robustness index that describes to what extent the dynamic load carrying capacity can be increased compared to the ordinary value for quasi-static actions and small deflections.

The real core of the thesis is the systematic description of subsequent stages of the behaviour of rec-tangular slabs subjected to distributed loads. Uncracked, cracked, linear elastic and ideal plastic behaviour are conventional; pursuing the kinematic for large deflections, however, shows that the different parts of a slab do not only rotate about the intersecting yield lines, but also about vertical axes. Because such rotations are restrained by the adjacent parts, the yield lines are not only subjected to bending moments but also to tension and compression forces. Additional cracks in the axes of symmetry are necessary to comply with kinematics and additional failure modes may occur. Herraiz developed a software to numerically describe the different stages and the transitions between them. He compared his approach with those of other au-thors and with documented physical tests from literature. Since quite a few test are documented, statistical analyses were possible.

With this work, which has been accepted as thesis for his doctorate, Borja Herraiz has delivered a considerable contribution towards judging the structural capacity of flat slabs that have suddenly lost a support. This is a prerequisite to perform codified robustness verifications in engineering practice in mid term.

Zurich, October 2016 Thomas Vogel

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TABLE OF CONTENTS

PREFACE ........................................................................................................................... iii 

TABLE OF CONTENTS .................................................................................................... v 

NOMENCLATURE ........................................................................................................... xi 

ABSTRACT ....................................................................................................................... xxi 

KURZFASSUNG ............................................................................................................. xxiii 

RESUMEN ...................................................................................................................... xxv 

1. INTRODUCTION 1 

1.1. Context of the research topic .......................................................................................................................... 1 

1.2. Problem statement ............................................................................................................................................ 2 

1.3. Motivation and objectives ................................................................................................................................ 2 

1.4. Scope and limitations ........................................................................................................................................ 3 

1.5. Outline of the thesis .......................................................................................................................................... 3 

2. ROBUSTNESS OF BUILDING STRUCTURES 5 

2.1. Introduction ....................................................................................................................................................... 5 

2.2. Definition of key terms .................................................................................................................................... 5 2.2.1. Structural robustness ................................................................................................................................. 5 2.2.2. Progressive and disproportionate collapse ............................................................................................ 6 

2.3. Historical case studies ....................................................................................................................................... 6 2.3.1. Ronan Point ................................................................................................................................................ 6 2.3.2. Alfred P. Murrah Federal Building ......................................................................................................... 7 2.3.3. World Trade Center .................................................................................................................................. 8 2.3.4. Other progressive and disproportionate collapse events .................................................................... 8 

2.4. Strategies and design methods for enhancing structural robustness ......................................................... 8 

2.5. Structural robustness in the standards ........................................................................................................ 11 2.5.1. European, British and Swiss standards ............................................................................................... 11 2.5.2. Guidelines from the United States ....................................................................................................... 12 

American Society of Civil Engineers ......................................................................................................... 12 General Services Administration ................................................................................................................ 13 Department of Defense ............................................................................................................................... 14 

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2.6. Quantifying structural robustness ................................................................................................................. 15 

2.7. The ALP method and the sudden column failure scenario ...................................................................... 15 

2.8. Membrane action as reserve of strength ...................................................................................................... 17 2.8.1. Membrane action in laterally restrained slabs ..................................................................................... 17 2.8.2. Membrane action in laterally unrestrained slabs ................................................................................. 19 

3. ROBUSTNESS ASSESSMENT PROCEDURE OF FLAT SLAB STRUCTURES SUBJECTED TO A SUDDEN COLUMN FAILURE 21 

3.1. Introduction ..................................................................................................................................................... 21 

3.2. Problems arising when modelling and analysing building structures subjected to a sudden column removal .............................................................................................................................................. 21 

3.3. Damage tolerance of reinforced concrete flat slab structures .................................................................. 22 

3.4. Degree of idealisation of the structural system .......................................................................................... 23 3.4.1. Modelling of a flat slab structure subjected to a column removal ................................................... 23 3.4.2. Modelling the boundary conditions ...................................................................................................... 25 

3.5. Extent of damage ............................................................................................................................................ 26 

3.6. Modelling the dynamic problem ................................................................................................................... 27 3.6.1. Nonlinear time history dynamic analysis ............................................................................................. 27 3.6.2. Linear elastic static analysis with Dynamic Amplification Factors .................................................. 27 3.6.3. Nonlinear static analysis with Dynamic Amplification Factors ....................................................... 27 3.6.4. Energy-based analysis ............................................................................................................................. 28 

3.7. Modelling the material and geometrical nonlinearities .............................................................................. 29 3.7.1. Critical failure modes of the reduced system ...................................................................................... 29 3.7.2. Flexural behaviour of laterally unrestrained reinforced concrete slabs ........................................... 31 3.7.3. Experimental tests on laterally unrestrained slabs .............................................................................. 33 3.7.4. Simplified approaches for analysing the response of laterally unrestrained slabs ......................... 33 

Slabs subjected to a peripheral column removal ...................................................................................... 33 Slabs subjected to an internal column removal ........................................................................................ 34 

3.7.5. Validation of the new design approach ................................................................................................ 35 

3.8. Proposal for a structural robustness index .................................................................................................. 35 

4. ENERGY BALANCE METHOD 37 

4.1. Introduction ..................................................................................................................................................... 37 

4.2. Physical, theoretical and graphical interpretation ....................................................................................... 37 

4.3. Assumptions and inherent errors of the EBM: parameter studies .......................................................... 38 4.3.1. Step gravity load as a sudden column removal ................................................................................... 39 4.3.2. The affected substructure behaves like a SDOF system ................................................................... 43 4.3.3. Damping and energy dissipation mechanisms .................................................................................... 47 4.3.4. Strain-rate effects ..................................................................................................................................... 50 4.3.5. Conclusions of the parameter studies .................................................................................................. 52 

4.4. Comparison with experimental results ........................................................................................................ 53 4.4.1. Methodology ............................................................................................................................................ 53 

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4.4.2. Description of the test specimens ........................................................................................................ 53 4.4.3. Experimental results and application of the energy-based approach ............................................. 54 

Corner column removal ............................................................................................................................... 54 Penultimate column removal ...................................................................................................................... 55 

4.4.4. Conclusions of the comparison with experimental tests .................................................................. 56 

5. NOVEL DESIGN APPROACH FOR THE STATIC ANALYSIS OF LATERALLY UNRESTRAINED SLABS CONSIDERING MEMBRANE ACTION 57 

5.1. Introduction .................................................................................................................................................... 57 

5.2. Scope and adopted assumptions .................................................................................................................. 57 

5.3. Material assumptions ..................................................................................................................................... 58 

5.4. Pre-yielding stage ............................................................................................................................................ 59 5.4.1. Introduction ............................................................................................................................................. 59 5.4.2. Linear stage .............................................................................................................................................. 59 5.4.3. Parabolic stage ......................................................................................................................................... 61 

5.5. Transitional stage ............................................................................................................................................ 63 5.5.1. Introduction ............................................................................................................................................. 63 5.5.2. Critical yield line pattern and ultimate flexural load .......................................................................... 63 5.5.3. Description of the kinematic model .................................................................................................... 64 5.5.4. Ultimate vertical deflection ................................................................................................................... 68 5.5.5. Transition curve ...................................................................................................................................... 69 

Exponential function.................................................................................................................................... 69 Fourth order polynomial function ............................................................................................................. 71 Domain of application ................................................................................................................................. 71 

5.6. Membrane action stage .................................................................................................................................. 72 5.6.1. Introduction ............................................................................................................................................. 72 5.6.2. Distribution of concrete depths along the yield lines ....................................................................... 73 5.6.3. Yield and failure criterion ...................................................................................................................... 74 5.6.4. Relative lateral displacements along the axial crack .......................................................................... 77 5.6.5. Bond-slip model ...................................................................................................................................... 78 5.6.6. Equilibrium relationships ...................................................................................................................... 83 

Equilibrium of the membrane forces along the flexural yield lines ...................................................... 85 Equilibrium of the in-plane bending moments at the compression centre of the axial crack ......... 86 

5.6.7. Estimation of the load-bearing capacity of the slab .......................................................................... 87 

5.7. Failure criteria ................................................................................................................................................. 89 5.7.1. Compressive failure due to concrete crushing ................................................................................... 89 5.7.2. Reinforcement rupture at the axial crack ............................................................................................ 90 

5.8. Computational implementation ................................................................................................................... 90 5.8.1. Introduction ............................................................................................................................................. 90 5.8.2. Input parameters ..................................................................................................................................... 91 5.8.3. Additional parameters ............................................................................................................................ 91 5.8.4. Pre-yielding stage .................................................................................................................................... 92 5.8.5. Transitional stage .................................................................................................................................... 93 5.8.6. Membrane action stage .......................................................................................................................... 93 

Table of contents

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6. OVERVIEW AND EVALUATION OF PREVIOUS APPROACHES FOR THE ANALYSIS OF LATERALLY UNRESTRAINED SLABS 95 

6.1. Introduction ..................................................................................................................................................... 95 

6.2. Overview of the most relevant previous approaches ................................................................................ 95 6.2.1. Sawczuk and Winnicki (1965) ................................................................................................................ 95 6.2.2. Taylor (1965) ............................................................................................................................................ 96 6.2.3. Kemp (1967)............................................................................................................................................. 98 6.2.4. Morley (1967) ........................................................................................................................................... 98 6.2.5. Hayes (1968) ............................................................................................................................................. 98 6.2.6. Bailey (2001) ............................................................................................................................................. 99 6.2.7. Omer et al. (2006) ................................................................................................................................. 100 6.2.8. Bailey and Toh (2007) .......................................................................................................................... 102 6.2.9. Numerical approaches ......................................................................................................................... 102 

6.3. Parameter study of the previous approaches ........................................................................................... 103 6.3.1. Square isotropic simply supported reinforced concrete slab ......................................................... 103 6.3.2. Rectangular isotropic simply supported reinforced concrete slab ................................................ 105 6.3.3. Rectangular orthotropic simply supported reinforced concrete slab ........................................... 107 

6.4. Comparison of the different proposed collapse mechanisms ............................................................... 109 

6.5. Comparison of the new design approach with previous approaches .................................................. 110 6.5.1. Square isotropic simply supported reinforced concrete slab ......................................................... 111 6.5.2. Rectangular isotropic simply supported reinforced concrete slabs .............................................. 111 6.5.3. Rectangular orthotropic simply supported reinforced concrete slabs ......................................... 112 

7. COMPARATIVE ASSESSMENT OF THE DESIGN APPROACHES WITH EXPERIMENTAL TESTS 115 

7.1. Introduction .................................................................................................................................................. 115 

7.2. Small-scale tests performed by Cashell et al. ........................................................................................... 115 7.2.1. Methodology and description of the experimental set-up and specimens .................................. 115 7.2.2. Experimental and simulation results ................................................................................................. 118 7.2.3. Application of the EBM ...................................................................................................................... 119 

7.3. Small-scale tests performed by Bailey and Toh ....................................................................................... 123 7.3.1. Methodology and description of the experimental set-up and specimens .................................. 123 7.3.2. Experimental and simulation results ................................................................................................. 125 7.3.3. Application of the EBM ...................................................................................................................... 127 

7.4. Full-scale test conducted by Bailey et al. .................................................................................................. 131 7.4.1. Methodology and description of the experimental set-up and specimens .................................. 131 7.4.2. Experimental and simulation results ................................................................................................. 133 7.4.3. Application of the EBM ...................................................................................................................... 134 

7.5. Comparison and discussion of the experimental and predicted load-deflection relationships........ 134 

7.6. Comparison and discussion of the application of the EBM and the corresponding pseudo-dynamic results ............................................................................................................................................. 135 

7.7. Comparison and discussion of the experimental and predicted failure loads and deflections ........ 136 7.7.1. Description of the statistical analysis ................................................................................................. 136 7.7.2. Description of the graphical representation ..................................................................................... 137 

Table of contents

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7.7.3. Comparison of the experimental and predicted relative central vertical deflections at failure ..................................................................................................................................................... 138 

7.7.4. Comparison of the experimental and predicted failure static relative bearing capacities .......... 139 7.7.5. Comparison of the experimental and predicted failure dynamic bearing capacities .................. 140 7.7.6. Discussion of the statistical analysis .................................................................................................. 141 7.7.7. Discussion of the graphical representation ...................................................................................... 142 

7.8. Conclusions ................................................................................................................................................... 144 

8. CONCLUSIONS AND FURTHER RESEARCH 145 

8.1. Introduction .................................................................................................................................................. 145 

8.2. Summary and conclusions ........................................................................................................................... 145 

8.3. Further research ............................................................................................................................................ 147 8.3.1. Theoretical investigation ...................................................................................................................... 147 8.3.2. Experimental investigation .................................................................................................................. 148 

BIBLIOGRAPHY ............................................................................................................. 149 

xi

NOMENCLATURE

Abbreviations

ALP Alternative Load PathASCE American Society of Civil EngineersBRE British Research EstablishmentCC Consequence Class CI Confidence Interval DAF Dynamic Amplification FactorDCR Demand Capacity RatioDL Dead Load DoD Department of DefenseEBM Energy Balance MethodFEM Finite Element MethodFM Failure Mode GSA General Service AdministrationKED Key Element Design LL Live Load LOP Level Of Protection MAPE Mean Absolute Percentage ErrorMDOF Multi Degree Of FreedomM-N Moment–Normal force interactionOLS Ordinary Least SquaresPI Prediction Interval RC Reinforced Concrete RIF Residual Influence FactorRSR Reserve of Strength RatioSDOF Single Degree Of FreedomSL Snow Load SLR Specific Load ResistanceWL Wind Load 2D 2 Dimensional 3D 3 Dimensional

Nomenclature

xii

Uppercase Roman letters

A cross-sectional area per unit width A n integration constant A s area of the tension layer of reinforcing steel per unit width A s´ area of the compression layer of reinforcing steel per unit width A sx area of the tension layer of reinforcing steel per unit width in the x-direction A sx´ area of the compression layer of reinforcing steel per unit width in the x-direction A sy area of the tension layer of reinforcing steel per unit width in the y-direction A sy´ area of the compression layer of reinforcing steel per unit width in the y-direction B n integration constant C viscous damping coefficient C F resultant of the cross-sectional compression forces C r consequences (costs) associate to a global failure or structural collapse D uncracked flexural stiffness of the reinforced concrete slab D cr fully cracked flexural stiffness of the reinforced concrete slab E energy E cm Young’s modulus for concrete E k kinetic energy of a structural system  P rate of work done by a point-load arrangement  q rate of work done by a uniformly distributed load

E s absorbed strain energy of a structural system E se elastic Young’s modulus for reinforcing steel E s, eff effective Young’s modulus for reinforcing steel in the bond-slip model E sp post-yielding hardening modulus for reinforcing steel F C resultant of the compression axial forces along the diagonal yield lines F C, c resultant of the compression forces along the axial crack F lim,t total experimental load causing failure F T resultant of the tensile axial forces along the diagonal yield lines F T ´ resultant of the tensile forces along the yield line parallel to the longer span F T, c resultant of the tensile forces along the axial crack F u total theoretical ultimate bearing capacity of the slab given by the classical yield-line the-

ory I cr moment of inertia of the cracked cross section I eff effective moment of inertia of the cross section at the corners K associated structural stiffness K 1 elastic structural stiffness K 2 post-yielding structural stiffness L longer span of the rectangular slab L b length of the beam M associated structural mass M C in-plane moments caused by the compression forces along the diagonal yield lines M CC resultant of the in-plane moments at the compression centre of the axial crack M p plastic moment M T in-plane moments caused by the tensile forces along the diagonal yield lines M T´ in-plane moments caused by the tensile forces along the parallel yield line M T, c in-plane moments caused by the tensile forces along the axial crack

Nomenclature

xiii

M V in-plane moments caused by the in-plane shear forces along the diagonal yield lines P applied point load P(E ) probability of occurrence of hazard E (accidental action) P(F ) probability of global failure or structural collapse P(F |LE ) probability of collapse given that L and E both occur P(L|E ) probability of local damage, L, given that E occurs Q general load acting on a structure Q d dynamic load given by the dynamic capacity curve Q Ld static load given by the nonlinear pushover response Q 1, Q 2 statically equivalent nodal vertical forces acting at the intersection of the yield lines R structural capacity R F robustness factor R y yielding capacity of the structure R 2 coefficient of determination for a linear regression S y standard error of the estimate T natural vibration period T F resultant of the cross-sectional tensile forces T u, x reinforcement ultimate strength per unit width in the x-direction T y, x reinforcement yield strength per unit width in the x-direction T 1 fundamental natural vibration period V resultant of the in-plane shear forces along the diagonal yield lines W L work done by the acting loads X parameters defining the ratio of negative to positive ultimate moment in the x-direction Y parameters defining the ratio of negative to positive ultimate moment in the y-direction

Lowercase Roman letters

a p parameter defining the parabolic curve of the pre-yielding stage b reference unit width of a cross section b p parameter defining the parabolic curve of the pre-yielding stage c compressive concrete depth (distance to the neutral axis) c (x) concrete depth distribution along the yield line parallel to the longer span c (y) concrete depth distribution along the diagonal yield lines c A B concrete depth for simultaneous failure of concrete and reinforcement c bar concrete depth of the cross section’s barycentre c c concrete depth at the corners of the slab c c,1 concrete depth causing concrete reaching the strain for maximum strength ε c 1 c n averaged concrete depth of the neutral axis for the ultimate pure positive bending moment

(n = 0) c n´ averaged concrete depth of the neutral axis for the ultimate pure negative bending moment

(n = 0) c p parameter defining the parabolic curve of the pre-yielding stage c y,A concrete depth corresponding to the yielding under tension of the compression reinforce-

ment, for the ultimate curvature causing tension reinforcement failure c y,A,c concrete depth corresponding to the yielding under compression of the compression re-

inforcement, for the ultimate curvature causing tension reinforcement failure

Nomenclature

xiv

c y,B,t concrete depth corresponding to the yielding under tension of the compression reinforce-ment, for the ultimate curvature causing concrete failure

c y,B,t,c concrete depth corresponding to the yielding under compression of the compression re-inforcement, for the ultimate curvature causing concrete failure

c y,B,b concrete depth corresponding to the yielding under tension of the tension reinforcement, for the ultimate curvature causing concrete failure

c y,B,b,c concrete depth corresponding to the yielding under compression of the tension reinforce-ment, for the ultimate curvature causing concrete failure

d effective depth of the bottom layer reinforcement d´ effective depth of the top layer reinforcement d cc height of the axis of rotation d r lever arm of the neutral axis d x effective depth of the bottom layer reinforcement in the x-direction d x´ effective depth of the top layer reinforcement in the x-direction d y effective depth of the bottom layer reinforcement in the y-direction d y´ effective depth of the top layer reinforcement in the y-direction e B enhancement of the load-bearing capacity due to membrane action predicted by Bailey

and Toh e H enhancement of the load-bearing capacity due to membrane action predicted by Hayes e lim,s maximum static capacity enhancement due to membrane action predicted by the consid-

ered approach e lim,s, EBM maximum dynamic capacity enhancement due to membrane action estimated by means of

the energy balance method, calculated based on a simulated pushover response e lim,t maximum static capacity enhancement due to membrane action observed in the experi-

mental tests e lim,t, EBM maximum dynamic capacity enhancement due to membrane action estimated by means of

the energy balance method, calculated based on an experimental pushover response e m enhancement of the load-bearing capacity due to membrane action err relative error on the maximum dynamic displacement estimation of the energy balance

method e x (y) partial elongation of the diagonal yield lines in the x-direction at the neutral axis for the

positive ultimate moment e y (x) partial elongation of the parallel yield line in the y-direction at the neutral axis for the

positive ultimate moment e θ´ partial elongation orthogonal to the diagonal yield lines for the triangular slab regions e φ´ partial elongation orthogonal to the diagonal yield lines for the trapezoidal slab regions f ck characteristic compressive cylinder strength of concrete f cm average compressive cylinder strength of concrete f ctm average tensile strength of concrete f s,u average ultimate tensile strength of reinforcing steel f s,y average yield strength of reinforcing steel g gravitational acceleration h total slab thickness h min minimum slab thickness of a profiled cross section i, j summation indices k, k1, k2, k3 auxiliary parameters used for the geometrical relationships of the slab l shorter span of the rectangular slab l b reinforcing bar bonding length l b, y reinforcing bar yielded bonding length m  internal bending moment of a cross section per unit width

Nomenclature

xv

m cr cracking bending moment of a cross section per unit width m cr,x cracking bending moment in the x-direction of a cross section per unit width m cr,y cracking bending moment in the y-direction of a cross section per unit width m n, c bending moment orthogonal to the diagonal yield lines per unit width at the corners of

the slab m q bending moment along the edges of the slab produced by uniform vertical loading m u positive bending moment of a cross section per unit width, caused by an ultimate curvaturem u´ negative bending moment of a cross section per unit width, caused by an ultimate curvaturem u, n ultimate bending moment orthogonal to the yield lines per unit width, caused by an ulti-

mate curvature m u, nt ultimate torsional moment orthogonal to the yield lines per unit width, caused by an ulti-

mate curvature m u, x positive ultimate bending moment in the x-direction per unit width, caused by an ulti-

mate curvature m u, x´ negative ultimate bending moment in the x-direction per unit width, caused by an ulti-

mate curvature m u, y positive ultimate bending moment in the y-direction per unit width, caused by an ulti-

mate curvature m u, y´ negative ultimate bending moment in the y-direction per unit width, caused by an ulti-

mate curvature m u, 0 ultimate pure bending moment of a cross section per unit width (n = 0) m u, 0, x positive ultimate pure bending moment in the x-direction per unit width (n = 0) m u, 0, x´ negative ultimate pure bending moment in the x-direction per unit width (n = 0) m u, 0, y positive ultimate pure bending moment in the y-direction per unit width (n = 0) m u, 0, y´ negative ultimate pure bending moment in the y-direction per unit width (n = 0) m x, max overall maximum elastic bending moments in the x-direction per unit width m y yielding bending moment of a cross section per unit width m y, max overall maximum elastic bending moments in the y-direction per unit width m y, x yielding bending moment in the x-direction of a cross section per unit width m y, y yielding bending moment in the y-direction of a cross section per unit width n internal axial force acting on a cross section per unit width n E ratio of the modulus of elasticity of reinforcing steel to concrete n , c axial force orthogonal to the diagonal yield lines per unit width at the corners of the slab n u axial force acting on a cross section per unit width, caused by an ultimate curvature n u,n axial force orthogonal to the yield lines per unit width, caused by an ultimate curvature n u,t in-plane shear force parallel to the yield lines per unit width, caused by an ultimate curva-

ture n u, x axial force in the x-direction per unit width, caused by an ultimate curvature n u, y axial force in the y-direction per unit width, caused by an ultimate curvature p total number of observations considered for the evaluation q applied uniform load/load bearing capacity of the slab q cma maximum uniform load developed by compressive membrane action q cr uniform load causing first cracking q e,max maximum uniform load at the end of the pre-yielding stage corresponding to w e,max q lim uniform static load causing failure q lim,EBM uniform load corresponding to the estimated maximum dynamic capacity by means of the

energy balance method q lim,s uniform static load causing failure predicted by the considered approach

Nomenclature

xvi

q lim,s,EBM uniform load corresponding to the estimated maximum dynamic capacity by means of the energy balance method, calculated based on a simulated pushover response

q lim,t uniform static load causing failure observed in the experimental testsq lim,t,EBM uniform load corresponding to the estimated maximum dynamic capacity by means of the

energy balance method, calculated based on an experimental pushover response q qp quasi-permanent combination of loads acting on the considered structureq u uniform ultimate flexural load calculated by means of the classical yield-line theory q y uniform load causing first yielding q κ,cr uniform load of the pre-yielding stage corresponding to the vertical deflection w κ,cr q0 gravity loads acting on the considered substructure before the column removal r associated risk to a global failure or structural collapses (y) lateral displacements along the axial cracks ini (y) lateral displacements along the axial crack before membrane forces develop s r (y) relative lateral displacement along the axial cracks t, b average spacing between reinforcing bars in the y-directions y r relative displacement/slip at the axial crack causing yielding of the reinforcing bars s u r relative displacement/slip at the axial crack causing rupture of the reinforcing bars t time t r column removal time t α,p-2 Student’s t critical two-tailed values for a 1-α confidence level and p-2 degrees of freedom t 0 time moment of the sudden column removalu (y) lateral displacements of the slab at the axis of rotation in the x-direction u b slip of the embedded reinforcing bar along the axial mid-cracku n (y) lateral displacement of the slab at the neutral axis for the pure positive bending ultimate

moment in the x-direction v (x) lateral displacements of the slab at the axis of rotation in the y-directionv n (x) lateral displacement of the slab at the neutral axis for the pure positive bending ultimate

moment in the y-direction w generic vertical deflection of the structural systemw cr central vertical deflection corresponding to first cracking w d,e maximum dynamic deflection given by the considered approachw d,EBM maximum dynamic deflection given by the energy balance methodw e,max maximum central vertical deflection at the end of the pre-yielding stagew e,max,inf maximum value of w e,max that assures that the second point of the fourth order polynomial

curve is at the end of the transitional stage w ini central vertical deflection at which a specific point of the axial crack starts opening w lim central vertical deflection corresponding to failurew lim,s central vertical deflection corresponding to failure predicted by the considered approach w lim,t central vertical deflection corresponding to failure observed in the experimental tests w p,st preliminary static deflections due to the gravity loads prior to the column removal w u central ultimate vertical deflection for which membrane action is assumed to begin in the

new design approach w u,t approximate experimental vertical deflection for which membrane action begins in the slab w y central vertical deflection corresponding to first yieldingw y, c central vertical deflection for which strains begin to develop at the corners w y, c, 2 central vertical deflection for which the cross section at the corners completely yields w y,qy maximum central vertical deflection for the pre-yielding stage that assures that the yielding

load is not exceeded

Nomenclature

xvii

w κ,cr maximum central vertical deflection for the pre-yielding stage that assures that the mini-mum flexural stiffness at the end of the stage is at least the fully-cracked flexural stiffness

w 0 central vertical deflection w 0, inf´ deflection coordinate of the second inflection point for the fourth order polynomial tran-

sition curve x parameter defining dimensions in the longer span direction

independent coordinates of the ordinary least squares regression line x b coordinate defining positions along the bonding length of the reinforcing bars x d bond-slip length x t experimental result of the correspondent parameter ̅  t average value of the experimental results of the correspondent parameter by the consid-

ered approach x 0 parameter defining the point of zero axial forces along the diagonal yield lines in the x-

direction y parameter defining dimensions in the shorter span direction

dependent coordinates of the ordinary least squares regression line y cc parameter defining the position of the compression centre at the axial crack y cr length of the tensile zone of the axial crack y s predicted result of the correspondent parameter by the considered approach  s average value of the predicted results of the correspondent parameter by the considered

approach y 0 parameter defining the point of zero axial forces along the diagonal yield lines in the y-

direction z parameter defining dimensions in the vertical direction z cr parameter defining the position of the neutral axis on a cracked reinforced concrete cross

section (centroid of the transformed cross section taking in account the different elasticity modulus of steel and concrete)

z n parameter defining the position of the neutral axis on a generic cross section

Uppercase Greek letters

Γ ratio of ι st to χ st Δ parameter defining the position of the suddenly removed column Λ rotation ratio (φ/θ ) Υ dynamic amplification factor value Ψ ratio of slab length representing the lateral displacements of the slab Ω y parameter describing the effect of reinforcement yielding on the bond strength

Lowercase Greek letters

α angle between the diagonal yield lines of the slab and the x-direction α 1, 2, 3 parameters used for defining the relationship between the applied uniform load and the

elastic deflections and bending moments of a slab for different boundary conditions β, β’ angles defining elongations of the yield lines on the horizontal plane at the axis of rota-

tion γ aspect ratio of the slab (L/ l ) ε  generic cross-sectional axial strain

Nomenclature

xviii

  generic cross-sectional strain rates ε c  maximum cross-sectional strains of concrete ε c  maximum compression strains at the corners of the slab ε  c, u ultimate compressive strain of concrete ε c 1 concrete strain at the maximum compressive strength ε s cross-sectional strains of reinforcing steel ε s,c  longitudinal strain distribution of the reinforcing bars along the bonding length and axial

crack length ε s,c m assumed average yield strain of the reinforcement along the yielded bonding length ε s,c  (0) maximum strains along the tensile region of the axial crack ε s,c  (l/2) maximum strains along the compression region of the axial crack ε s, u ultimate tensile strain of reinforcing steel ε s, y yield strain of reinforcing steel η coefficient of orthotropy (m u,0, y/m u, 0, x) θ rotation relative to the horizontal plane of the triangular slab regions for the x-direction θ´ rotation relative to the horizontal plane of the triangular slab regions orthogonal to the

diagonal yield lines ι relative dynamic displacement of the structure measured from the position of static equi-

librium ̅ absolute dynamic displacement of the structure ̅ velocity of the structure ̅ acceleration of the structure  ̅d absolute maximum dynamic displacement of the structure  ̅d,e absolute maximum dynamic displacement given by the considered approach  ̅d,EBM absolute maximum dynamic displacement given by the energy balance method ι st displacement of the structure at the position of static equilibrium κ cr slope of the fully cracked stiffness on the load-deflection relationship κ f slope of the uncracked stiffness on the load-deflection relationship λ parameter describing the effect of transverse cracking in the bond strength μ (y) parameter describing the variation of concrete depth with respect to the neutral axis for

pure bending (n = 0) μ d ductility value of the structure ξ viscous damping ratio ρ reinforcement ratio of the tension layer of reinforcing steel ρ´ reinforcement ratio of the compression layer of reinforcing steel ς ratio of the total to the uniformly distributed bearing capacity of the slab given by the

classical yield-line theory σ generic cross-sectional normal stress τ bond stress between concrete and the reinforcing bars τ b average effective bond strength between concrete and the reinforcing bars τ b, m average bond strength considering the effect of transverse cracking τ b, u average bond strength when ε s, u is reached τ b, y average bond strength considering the effect of yielding of reinforcement υ Poison’s ratio of the reinforced concrete slab φ rotation relative to the horizontal plane of the trapezoidal slab regions for the y-direction φ´ rotation relative to the horizontal plane of the trapezoidal slab regions orthogonal to the

diagonal yield lines χ curvature of a cross section

Nomenclature

xix

curvature rates of a cross section χ A ultimate curvature corresponding to tension reinforcement failure χ B ultimate curvature corresponding to concrete crushing χ st curvature of a structure’s cross section at the position of static equilibrium ψ parameter defining the geometry of the yield line pattern ω slope of the load-deflection curve at the end of the pre-yielding stage ω n nth natural vibration frequency of the structure ω 1 fundamental vibration frequency of the structure ϖ parameter defining the ratio of the rate of work done by a point-load to the total load

Other symbols

ϕ b average value of the diameter of the reinforcing bars in the x-direction ϕ n nth vibration mode of the structure ϕ 1 fundamental vibration mode of the structure

xxi

ABSTRACT

Several catastrophic building collapses during the last decades have emphasised the necessity for increasing the progressive collapse resistance of structures. In response, several widely used design codes have included the requirement that a structure should be capable of surviving the removal of a load-bearing element. This approach, often referred to as the sudden column removal scenario, is hazard-independent and focuses solely on the capacity of a structure to utilise alternative load paths and redistribute gravity loads after the loss of a column.

The behaviour of steel and reinforced concrete frame structures subjected to this hazard scenario has been the topic of extensive research, and design requirements have been proposed. Such investigations for reinforced concrete flat slabs, however, are much scarcer. It is important to consider the behaviour of such slabs in more detail since they are commonly used in buildings where large agglomerations of human beings occur. This is primarily due to the benefits of their design flexibility and low construction height. The analysis of flat slab structures subjected to a sudden column removal scenario is a challenging task due to the large resulting deformations, and simplified structural models need to be developed.

This research project focuses on the response of flat slab structures to a sudden column loss and proposes a global procedure to evaluate and quantify the robustness of such structures subjected to this hazard scenario. This procedure addresses the indirect issues related to the idealisations of the structural system modelling, the corresponding boundary conditions, the critical failure modes and the extent of dam-age in flat slab buildings subjected to a column removal. Direct problems arising from the analysis of the bay directly affected by the sudden column loss, such as material and geometrical nonlinearities and dynamic effects, were investigated in detail. In addition, an indicator is proposed to quantify the structural robustness of flat slab structures.

The dynamic component of this problem was simplified through the implementation of an energy-based method, which reduces the analysis of the system to that of a push-over static response. This approach is based on several simplifications, which lead to approximate, yet accurate values of the maximum dynamic response of the structural system. A parameter study was performed to investigate the sensitivity of this method to different variables, and results from the method were compared to experimental results. The analyses indicate that this approach consistently leads to moderately conservative predictions for typical slab configurations. Additional experimental results are required, however, to validate the accuracy of the pro-posed method.

The calculation of the push-over static response requires consideration of the influence from material and geometrical nonlinearities. In order to solve this problem, a new approach was developed to estimate the static response of laterally unrestrained reinforced concrete slabs considering membrane action. This method is based on the kinematics of a rigid perfectly-plastic slab model and equilibrium of the internal forces originating from the corresponding deformations. Typical failure criteria were implemented to esti-mate the load-bearing capacity and corresponding failure deflection. Response predictions from the devel-oped method were compared to experimental results from a set of 45 specimens, originating from different experimental campaigns, and simulations utilising other approaches. A comparative assessment suggests that the proposed method generally leads to more precise, accurate, and therefore reliable predictions than the other approaches considered. The errors and deviations of the results obtained with this method are small, and the proposed method represents a promising new alternative to more complex and time-con-suming numerically-based, approaches without losing significant accuracy. In addition, it was found that the decrease on load-bearing capacity due to dynamic effects after the sudden column loss was practically bal-anced by a corresponding increase of the bearing capacity due to membrane action for most of the consid-ered specimens. Therefore, it would be possible to neglect both these effects in future calculations without incurring unacceptably large errors.

Abstract

xxii

In conclusion, the proposed global procedure enables performing accurate, quick, homogeneous and simple robustness assessments for flat slab structures subjected to a sudden column removal scenario. This procedure can also easily be extended in order to consider different dynamic effects, boundary conditions, failure modes, structural configurations and materials.

xxiii

KURZFASSUNG

In den letzten Jahrzehnten haben mehrere katastrophale Gebäudeeinstürze schmerzlich aufgezeigt, dass Tragwerke hinsichtlich eines progressiven Kollapses nicht ausreichend dimensioniert werden. Als Folge wurden in mehreren weitverbreiteten Normen die Konstruktionsanforderungen insofern angepasst, dass Tragwerke den Ausfall eines einzelnen Tragelements überstehen sollen. Dieser Ansatz, der oft als Gefähr-dungsbild plötzlicher Stützenausfall bezeichnet wird, ist gefahrenunabhängig und basiert auf der Fähigkeit von Tragwerken alternative Lastpfade zu schaffen und dadurch die bestehenden Lasten nach einem Ausfall umzuverteilen.

Ausführliche Forschungsvorhaben über das Verhalten von Rahmenkonstruktionen aus Stahl und Stahlbeton infolge plötzlicher Stützenausfall wurden durchgeführt und darauf basierend Konstruktionsan-forderungen vorgeschlagen. Untersuchungen an Stahlbetonflachdecken sind deutlich seltener. Flachdecken werden jedoch gerne auch in Gebäuden mit grossen Menschenansammlungen genutzt, da sie flexibel ein-setzbar sind und eine relativ niedrige Konstruktionshöhe haben. Daher ist es wichtig, ihr Verhalten beim Gefährdungsbild plötzlicher Stützenausfall im Detail zu betrachten. Eine Analyse ist infolge der zu erwar-tenden grossen Verformungen schwierig, und daher sind vereinfachte Tragwerksmodelle erforderlich.

Das vorliegende Forschungsprojekt konzentriert sich auf das Verhalten von Flachdecken infolge eines plötzlichen Stützenausfalls und stellt ein globales Vorgehen vor, um die Robustheit solcher Tragwerke zu beurteilen. Das vorgeschlagene Vorgehen befasst sich mit den indirekten Problemen bezüglich der Ideali-sierung des Tragwerks, den entsprechenden Randbedingungen, des kritischen Ausfallmodus und der Schad-stelle in der Flachdecke. Die direkten Probleme wie nichtlineares Materialverhalten, geometrische Nichtli-nearitäten und dynamische Einwirkungen, die aus der Analyse der direkt betroffenen Flachdecke entstehen, werden im Detail untersucht. Zudem wird ein Faktor vorgeschlagen, um die Robustheit von Flachdecken-konstruktionen zu bewerten.

Um die dynamische Komponente des Problems zu vereinfachen, wurde eine Energiebilanzmethode entwickelt. Dieser Ansatz erlaubt es, die dynamische Komponente durch das statische Tragverhalten des Systems zu bestimmen. Durch die vorausgesetzten Annahmen können damit die tatsächlichen Werte der maximalen Verschiebungen des Systems approximiert werden. Zur Verifikation der Methode wurde einer-seits eine Parameterstudie durchgeführt und andererseits die bestimmten Werte mit Ergebnissen von expe-rimentellen Versuchen verglichen. Die Ergebnisse dieser Untersuchungen zeigen, dass für typische Flach-decken der gewählte Ansatz stets zu leicht konservativen Prognosen führt. Die Ergebnisse sind allerdings nicht eindeutig hinsichtlich der Genauigkeit des Verfahrens. Daher sind weitere experimentelle Ergebnisse wünschenswert, um die Methode zu validieren.

Um die Berechnung des statischen Tragverhaltens des Systems durchzuführen, müssen sowohl das nichtlineare Materialverhalten sowie die geometrischen Nichtlinearitäten berücksichtigen werden. Dazu wurde eine neue Methode entwickelt, um das statische Tragverhalten von seitlich nicht gehaltenen Stahlbe-tonplatten unter Berücksichtigung der Membranwirkung abzuschätzen. Dieses Verfahren beruht auf der Kinematik einer perfekt starr plastischen Platte und dem Gleichgewicht der inneren Kräfte. Realistische Ausfallkriterien wurden angenommen, um die Tragfähigkeit und die auftretenden Durchbiegungen beim Versagen abzuschätzen. Das neue Verfahren wurde mit den Ergebnissen von 45 verschiedenen experimen-tellen Versuchen verglichen. Zusätzlich erfolgte ein Vergleich mit Simulationen basierend auf anderen be-reits vorhandenen Ansätzen. Die Ergebnisse dieser Analyse deuten darauf hin, dass die hier vorgestellte Methode in der Regel zu präziseren und deshalb verlässlicheren Prognosen führt als die bekannten Ansätze. Die Abweichungen des Verfahrens sind moderat, weshalb es eine viel versprechende Alternative zu kom-plexen und zeitraufwendigen numerischen Methoden ist, ohne bedeutend an Genauigkeit zu verlieren. Eine weitere Erkenntnis ist, dass die Abnahme der Tragfähigkeit aufgrund der dynamischen Einwirkungen nach einem plötzlichen Stützenausfall durch die Zunahme der Tragfähigkeit aufgrund der Membranwirkung

Kurzfassung

xxiv

– für die meisten der untersuchten Exemplaren – praktisch ausgeglichen ist. Deshalb kann ein Ansatz darin liegen, zukünftig beide Effekte bei der Analyse zu vernachlässigen.

Das vorgeschlagene einfache globale Vorgehen, um Flachdecken infolge eines plötzlichen Stützenaus-falls zu analysieren, ermöglicht die zügige und homogene Durchführung von Robustheitsbeurteilungen. Das vorgeschlagene Vorgehen kann weiter entwickelt werden, um andere Randbedingungen, dynamische Ein-wirkungen, Ausfallmodi, Tragwerksgestaltungen oder Materialien zu berücksichtigen.

xxv

RESUMEN

Durante las últimas décadas, diversos edificios han colapsado con consecuencias catastróficas, lo cual ha acentuado la necesidad de incrementar la resistencia estructural frente a colapsos progresivos. Consecuen-temente, varias normativas internacionales han incorporado como un requisito para el diseño que las estruc-turas sean capaces de resistir la pérdida de un elemento portante. Esta estrategia, comúnmente conocida como supresión repentina de una columna, es independiente del efecto o acción que produce este suceso y se centra únicamente en la capacidad de la estructura de establecer rutas alternativas de carga así como redis-tribuir las cargas gravitatorias tras la supresión de una columna.

La respuesta a tal extraordinario suceso de estructuras reticuladas tanto de acero como de hormigón armado, ha sido objeto de amplias investigaciones. Al mismo tiempo, se han propuesto requisitos para el diseño de estas estructuras. Sin embargo, las estructuras formadas por forjados planos de hormigón armado apenas han sido investigadas en este aspecto. Este tipo de configuración estructural, muy popular debido a la flexibilidad que ofrece en la distribución de espacios y a la reducida altura de canto, se utiliza en construc-ciones donde se producen grandes aglomeraciones, y por ello, resulta necesario estudiar su respuesta estruc-tural en detalle. Las grandes deformaciones a las que se ven sometidos los forjados planos de hormigón a consecuencia de la supresión repentina de una columna, hacen que el análisis estructural sea altamente com-plejo. Por ello, resulta necesario el uso de modelos simplificados.

El presente proyecto de investigación se centra en la respuesta estructural de forjados planos de hor-migón armado tras la supresión repentina de una columna, incluyendo un procedimiento para evaluar y cuantificar la robustez de este tipo de estructuras. Este procedimiento se ocupa, por un lado, de los proble-mas indirectos correspondientes a la idealización estructural y sus condiciones de contorno, las tipologías críticas de colapso y la extensión de los daños tras el suceso. Por otro lado, las cuestiones derivadas del análisis estructural correspondientes al vano directamente afectado por la columna eliminada, tales como las no linealidades en las propiedades de los materiales y en la geometría del sistema así como los efectos diná-micos, han sido investigadas en detalles. Además, este proyecto de investigación incluye un nuevo factor para la evaluación de la robustez estructural de forjados planos.

Con el fin de simplificar estos efectos dinámicos, se ha empleado un método basado en el equilibrio de energías. Este método permite calcular la respuesta dinámica a través de la respuesta incremental estática del sistema. Las diferentes simplificaciones y suposiciones que de forma inherente incluye este método, llevan a la predicción de resultados aproximados, pero suficientemente exactos, de la máxima respuesta dinámica. A fin de evaluar la robustez y exactitud de este método, se incluyen un estudio de parámetros de distintas variables y una comparación de las predicciones del método con resultados experimentales. Estos análisis demuestran que este método, para configuraciones estructurales típicas, siempre predice estimacio-nes que se encuentran moderadamente del lado de la seguridad. Sin embargo, para poder validar completa-mente la exactitud de este método, se requiere un mayor número de resultados experimentales.

El análisis estructural correspondiente a la respuesta incremental estática debe tener en cuenta los efec-tos de las no linealidades materiales y geométricas. Consecuentemente, un nuevo método ha sido desarro-llado para estimar la respuesta incremental estática de losas de hormigón armado sin restricciones laterales considerando los efectos membrana. Este método se basa en la cinemática de un modelo de losa ideal rígido-plástico y en el equilibrio de los esfuerzos originados por las correspondientes deformaciones. Diversos criterios de rotura fueron implementados con el fin de realizar una estimación realista de las capacidades de carga en rotura de la losa y sus correspondientes flechas. Las predicciones de este nuevo método han sido comparadas con los resultados experimentales de un conjunto de 45 muestras de diferentes campañas, in-cluyendo las predicciones de otros métodos ya existentes. Los resultados de esta comparación sugieren que el nuevo método predice, en general, estimaciones más exactas, precisas, y por lo tanto, más fiables que los otros métodos considerados. Los errores y desviaciones derivados de los resultados de este método son

Resumen

xxvi

pequeños y podría proponerse como una alternativa real a otros complejos y tediosos métodos numéricos, sin por ello perder demasiada exactitud. Adicionalmente, esta investigación sugiere que, para la mayoría de las losas estudiadas, la disminución de la capacidad de carga debido a los efectos dinámicos se contrarresta prácticamente con el aumento de la capacidad de carga debido al efecto membrana. Resultaría, por consi-guiente, posible ignorar estos efectos en futuros análisis sin por ello dar lugar a errores excesivos.

En conclusión, el procedimiento global propuesto permite la realización de evaluaciones exactas, rápi-das, homogéneas y simples de la robustez estructural de forjados planos de hormigón armado sometidos a la supresión repentina de una columna. Este procedimiento puede seguir desarrollándose fácilmente con el fin de incorporar otras tipologías estructurales, condiciones de contorno, efectos dinámicos, tipologías crí-ticas de colapso y materiales estructurales.

“Essentially, all models are wrong, but some are useful”

George E. P. Box

1

1. INTRODUCTION

1.1. Context of the research topic Structural engineers are meant to design cost-effective constructions and provide them with sufficient load-bearing capacity to ensure their operability during their life span. These constructions should be able to resist the most likely and typical loads without interrupting their normal usage, but also extraordinary loads without collapsing. Modern structures are designed following semi-probabilistic procedures, which by means of partial safety factors suggested by design codes, provide the constructions with additional bearing capacity and redundancy that might make them resist the effects of abnormal and/or unforeseen loads or events.

Despite all these foresights in design, catastrophic events and calamities have occurred in modern-designed structures. Partial and global collapse cases like the apartment building at Ronan Point in London in 1968, the Alfred P. Murrah Federal Building in Oklahoma City in 1995 and the World Trade Center in New York City in 2001, among others, have underlined the necessity to increase the requirements in design for accidental situations. Most of these events were initiated by a local damage in the structure, which further spread throughout the construction leading to additional failures, following a mechanism typically referred as progressive collapse (see Figure 1). The recommendations from design codes encouraging engineers to design structures that should maintain integrity even if a structural element is damaged or destroyed were, apparently, ineffective in these cases.

Accurate consideration of progressive collapse scenarios for each particular structure is often very complex, due to the chaotic nature of these scenarios and the large number of possible, but very unlikely threats that a building can be subjected to, that might lead to the failure of a single structural element. These events could be human-oriented threats such as human errors, vehicles impacts, large explosions, bombings and abnormal fires or extraordinary natural hazards. These events are usually beyond regular design condi-tions and therefore require more advanced analyses and modelling procedures.

Figure 1. Progressive collapse of a high-rise building. Adapted from [38].

Chapter 1. Introduction

2

1.2. Problem statement

During the last decades, multiple catastrophic failures of structures initiated by a large variety of localized damages followed by progressive collapse have occurred. Adequate design methods are therefore required to prevent such events. These requirements should not only concern extraordinary governmental or key structures, but also regular buildings with a large associated risk in case of a local failure. Medium-high reinforced concrete flat slab structures lie within this last category. Flat slabs are a very popular form due to their design flexibility and low storey heights and are used all over the world for both residential and office buildings, where large agglomerations of human beings occur and a collapse would lead to a significant number of casualties.

Extensive research has been performed in this topic, leading to different strategies to enhance the performance of reinforced concrete flat slabs with respect to progressive collapse. One of the most effective design methods for these structures is to ensure their redundancy, so alternative load paths are available that can redistribute the existing loads in case of a key structural element removal. In this direction, a sudden column removal is the most accepted hazard scenario in buildings and is included in several widely used design standards and guidelines. In order to ensure the progressive collapse resistance of a building, these standards encourage designing the structure so that the extent of damage after the sudden column removal is smaller than prescribed limits.

Designing and analysing flat slab structures subjected to a sudden column removal scenario and esti-mating the associated extend of damage is a challenging task. Several problems arise when performing this analysis, which still do not have a clear solution, and they can be classified as indirect and direct. Indirect problems are the ones arising once the bay directly affected by the column removal have collapsed. Such problems are the structural idealisation degree at which the building should be considered for the analysis and the corresponding boundary conditions, how to evaluate the extent of damage, how to consider the propagation of damage horizontally and vertically within the building, how to consider the effects of falling debris, etc. Direct problems are the ones arising from the analysis of the bay directly affected by the sudden column removal, such as the complex structural analysis including dynamic loading, geometrical and material nonlinearities as well as defining adequate and realistic failure and/or collapse criteria.

1.3. Motivation and objectives

Providing a progressive collapse resisting design for structures is becoming more relevant with every passing day. Higher, larger, more complex and vulnerable structures are being projected around the world and the threats, especially terrorism, are always growing and so is their associated risk. Securing these constructions from various hazards is becoming an essential step during the conceptual design process, and larger efforts are being done to make them structurally “robust”. New versions of several renowned, widely used stand-ards are currently under development and will probably increase the requirements and provisions for the robustness of collapse resisting structures.

Hence, structural robustness is currently a trending topic with huge research possibilities and uncount-able questions to be answered. Even though a large amount of research has already been conducted on structural robustness, this topic is so wide and involves so many strategies, mechanisms and parameters that it can be said that research is still in an early stage, and this topic therefore requires further investigation.

In order to efficiently investigate a specific topic, it is essential to work with the better possible tools. In structural engineering, these tools are the structural models, procedures and assumptions used to simplify often too complex calculations without losing significant accuracy of the results. These models allow isolat-ing the most relevant parameters that characterise the structural behaviour and performing faster and sim-pler calculations. This optimization is essential in research, in order to conduct parameter studies and better understand the occurrence of different phenomena.

The relevance of providing a sufficient progressive collapse resistance to reinforced concrete flat slab structures by designing them for a sudden column removal scenario has been previously justified, and the different problems arising during this process have been highlighted. Hence, it becomes clear that, in order to effectively conduct research on this topic, it is necessary to develop a proper “tool”, in other words, a proper model or procedure to easily calculate the progressive collapse resistance or robustness of flat slab structures.

1.4. Scope and limitations

3

Hence, the main aim of this research project is, globally, to investigate the behaviour of reinforced concrete flat slab structures subjected to a sudden column removal scenario; and particularly, to develop a straightforward procedure to evaluate and quantify the robustness of such structures subjected to this hazard scenario. The influence of various simplifications and different parameters as well as the dynamic effects and nonlinear behaviour were also examined in this investigation.

To achieve the aforementioned aim, the following objectives were fulfilled:

Investigation of the level of structural idealization including the correspondent boundary conditions that a flat slab structure subjected to a sudden column removal can be analysed without losing significant accuracy.

Proposal for a simplified procedure to deal with the estimation of the extent of damage in flat slab structures.

Comprehensive analysis of an existing approach to simplify the calculation of the dynamic compo-nent of the sudden column removal problem. A parameter study to investigate the influence in the accuracy of the approach with different assumptions and a comparison of numerical predictions with experimental results were conducted.

Development of a new simplified design approach to estimate the pushover nonlinear static re-sponse of laterally unrestrained reinforced concrete slabs considering membrane action. Compari-sons of the numerical predictions with the ones obtained with other approaches as well as with experimental results were performed.

Proposal of a new structural robustness indicator for reinforced concrete flat slab structures.

1.4. Scope and limitations The scope of the current research project is to perform a fundamental analysis of the robustness of flat slab structures subjected to a sudden column removal scenario and to propose a simplified procedure for their assessment.

Within the scope of a PhD thesis, it appears not possible to cover the complete topic of robustness of structures in all aspects. This work focuses on the robustness of building structures, other kind of construc-tions such as bridges are out of the scope of this investigation. Specifically, this research project investigates the behaviour of flat slab structures, ignoring other building configurations.

This research project is conceived within the framework of the Alternative Load Path (ALP) design method, which considers that a primary bearing element of the structure must be notionally removed. This project considers a notional sudden column removal as the most critical hazard scenario in buildings, re-gardless the event that may cause this damage. Effects of the removal of other structural elements such as walls or beams are not covered in this investigation.

The proposed simplified approach introduces the dynamic component of the sudden column removal in the analysis as a sudden application of the existing gravitational loads prior to the removal. Besides, this approach ignores the effects of damping and the strain-rate effects.

The current investigation reduces the calculation of a complete flat slab building subjected to a sudden column removal to the analysis of the bay directly affected by the column removal, assuming simplified boundary conditions. The proposed design approach for estimating the nonlinear static response of slabs is only valid for laterally unrestrained rectangular reinforced concrete slabs with orthotropic reinforcement in one or two layers subjected to a uniformly distributed load. The slabs might be simply supported or rota-tionally restrained, but must be vertically supported along all the edges.

All the experimental results used for the evaluation of the proposed simplified procedures were ex-tracted and processed from several experimental campaigns performed by different researchers; the author did not conduct or participate in any of them.

1.5. Outline of the thesis This thesis comprises eight chapters and includes the proposal for a simplified global procedure for the robustness assessment of flat slab structures. The approaches included in this procedure are described, and their accuracy is evaluated by comparing the obtained output with experimental results.

Chapter 1. Introduction

4

After the general introduction in the current Chapter 1, Chapter 2, “Robustness of building structures”, provides a full description of the progressive collapse and structural robustness issue. Key terms used throughout this thesis are defined and a review of historical case studies is included. The different existing strategies and design methods for enhancing structural robustness are presented, together with an introduc-tion on how structural robustness is implemented in the most relevant widely used design codes. An insight of the available factors to quantify the robustness as well as a description of the ALP method and the sudden column failure hazard scenario are also included. Finally, the relevance of the reserve of strength provided by membrane action in slabs is briefly introduced in this chapter.

In Chapter 3, “Robustness assessment procedure of flat slab structures subjected to a sudden column failure”, the proposed simplified procedure to evaluate and quantify the robustness of flat slab structures subjected to a sudden column failure scenario is introduced. In this chapter, the aspects of the procedure relative to the structural level idealization, the corresponding boundary conditions and the extent of damage are described in detail. In addition, the different approaches available to estimate the dynamic response of the structure are presented and the Energy Balance Method (EBM) is briefly introduced. Similarly, the pre-viously developed approaches to estimate the nonlinear static response of reinforced concrete slabs at large deflections are briefly described. Finally, a new robustness indicator for flat slab structures based on the presented global procedure is proposed.

Chapter 4, “Energy Balance Method”, contains a comprehensive description of the proposed simpli-fied approach to estimate the dynamic response of a structure subjected to a sudden column removal. A physical and graphical interpretation of this method are provided. In addition, the results of an extensive parameter study are included, in order to estimate the accuracy of the method and evaluate the influence of its several implicit simplifications of the EBM. At the end of this chapter, the approach is applied to an experimental test specimen and the output of this analysis is compared to the experimental results. A short discussion of this comparison is included.

In Chapter 5, “Novel design approach for the static analysis of laterally unrestrained slabs considering membrane action”, the proposed simplified approach to estimate the nonlinear static response of laterally unrestrained reinforced concrete slabs, based on kinematics and equilibrium, is described in detail. A com-prehensive explanation of the complete approach is included in this chapter, so the derivation process can be easily followed and the calculations, assumptions and simplifications can be clearly interpreted. Realistic failure criteria are proposed and a justification for their suitability is provided. In addition, specific guidance is given on how to implement the approach in a computer in order to easily perform straightforward and quick simulations.

The Chapter 6, “Overview and evaluation of previous approaches for the analysis of laterally unre-strained slabs”, contains a detailed description of the most relevant previously developed approaches for the analysis of laterally unrestrained slabs. Each method is presented, exposing its most significant features and characteristics and highlighting the differences between the considered approaches. Subsequently, the methods are all compared to each other for different slab configurations and the results are discussed. Finally, the previous approaches are compared with the new design approach described in Chapter 5 for different slab configurations. A discussion of these results is also provided.

In Chapter 7, “Comparative assessment of the design approaches with experimental tests”, the results predicted by the new design approach and by some of the approaches presented in Chapter 6 (the ones including failure criteria) are compared to the experimental results of different campaigns. In total, results of 45 simply supported laterally unrestrained slab tests from different experimental campaigns were used for the comparative assessment. This chapter includes a detailed description of the considered experimental campaigns including all the specific characteristics of the specimens, test setup, procedures, etc. Both the experimental results and the numerical predictions obtained with the different approaches are included. At the end, a comparative assessment of these results is provided, including a graphical representation and a statistical analysis. A discussion of this comparison is included.

Chapter 8, “Conclusions and further research”, summarises the conclusions drawn from each chapter, highlighting the results from Chapter 4, 6 and 7. The features, advantages and disadvantages of the proposed simplified procedure are formulated and discussed critically. An outlook regarding possible future applica-tions of this procedure and recommendations for further research on the robustness of flat slabs structures are provided.

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2. ROBUSTNESS OF BUILDING STRUCTURES

2.1. Introduction During the last five decades, multiple progressive collapse failures all over the world have underlined the necessity of assessing and increasing the structural robustness of buildings subjected to unforeseen actions and unexpected events. In this direction, several design codes and guidelines have slowly included require-ments and strategies in order to enhance the robustness of structures. Simultaneously, efforts on the devel-opment of approaches to quantify structural robustness have been made. As previously explained, this re-search project focuses on the robustness of buildings, disregarding other relevant structures such as bridges.

This chapter provides details of the most relevant progressive collapse failures that have influenced the investigations on structural robustness. An overview of the most relevant standards, requirements and strat-egies is included. Some of the most broadly-accepted procedures to quantify structural robustness are pre-sented. In addition, the Alternative Load Paths (ALP) strategy, together with sudden column failure scenar-ios are described in detail. Finally, the effects of membrane action are introduced, highlighting the relevance of the reserve of strength that this mechanism provides to the system.

2.2. Definition of key terms Throughout this document, different key terms are used quite often. In order to avoid ambiguity, the fol-lowing definitions are proposed for these terms.

2.2.1. Structural robustness Robustness is variously defined depending on the application field, but is referred as an inherent property of a system to exhibit a satisfactory response or performance when subjected to unexpected or extreme events.

For the structural engineering field, this general definition can be particularized. Structural robustness is broadly defined as the ability of a structural system to withstand local failure without suffering damage disproportionate to the event that caused it and being able to fulfil most of its original funct