robustness of steel framed buildings with pre-cast
TRANSCRIPT
THE UNIVERSITY OF MANCHESTER
ROBUSTNESS OF STEEL FRAMED
BUILDINGS WITH PRE-CAST
CONCRETE FLOOR SLABS
A thesis submitted to the University of Manchester for the degree of
Doctor of Philosophy
in the Faculty of Engineering and Physical Sciences
2014
Seyed Mansoor Miratashiyazdi
School of Mechanical, Aerospace and Civil Engineering
2
Table of Contents
List of Figures .............................................................................................................. 7
List of Tables.............................................................................................................. 11
Abstract ...................................................................................................................... 12
Declaration ................................................................................................................. 13
Copyright ................................................................................................................... 14
Acknowledgements .................................................................................................... 15
Chapter 1. Research Background........................................................................... 16
1.1 Structural Robustness .................................................................................. 16
1.2 Research Originality .................................................................................... 18
1.3 Research Objectives .................................................................................... 19
1.4 Research Methodology ................................................................................ 20
1.5 Thesis Structure ........................................................................................... 20
Chapter 2. Literature Review ................................................................................. 22
2.1 Major Accidents of Progressive Collapse ................................................... 22
2.1.1 Ronan Point Building ........................................................................... 22
2.1.2 Alfred P. Murrah Federal Building ...................................................... 26
2.2 Building Codes and Regulations on Robustness of Structures ................... 28
2.2.1 Implementation of the robustness requirements .................................. 32
2.2.1.1 Application of the Tying Force Method to Precast Reinforced
Concrete Structures ......................................................................................... 35
2.3 Research on Structural Tying System of PCFS ........................................... 36
2.3.1 Bending Tests on Tie Connections ...................................................... 37
2.3.2 Formulation of Tie Behaviour in Bending ........................................... 39
2.4 Construction Technology of PCFS .............................................................. 42
2.4.1 Design of PCFS .................................................................................... 43
2.4.1.1 Flexural capacity .............................................................................. 43
2.4.1.1.1 Serviceability limit state (SLS) ................................................... 43
3
2.4.1.1.2 Ultimate limit state of flexure ..................................................... 44
2.4.1.2 Shear capacity of PCFS .................................................................... 45
2.4.2 PCFS Manufacturing ............................................................................ 46
2.5 Providing Tying Resistance in Precast Concrete Floor Slabs ..................... 49
2.5.1 Precast Concrete Floor Slabs ............................................................... 49
2.5.1.1 Hollowcore floor units ...................................................................... 49
2.5.1.2 Solid precast floor units .................................................................... 50
2.5.2 Connections of precast concrete floor slabs ......................................... 50
2.6 Bond-Slip ..................................................................................................... 51
2.7 Summary and Objectives of Research ......................................................... 55
Chapter 3. Validation of Numerical Modelling ..................................................... 57
3.1 Choosing DIANA FE Package .................................................................... 57
3.2 Modelling concrete in DIANA .................................................................... 58
3.2.1 Compressive Behaviour ....................................................................... 59
3.2.1.1 Tresca ............................................................................................... 59
3.2.1.2 Von-Mises ........................................................................................ 59
3.2.1.3 Mohr-Coulomb ................................................................................. 60
3.2.1.4 Drucker-Prager ................................................................................. 60
3.2.2 Tensile Behaviour ................................................................................ 61
3.2.2.1 Brittle Cracking ................................................................................ 61
3.2.2.2 Linear Tension Softening ................................................................. 62
3.2.2.3 Moelands Tension Softening ............................................................ 63
3.2.2.4 Hordijk Tension Softening ............................................................... 63
3.2.3 Bond-Slip ............................................................................................. 65
3.2.4 Concrete Elastic Material Properties .................................................... 66
3.2.4.1 Tensile Strength ................................................................................ 66
3.2.4.2 Elastic Modulus ................................................................................ 67
3.3 Axially Restrained Beams (Su et al., 2009) ................................................ 67
3.3.1 Test Subassemblies & Setup ................................................................ 68
3.3.2 Typical Beam Behaviour to Reach Catenary Action ........................... 71
4
3.3.3 Finite Element Model ........................................................................... 72
3.3.4 Comparison between Simulation and Test Results .............................. 74
3.4 Verification of Bond-Slip Modelling .......................................................... 78
3.4.1 Test Specimens and Variables ............................................................. 78
3.4.2 Finite Element Model ........................................................................... 80
3.4.3 Comparison between Simulation and Test Results .............................. 82
3.5 Summary ..................................................................................................... 85
Chapter 4. Parametric study of 2D Restrained Slab .............................................. 86
4.1 Illustrative behaviour ................................................................................... 87
4.1.1 Slab with total axial restraint................................................................ 87
4.1.2 Slabs with Elastic Axial Restraint ........................................................ 94
4.1.3 Slabs with Partial Tie Bar .................................................................... 98
4.2 Accidental Load Calculation ..................................................................... 101
4.3 Parametric Study ....................................................................................... 102
4.3.1 Height of the Slab............................................................................... 103
4.3.2 Slab Span ............................................................................................ 107
4.3.3 Tie Bar Length ................................................................................... 107
4.3.4 Tie Bar Height .................................................................................... 110
4.3.5 Tie Bar Diameter ................................................................................ 112
4.3.6 Tie Bar Yield Stress ........................................................................... 114
4.3.7 Grouting Concrete Strength ............................................................... 115
4.3.8 Ultimate Strain of the Steel Tie Bar ................................................... 116
4.3.9 Summary ............................................................................................ 117
Chapter 5. 2D Slab Analytical Load-Displacement Relationship ....................... 119
5.1 Development of the Analytical Relationship ............................................ 119
5.1.1 Axially Restrained Slabs .................................................................... 119
5.1.2 Elastic Axially Restrained Slabs ........................................................ 121
5
5.2 Maximum Slab Displacement ................................................................... 123
5.2.1 Slab Height ......................................................................................... 126
5.2.2 Tie Bar Diameter ................................................................................ 127
5.2.3 Tie Bar Position ................................................................................. 128
5.2.4 Ultimate Strain of the Steel Tie Bar ................................................... 130
5.3 Validation of the Analytical Prediction for the Maximum Catenary Action
Resistance ............................................................................................................. 131
5.3.1 Comparison of FE and Analytical Results ......................................... 133
5.4 Summary ................................................................................................... 135
Chapter 6. Three-dimensional behaviour of PCFS with column removal ........... 137
6.1 Finite Element Model ................................................................................ 138
6.2 Effects of Loss of an Edge Column ........................................................... 140
6.2.1 Slab Height ......................................................................................... 141
6.2.2 Tie Bar Diameter ................................................................................ 146
6.2.3 Tie Bar Ultimate Tensile Strain ......................................................... 148
6.2.4 Transverse Tie Bars............................................................................ 149
6.2.5 Procedure of Improving Robustness of Precast Floor Systems ......... 153
6.3 Loss of a Centre Column ........................................................................... 153
6.3.1 Slab Height ......................................................................................... 154
6.3.2 Tie Bar Diameter ................................................................................ 157
6.3.3 Tie Bar Ultimate Strain ...................................................................... 159
6.3.4 Transverse Tie Bars............................................................................ 159
6.4 Loss of Corner Column ............................................................................. 160
6.5 Conclusions ............................................................................................... 161
Chapter 7. Conclusion and Further Studies ......................................................... 163
7.1 Literature Review ...................................................................................... 163
7.2 Finite Element Model & Validation .......................................................... 164
6
7.3 Two-Dimensional Analysis of Slabs ......................................................... 165
7.4 Predictive Analytical Relationship of the 2D Model ................................ 166
7.5 Three-Dimensional Simulation of the Floor System ................................. 167
7.6 Limitations of the Current Study ............................................................... 168
7.7 Future Studies ............................................................................................ 168
Bibliography ............................................................................................................. 170
Appendix 1: Assessment of tie bars designed according to British Standard
regulations ................................................................................................................ 176
Appendix 2: Evaluation of Analytical Relation ....................................................... 186
7
List of Figures
Figure 1-1: Effects of Losing an External Column to a Blast Loading: a) alternate
load path design: no progressive collapse; b) conventional design: progressive
collapse (Baldridge and Humay, 2003) ...................................................................... 17
Figure 1-2: Precast Floor Unit Connection Layout (FIB-Bulletin-43, 2008) ............ 18
Figure 2-1, Ronan Point Building after collapse (MacLeod, 2005) ........................... 24
Figure 2-2: Estimation of tie force in catenary action................................................ 25
Figure 2-3, Alfred P. Murrah Building, before the attack (Suni, 2005) ..................... 26
Figure 2-4, Alfred P. Murrah Building, after the attack (Chernoff, 2009) ................ 27
Figure 2-5: Floor ties in a concrete structure (Brooker, 2008) .................................. 33
Figure 2-6: Area of the structure susceptible to collapse (Approved Document A) .. 34
Figure 2-7: Vertical and Horizontal tying in a structure (NIST, 2007) ..................... 35
Figure 2-8: Lifting tests on the connection between PCFSs (Engström, 1992) ......... 39
Figure 2-9: Pure suspension mode of action in a precast floor after loss of an interior
column (Engström, 1992)........................................................................................... 40
Figure 2-10: Resistance mechanism of tie connection between PCFS in the case of
lost column ................................................................................................................. 42
Figure 2-11: Installed Casting Beds (Spiroll, 2014) .................................................. 46
Figure 2-12: Extruder Components (Elematic, 2014) ................................................ 47
Figure 2-13: Different types of cut on PCFS (Spiroll, 2014) ..................................... 48
Figure 2-14: Slab lifting from its side grooves (Ultra-Span, 2012) ........................... 48
Figure 2-15: Hollowcore unit profile on steel structure (Hanson, 2014) ................... 49
Figure 2-16: Placement of tie bar in hollowcores (CCIP-030) .................................. 50
Figure 2-17: Placement of tie bar in between units (CCIP-030) ................................ 51
Figure 2-18: Generic bond-slip relationships (CEB-FIP 2010) ................................. 53
Figure 3-1: Tresca and Mohr-Coulomb yield criteria (TNO-DIANA, 2010) ............ 59
Figure 3-2: Mohr-Coulomb and Drucker-Prager yield criteria (TNO-DIANA, 2010)
.................................................................................................................................... 61
Figure 3-3: Brittle Cracking Behaviour (TNO-DIANA, 2010) ................................. 61
Figure 3-4: Linear tension softening (TNO-DIANA, 2010) ...................................... 62
Figure 3-5: Moelands Tension Softening (TNO-DIANA, 2010)............................... 63
8
Figure 3-6: Hordijk tension softening (TNO-DIANA, 2010) .................................... 64
Figure 3-7: Comparison of tension softening models ................................................ 64
Figure 3-8: Cubic function of Dörr model (TNO-DIANA, 2010) ............................. 65
Figure 3-9: Power Law of Noakowski (TNO-DIANA, 2010) ................................... 65
Figure 3-10: Test Subassembly and Reinforcement Layout (Su et al 2009) ............. 68
Figure 3-11: Test Specimen and Schematic Illustration of Test Setup (Su et al, 2009)
.................................................................................................................................... 70
Figure 3-12: Behaviour of Stages of model A3 ......................................................... 72
Figure 3-13: Results of mesh sensitivity study .......................................................... 73
Figure 3-14: Finite element model ............................................................................. 74
Figure 3-15: Comparison of experiment and FE results for A-series ........................ 75
Figure 3-16: Comparison of experiment and FE results B-series .............................. 77
Figure 3-17: Test specimens of type I and II (Engström et al., 1998) ....................... 79
Figure 3-18: Concrete block FE model for simulation of bond-slip tests .................. 81
Figure 3-19: Comparison of experiments with FE results ......................................... 83
Figure 3-20: Comparison of experiment and FE results, Type II .............................. 84
Figure 3-21: Bond-Slip specimens deformed mesh, Types I and II .......................... 84
Figure 4-1: Two-dimensional representation of the slabs with pinned BC ............... 86
Figure 4-2: Mesh view of half of the model (left slab) .............................................. 88
Figure 4-3: FE Results for the 2D slab Model ........................................................... 90
Figure 4-4: Axial tie bar stress in the connection ...................................................... 92
Figure 4-5: Axial Stress and Crack Pattern of Slabs' Connection .............................. 93
Figure 4-6: Schematic 2D Slabs with Elastic Axial Restraints .................................. 94
Figure 4-7: Columns acting as fixed beams ............................................................... 95
Figure 4-8: Calculation of stiffness for a fixed beam under a point load .................. 95
Figure 4-9: Effect of Horizontal Displacement on Vertical Deflection ..................... 96
Figure 4-10: Behaviour of slabs with elastic axial restraint ....................................... 97
Figure 4-11: Partial Tie Bar, development of Catenary Action ................................. 98
Figure 4-12: Crack Pattern, Partial Tie Bar ............................................................... 99
Figure 4-13: Crack Pattern, Partial Tie Bar, showing concrete cracking in the
unreinforced zone of grouting .................................................................................... 99
Figure 4-14: Partial Tie Bar, large tie bar size resulting in total crack of concrete
grouting in the unreinforced zone ............................................................................ 100
9
Figure 4-15: Axial Stress in Unreinforced Region, a) no crack through in the
unreinforced region (small tie bar); b) Crack through in the unreinforced region
(large tie bar) ............................................................................................................ 100
Figure 4-16: Slab's reference case dimensions ........................................................ 103
Figure 4-17: Hollowcore Sections (Bison, 2012) .................................................... 104
Figure 4-18: Effects of varying Slab Height on Vertical Load and Axial Reaction
Force ......................................................................................................................... 105
Figure 4-19: Force diagram ...................................................................................... 106
Figure 4-20: Variation of slab span affecting the connection response ................... 107
Figure 4-21: Variation of Tie Bar Length on the development of catenary action .. 109
Figure 4-22: Placing the Tie Bar between Hollowcore Units (CCIP-030) .............. 110
Figure 4-23: Effects of tie bar height (measured from bottom of slab) on catenary
action development .................................................................................................. 112
Figure 4-24: Effects of changing tie bar diameter ................................................... 113
Figure 4-25: Effects of changing tie bar yield stress................................................ 114
Figure 4-26: Effect of concrete strength on load-displacement of slabs' connection
(stresses in MPa) ...................................................................................................... 115
Figure 4-27: Effects of changing Tie Bar Ultimate Strain ....................................... 116
Figure 5-1: Free body diagram of one slab .............................................................. 120
Figure 5-2: Comparison of FE model with analytical relationship, slab height 265
mm, width 1200 mm, span 5 m, tie bar height 45mm, diameter 16 mm, steel yield
stress 500 MPa ......................................................................................................... 121
Figure 5-3: Slab deflection with axial displacement ................................................ 122
Figure 5-4: Comparison of the FE model with elastic BC with the analytical
relationship, slab span: 7 m ...................................................................................... 123
Figure 5-5: Strain distribution at the connection between PCFS ............................. 124
Figure 5-6: Strain distribution along the tie bar in the FE model ............................ 124
Figure 5-7: Variation of tie bar strain distribution with slab height ........................ 126
Figure 5-8: Effect of tie bar diameter on strain distribution length ......................... 128
Figure 5-9: Variation of strain distribution with tie bar position ............................. 129
Figure 5-10: Effect of tie bar ultimate strain on connection behaviour ................... 130
Figure 5-11: Comparison of FE base case with the analytical calculation .............. 132
Figure 6-1: Plan of a floor system ............................................................................ 137
Figure 6-2: Modelled slab resting on steel structure ................................................ 138
10
Figure 6-3: Floor arrangement and three dimensional finite element model
representation ........................................................................................................... 139
Figure 6-4: Possible structural behaviour after loss of an edge column, (Baldridge
and Humay, 2003) .................................................................................................... 141
Figure 6-5: Slab Height effect on the connection behaviour ................................... 142
Figure 6-6: Plate structure with 3 fixed and 1 free edges (Moody, 1990) ............... 144
Figure 6-7: Effect of tie bar diameter on floor behaviour ........................................ 147
Figure 6-8: Effect of changing tie bar ultimate tensile strain................................... 149
Figure 6-9: Arrangement of transverse ties .............................................................. 150
Figure 6-10: Effects of introducing transverse tie bars ............................................ 151
Figure 6-11: Strain in main longitudinal tie (b) and two of the transverse ties (c) .. 152
Figure 6-12: Loss of centre column in a floor system ............................................. 154
Figure 6-13: effect of slab height when central column is lost ................................ 155
Figure 6-14: Plate structure with point load at centre (Timoshenko, 1959) ............ 156
Figure 6-15: Effect of tie bar diameter when central column is lost ........................ 158
Figure 6-16: effect of tie bar strain when central column is lost ............................. 159
Figure 6-17: Transverse tie bars in loss of a centre column .................................... 159
Figure 6-18: Deformed shape of the model with loss of a corner column ............... 160
Figure 6-19: Load-displacement and tie bar strain diagram under corner column loss
.................................................................................................................................. 161
Figure A1-1: comparison between simulation load-deflection curves and accidental
load, with BS tie bar ................................................................................................. 180
Figure A1-2:Comparisonofsimulations’load-deflection curve with accidental load,
with recommended tie bar ........................................................................................ 184
11
List of Tables
Table 2-1: Building classes (Approved Document A) ............................................... 30
Table 2-2: Differences in building classification in BS and EN ................................ 32
Table 2-3: Load/Span Table (Bison, 2012) ................................................................ 46
Table 2-4: Type I (tie bar pull-out) bond-slip relationship parameters (CEB-FIP
2010) .......................................................................................................................... 53
Table 2-5: Parameters for Type II (tie bar yield) bond-slip relationship ................... 54
Table 3-1: Crack band width calculation (TNO-DIANA, 2010) ............................... 62
Table 3-2: Specimen Properties ................................................................................. 69
Table 3-3: FE models divisions along 3D axis for mesh sensitivity study ................ 73
Table 3-4: Characteristics of the test specimen (Engström et al., 1998) .................... 80
Table 3-5: Calculated concrete material properties (based on CEB-FIP Model Code
2010) .......................................................................................................................... 80
Table 4-1: Material Properties used in FE Model ...................................................... 88
Table 5-1: Calculation of strain distribution for the base case................................. 131
Table 5-2: Comparison of FE and analytical results, Tie Bar height ....................... 133
Table 5-3: Comparison of FE and analytical results, Tie Bar Ult. Tensile Strain ... 134
Table 5-4: Comparison of FE and analytical results, Slab Height ........................... 134
Table 5-5: Comparison of FE and analytical results, Tie Bar Diameter .................. 135
Table 6-1: Material property of concrete and steel used in the FE model ............... 140
Table 6-2: Accidental load based on the ultimate bending resistance of the slabs .. 145
Table 6-3: Position (from bottom of the section) and lengths of the transverse tie bars
.................................................................................................................................. 151
Table 6-4: Equivalent of accidental point load on the connection ........................... 157
Table A1-1: Simulations based on Bison (2012) load-span table ............................ 177
Table A1-2: Models with recommended tie diameters ............................................ 181
Table A1-3: Comparison of tie forces of the BS and recommended ties ………....185
Table A2-1: Models with tie bar of diameter 10 mm .............................................. 187
Table A2-2: Models with tie bar of diameter 20 mm .............................................. 188
12
University of Manchester
Seyed Mansoor Miratashiyazdi
Doctor of Philosophy
Robustness of steel framed buildings with precast concrete floor slabs
2014
Abstract
Following some incidents in high-rise buildings, such as Ronan Point London 1968,
in which collapse of a limited number of structural elements progressed to a failure
disproportionate to the initial cause, consideration of robustness was introduced in
British Standard. The main method of preventing progressive collapse for providing
robustness to steel framed buildings with precast concrete floor slabs focuses on the
allowable tying forces that the reinforcement in between the slabs and in hollowcores
should carry. However there are uncertainties about the basis of the practical rules
associated with this method. This thesis presents the results of numerical and
analytical studies of tie connection behaviour between precast concrete floor slabs
(PCFS). It is shown that under current design regulations the tie connection is not
able to resist the accidental load limit applied on the damaged floor slabs.
By establishing the capability of a finite element model to depict and predict the
behaviour of concrete members in situations such as arching and catenary action
against several experimental tests, an extensive set of parametric studies was
conducted in order to identify the effective parameters in enhancing the resistance of
the tie connection between PCFSs. These parameters include: tie bar diameter,
position, length, yield stress and ultimate strain; the slab’s height, length; and the
compressive strength of the grouting concrete in between the slabs that encases the
tie bar. Recommendations are made based on the findings of this parametric study in
order to increase the resistance of the tie connection. Based on the identified
effective parameters in the parametric study a predictive analytical relationship is
derived which is capable of determining the maximum vertical displacement and
load that the tie connection is able to undergo. This relationship can be used to
enable the connection to capture the accidental limit load on a damaged slab.
The identified parameters are examined in a three dimensional finite element model
to assess their effect when columns of the structure are lost in different locations
such as an edge, corner or internal column. Based on the findings of this study
methods for improving the connections performance are presented. Also the effect of
alternative transverse tying method is evaluated and it is concluded that although this
kind of tie increases the load carrying capacity of the connection, its effect on the
catenary action is not significant.
13
Declaration
No portion of the work referred to in this thesis has been submitted in support of an
application for another degree or qualification of this or any other university or other
institute of learning.
14
Copyright
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andinTheUniversity’spolicyonPresentationofTheses.
15
Acknowledgements
I would like to express my sincere gratitude to my supervisor Professor Yong Wang.
It was with the help of his knowledge, experience, and affection of work that writing
this thesis became possible and his continuous guidance that always inspired new
ways of thinking for me. What I learnt from him in terms of problem tackling and
engineering judgement, was far more important than things about precast concrete
floor slabs.
Also I would like to thank Dr Parthasarathi Mandal, for his kindness and patience
that were of great help for me in difficult situations. It would not been possible to go
over the obstacles in this research without his aid.
And last but not the least I want to thank my father Dr Sam Miratashi, for his support
and encouragement throughout these years.
16
Chapter 1. Research Background
1.1 Structural Robustness
When subjected to any unpredicted and accidental loading, the ability of a structure
not to suffer collapse disproportionate to the original cause is called robustness.
Upon achieving robustness in a structure, accidental actions which are beyond the
engineering design values should not cause damage to the structure which is not
proportionate to the direct initial and local damage caused by the accidental actions
(Menzies, 2005). In the literature the meaning of disproportionate collapse is not
usually distinct from the meaning of progressive collapse, though the result of a
progressive collapse usually leads to a disproportionate collapse, but their concepts
are different. A progressive collapse happens when localised damage in a part of a
structure, due to weakness in joints and linking elements, leads to further damage
and progresses to other parts of the structure. Since the result of the first damage has
widened its effect in other parts of the structure (i.e. progressed) and so is not
proportionate to the initial damage, it is also a disproportionate collapse (Hai, 2009).
The concept of progressive collapse has been considered in codes and building
regulations since the partial collapse of the Ronan Point Building in East London in
1968 (Moore, 2002), where a gas explosion on the 18th
floor caused the collapse of
the whole corner of a 22 storey building. In this incident 4 people were killed. The
explosion threw a wall panel out; consequently, without support, the upper floor
structural elements fell down on the lower floors and this phenomenon made
progress throughout the height of the building. As a result, in the early 1970s the
concept of robustness was introduced to the building codes and regulations in the
UK and later on the European and American codes (Pearson and Delatte, 2005).
Figure 1-1 shows a schematic example of an accidental action and the possible
scenarios that may happen afterwards. If the connections between the structural
components are able to withstand the extra applied load (accidental load), in this case
by bridging over the lost column, the floors above the lost column retain their
integrity and the structure remains robust (Figure 1-1-a), otherwise progressive
collapse occurs (Figure 1-1-b).
17
(a) (b)
Figure 1-1: Effects of Losing an External Column to a Blast Loading: a) alternate load path
design: no progressive collapse; b) conventional design: progressive collapse (Baldridge and
Humay, 2003)
Different types of structures have different rules for supplying robustness to the
structure. Among them, due to their intrinsic discrete nature, structures constructed
using precast concrete floor slabs (PCFS) are one of the most vulnerable to
disproportionate collapse. The scope of this study will be to examine methods of
providing integrity of precast concrete floor slabs in steel framed buildings.
The highest risk of progressive collapse occurs when a supporting element in the
structure, most commonly a column, is lost due to an accidental action. In this
situation the beams that were supported by the removed column may act like a
hanging chain (catenary action), but the exact reaction of the connections of the
beams and the column above, and the connections between the precast concrete
elements that form the floor of which the column is lost, are most influential in
determining the behaviour of the remaining structural elements (Liu, 2010). To
prevent progressive collapse, the PCFSs should be connected to each other and to the
supporting frame, in such a way that the integrity and continuity of the floor remains
18
in an acceptable degree so that the first link of the progressive collapse chain does
not form.
For steel structures with precast concrete floor system, the required integrity of the
connection between precast units is provided by the steel tie bars which are placed
both in the hollowcore units and in between them in the longitudinal joints
(Figure 1-2).
Figure 1-2: Precast Floor Unit Connection Layout (FIB-Bulletin-43, 2008)
However, although the tying method is commonly used in construction as a means of
providing structural robustness, there are uncertainties about the basis of the practical
rules associated with this method. Understanding and improving the effectiveness of
these practical methods are the main aims of this research.
1.2 Research Originality
The present regulations for providing robustness to steel framed buildings with
precast concrete floor slabs focus on the allowable tying forces that the
PCFS
Longitudinal Tie
Transverse Tie
Tie in the Hollowcore
19
reinforcement in between the slabs and in hollowcores should carry. This
information is based on investigations on collapsed buildings such as Ronan Point,
London (see section 2.1.1 for further details). The derivation of tying force
relationship was based on equilibrium of catenary action force in the beam/slab and
the applied load and assuming a vertical displacement of
(L being the beam/slab
span for a span of 5m) and typical loading condition. This derivation did not
consider whether the structure has sufficient deformation capacity. Neither did it
consider whether the surrounding structure would be able to provide the necessary
axial restraint that is required to activate catenary action.
In the lack of research on performance of ties connecting the PCFS, the present study
attempts to develop a thorough understanding of the response of steel framed
structures with precast floor slabs on column removal and the fundamental
mechanisms of catenary action in precast floor slabs. Through such a study, better
methods of providing robustness to this type of structure will be recommended.
1.3 Research Objectives
The objectives of this study can be summarised as:
Establish the foundation of a reliable FE model to examine the factors
affecting the behaviour of the tie connection between two Precast Concrete
Floor Slabs (PCFS) until fracture;
Identifying the influential parameters that affect PCFS tie connections
behaviour;
Developing a predictive analytical method to predict PCFS tie connection
behaviour, validated by the parametric study results;
Assessing the effectiveness of current building code regulations and practical
construction methods for providing robustness of PCFS in steel framed
buildings;
Suggesting methods for improving robustness of precast concrete floors in
steel frame buildings;
Studying the mechanisms of collapse, due to the loss of a column in different
locations, in a representative steel framed building with PCFS.
20
1.4 Research Methodology
Due to resource limits, this research will be conducted through numerical
simulations using the finite element method (FEM). In this study the commercial
FEM package TNO DIANA has been utilised owing to its powerful material models
for concrete. The numerical modelling will be validated by comparison with
available experiments on concrete structural elements that undergo arching and
catenary actions which are similar to the expected behaviour of PCFS slabs under
accidental loading.
The validated modelling method will then be used to conduct extensive parametric
studies, one set for precast concrete slabs for thorough understanding of the catenary
action mechanism, and one set for steel framed structures with precast concrete slabs
for understanding of realistic whole structural behaviour. These parametric study
results are then used to formulate an analytical predictive method which may be used
in practical design. The parametric study results are also used to assess effectiveness
of the current construction methods and to identify methods that can improve
robustness of this type of construction.
1.5 Thesis Structure
Chapter 2 covers a review of literature in the field of robustness with special focus
on precast concrete floor slabs in steel frame buildings, and explains the different
approaches in building code regulations regarding the robustness concept and
critically assesses the shortcomings of these methods, leading to justification for the
current research.
In chapter 3 tests, reported in literature, on concrete structural elements that undergo
a very similar behaviour to the interest of the present study, are simulated. Using the
validated numerical model, an extensive parametric study is carried out and the
results are reported in chapter 4. Variables examined in the parametric study include:
precast concrete slab’s height and length; connecting tie bar’s position, length,
diameter, yield stress, and ultimate strain; and grouting concrete’s compressive
strength.
21
Based on the influential parameters identified in chapter 4, a mathematical
relationship is formulated in chapter 5 to analytically predict the precast concrete
slab load-deflection behaviour under catenary action until failure which is
characterised by reinforcement fracture.
Chapters 3-5 deal with catenary action in precast concrete slab elements in the
direction of the span. In realistic structures, the slabs interact in directions
perpendicular to their span and also with the surrounding structural elements.
Chapter 6 reports the results of a numerical parametric study examining how this
type of structure behaves with the removal of a supporting column in different
locations. Comparisons will be made between structures using the existing practical
construction details with the alternative details that have been shown in chapter 4 to
provide improved robustness.
Chapter 7 summarises the results of this study and presents topics for further
investigation in this field.
22
Chapter 2. Literature Review
This chapter presents a detailed introduction to background information related to
current UK rules governing the design and construction of precast floor systems
(PCFS) to meet the requirements of structural robustness. Further details related to
technical investigations, including testing, numerical simulation and analytical
calculation methods, will be presented in relevant chapters.
First this chapter reviews major incidents that led to changes in the UK building
regulations concerning robustness of structures. This is then followed by a summary
of building rules and regulations that are intended to control disproportionate
collapse of building structures. Among these, providing sufficient tying between the
primary components of a structure is the main mean by which the regulations on
structural robustness are achieved.
This chapter will review the ties connecting PCFS and explain why the current tie
force regulations may not be effective. To help this review, this chapter will first
provide a brief review of the design and construction technology of this type of
structural units including placement of ties.
Since the most important parameter affecting behaviour of the ties between the PCFS
components is the bond between the steel reinforcement with the surrounding
concrete, this chapter provides a review of research on this phenomenon as well.
2.1 Major Accidents of Progressive Collapse
2.1.1 Ronan Point Building
All of the current rules and regulations, governing the design and construction of
buildings to control disproportionate collapse in the UK and elsewhere, can trace
their origin to the Ronan Point accident. On May 16 1968, there was a gas explosion
in an apartment on the 18th
floor of 22-storey Ronan Point building in east London,
which commenced the partial collapse of the whole corner of the structure. The
23
pressure from the explosion blew out the walls of the apartment, which were the sole
support for the walls directly above. The unsupported walls of the 19th
floor and the
three other floors above fell down to the 18th
floor, and this sudden impact loading
was much greater than its resistance. The corner of the 18th
floor collapsed and sent
debris cascading down the corner of the building, causing damage to each of the
floors below (Figure 2-1). Later, in investigations carried out on this building, it was
found that there had been many flaws in both design and construction quality (Choi
and Chang, 2009). However, it was the progressive and disproportionate manner in
which the corner of the building completely failed that caused all those concerns
(including public authorities, engineers, and academics) about the building
regulations of the time.
These investigations found that the pressure required for displacing the internal walls
in the Ronan Point building was 1.7 kPa and the pressure which could displace the
exterior walls was only 21 kPa (Griffiths, 1968). This showed the extremely poor
workmanship applied to this structure, and also the weakness of the building codes
used at that time, which were dated back to 1952. Later on it was also discovered
that the load applied by winds to this structure could cause a progressive collapse,
since the building code that the Ronan Point was designed according to, had not
considered structures with that height (Pearson and Delatte, 2005).
Continuing concerns over the structural integrity of the Ronan Point Building
eventuated in its demolition in May of 1986. In order to study the joints of this
structure carefully, Ronan Point was not demolished in the traditional fashion, it was
dismantled floor by floor. During these investigations the extensive scale of the poor
quality of the connections was evident throughout the building (Wearne, 2000).
There were some connections where it was found that the necessary force to break
them was as low as 15.6 kN (Hendry, 1979).
After this incident, and considering the fact that the Ronan Point building was
designed to comply with statutory building regulations of the time, the government
investigations concluded that the codes did not provide secure and robust structures
capable of resisting accidental actions. According to (Hendry, 1979), “new
regulations … require that under specified loading conditions a structure must
remain stable with a reduced safety factor in the event of a defined structural
24
member or portion thereof being removed. Limits of damage are laid down and if
these would be exceeded by the removal of a particular member, that member must
be designed to resist a pressure of 34 kN/m2 (5 lb/in
2) from any direction. Of special
importance in relation to load bearing wall structures is that these conditions should
be met in the event of a wall or section of a wall being removed, subject to a
maximum length of 2.25 times the storeyheight”(Hendry, 1979).
Figure 2-1: Ronan Point Building after collapse (MacLeod, 2005)
The values that appeared in the robustness regulations (tying force of 60 kN/m for
concrete structures, and the pressure of 34 kPa to be applied on the key element)
have their origins in investigations of the Ronan Point incident. The value of 34 kPa
was related to a severe gas explosion and is considered by some authors as overly
conservative (Burnett, 1975).
25
By considering the weak connections between the precast concrete members of the
Ronan Point building, it was assumed that if the structural elements were tied
together in a better fashion, the extent of the damage would have been far more
limited. The tying force resistance was calculated based on the typical loading in
concrete buildings with following assumptions:
gk and qk: permanent and variable load of 3.8 kN/m2 (Burnett, 1975 assumes 3.6
kN/m2)
L: beam span before loss of a column: 5m
Δ: Allowable deflection of the span in catenary action: L/5
Accidental loading condition: gk +
The moment equilibrium of the tie connection in the middle of the span in catenary
action would dictate (Figure 2-2):
Figure 2-2: Estimation of tie force in catenary action
Δ
Ft
Ft
gk + (qk/3)
L L
26
Equation 2-1
By substituting the various assumed values, Equation 2-1 yields the tie force (Ft) of
60 kN/m.
2.1.2 Alfred P. Murrah Federal Building
On April 19, 1995 in Oklahoma City a truck containing approximately 5000lbs of
explosives was parked near the north side of the Alfred P. Murrah building
(Figure 2-3), close to the middle point, where it was detonated. Roughly 30% of the
building was destroyed by the explosion, over 300 buildings nearby were destroyed
or damaged, and 168 lives were lost (Piotrowski and Perdue, 1995). The explosion
caused the destruction of the three columns adjacent to the blast on the ground floor,
and some other floor slabs and walls in the vicinity of the blast. But the final
destruction (Figure 2-4), being disproportionate to the initial incident, was due to a
progressive collapse (Corley, 1998).
Figure 2-3: Alfred P. Murrah Building, before the attack (Suni, 2005)
27
Figure 2-4: Alfred P. Murrah Building, after the attack (Chernoff, 2009)
The structure of the Alfred P. Murrah building was an ordinary concrete frame
designed in accordance with the ACI 318-71 code. The construction of the building
was of high quality and very well detailed. However, it was not designed to resist
any abnormal loading such as earthquake or explosion similar to many other federal
and office buildings of the time in that region (Corley, 1998).
Investigations carried out following the incident pointed out some design and
construction methods that could mitigate the effects of unpredicted accidental
loading (Corley, 1998):
a) If the building had been designed as a special moment frame or dual system
with special moment frame, among the three destroyed columns adjacent to
the blast loading, only the one which was the closest would have collapsed
due to brisance, and the other two could have survived. The presence of more
reinforcement in the concrete members of a special moment frame would
have facilitated higher energy dissipation. This shows the key role of the steel
bars in concrete members for prevention of progressive collapse which is the
subject of the present study.
b) Increased redundancy, in general, would have increased the chances of
preventing a bad situation getting worse (progressive collapse). “There
should be no single critical element whose failure would start a chain reaction
of successive failures that would take down a building. Each critical element
28
should have one or more redundant counterparts that can take over the critical
load in case the first should fail.”
c) Compartmentalized construction: this type of construction has proved to be
able to resist progressive collapse to a good level. But the inflexible and
rather small spaces that emerge by construction of this type, limits its
application to office buildings.
d) Dual systems (with special moment frame): the investigations concluded that
if the A. P. Murrah building had been designed for a seismic area, its level of
destruction could have been reduced by up to 85%. Utilizing more
reinforcement and different types of connection in the dual systems with
special moment frames would have given the building the capacity of
absorbing more energy.
Other investigations (Osteraas, 2006) have pointed out some general
recommendations to enhance robustness of structures:
a) Toavoidirregularitiesinthestructure’splanandto have a three dimensional
space frame.
b) To avoid anti-redundant features (such as transfer girders) and to provide the
structure with enough redundancy to form alternate load paths.
c) To provide the structural frame with mechanical fuses which allows the walls
and slabs to collapse without affecting other parts of the structural frame.
d) To provide the structural frame with enough ductility for energy absorption
(such as design of structures in highly seismic areas).
2.2 Building Codes and Regulations on Robustness of
Structures
The risk of progressive structural collapse due to damages caused by accidental
loading is different depending on the nature, the size and occupancy of the building.
It is therefore important to strike an appropriate balance between the cost of
providing a robust structure and the benefit of reduced risk of progressive collapse.
In many parts of the world, this is done by dividing buildings into a number of
classes and specifying different requirements for different classes. For example, in
29
the British Standard Approved Document A, buildings are divided into the following
three classes (Table 2-1):
30
Table 2-1: Building classes (Approved Document A)
Class Building type and occupancy
1 •Housesnotexceedingfourstoreys.
•Agricultural buildings.
•Buildingsintowhichpeoplerarelygo,providednopartofthebuildingis
closer to another building, or area where people do go, than a distance of 1.5
times the building height.
2A •5storeysingleoccupancyhouses.
•Hotelsnotexceeding 4 storeys.
•Flats,apartmentsandotherresidentialbuildingsnotexceeding4storeys.
•Officesnotexceeding4storeys.
•Industrialbuildingsnotexceeding3storeys.
•Retailingpremisesnotexceeding3storeysoflessthan2000m2 floor area
in each storey.
•SinglestoreyEducationalbuildings.
•Allbuildingsnotexceeding2storeystowhichmembersofthepublicare
admitted and which contain floor areas not exceeding 2000 m2 at each
storey.
2B • Hotels, flats, apartments and other residential buildings greater than 4
storeys but not exceeding 15 storeys.
•Educationalbuildingsgreaterthan1storeybutnotexceeding15storeys.
•Retailingpremisesgreaterthan3storeysbutnotexceeding15storeys.
•Hospitalsnotexceeding3storeys.
•Officesgreaterthan4storeysbutnotexceeding15storeys.
• All buildings to which members of the public are admitted and which
contain floor areas exceeding 2000 m2 for the notional but less than 5000 m
2
at each storey.
3 •Allbuildingsdefinedabove as Class 2A and 2B that exceed the limits on
area and/or number of storeys.
•Allbuildingscontaininghazardoussubstancesand/orprocesses.
•Grandstandsaccommodatingmorethan5000spectators.
31
For building class 1, provided that they are designed and constructed based on other
buildings codes and construction regulations there is no need for any action to assure
their robustness. For class 2A buildings it is said that effective horizontal ties or
effective anchorage of suspended floors to walls for framed and load bearing walls
should be provided according to BS 8110-1:1997 and BS 8110-2:1985 for concrete
structures, BS 5628-1:1992 for unreinforced masonry structures and BS 5950-1:2000
for steel structures. For class 2B buildings there should be effective vertical ties for
all supporting columns and walls and horizontal ties for frame and load bearing
walls. For class 3 buildings a systematic risk assessment should be undertaken while
considering all the normal predictable and unpredictable hazards and all the
structural elements should be designed based on the aforementioned building codes
and regulations.
The European code EN 1991-1-7:2006 takes a similar approach. Table 2-2 compares
the regulations between the British Standard and EuroCode for building
classification and precast concrete structures.
32
Table 2-2: Differences in building classification in BS and EN
Bld. Class Type Approved
Document A- A3
EuroCode 1991-1-7
1 Houses not
exceeding four
storeys
Single Occupancy
houses not exceeding
four storeys
2A Retailing premises:
less than 2000 m2
Retailing premises: less
than 1000 m2
2B Admissible damaged
area: 15% or 70 m2
whichever smaller
Admissible damaged
area 15% or 100 m2
whichever smaller
3 Bld. Containing
hazardous
substances
N/A
The value of 70 m2
for the admissible damage area is an estimation based on two 6 m
× 6 m structural bays that at the time of drafting the British Standard was a typical
bay size. But as the modern structural systems came into practice this size was
augmented to 7.5 m × 7.5 m which is almost the 100 m2 recommended by the
EuroCode considering two adjacent floor bays close to the lost column (CPNI,
2011).
2.2.1 Implementation of the robustness requirements
The British standards were the first building code to recommend regulations for
avoiding the progressive collapse of buildings, of which the catastrophic accident of
Ronan Point Building in 1968 was the main motivation. In a number of publications
of the British Standard such as those for steel, concrete, masonry and composite
33
structures, also steel and concrete bridges, the concept of robustness and
recommendations for achieving it are given.
The main methods proposed in the British Standards to ensure that the structure will
not suffer from an accidental action in an amount which is not proportional to the
cause, can briefly be described as follows (Moore, 2002):
The tying method: vertical and horizontal ties should be provided between
the primary structural components (Figure 2-5). The assumption is that the
provision of ties creates a structure with a degree of redundancy that
increases the structural continuity, and thus provides the building with
alternative load paths if part of the structure is removed by an accidental
action. In general the ties are steel members or rebar, also the beam to
column joint is considered to carry the tying force. The minimum value for
tying force resistance is 75 kN is steel structures and 60 kN in concrete.
Figure 2-5: Floor ties in a concrete structure (Brooker, 2008)
The bridging method: wherever tying is not feasible, the structure should be
designed to be able to bridge over the loss of a member which has not been
tied and the area of collapse should be limited and localised. To do so, each
time an untied member is notionally removed (including vertical load bearing
members and beams connected to one or more columns) and the area of the
34
affected zone in the immediate adjacent storeys, is checked so that the area at
risk of collapse should be limited to the smaller of these values: 15% of the
area of the considered storey, or 70 m2 (Figure 2-6).
Figure 2-6: Area of the structure susceptible to collapse (Approved Document A)
The key element method: if bridging over a missing member is not possible;
such a member should be designed as a key element which is capable of
resisting a pressure of 34 kN/m2 from any direction. The value of 34 kN/m
2
(5 lb/in2) was chosen based on the observational evidence on an estimation
that exterior wall panel would fail at Ronan Point (Hai, 2009). Such
accidental design loading is supposed to act simultaneously with one third of
all normal characteristic loading.
Following the above recommendations in BS is considered to produce robust
structures that can resist disproportionate collapse due to various accidental causes,
such as impact and gas explosions (Demonceau, 2008).
35
2.2.1.1 Application of the Tying Force Method to Precast Reinforced
Concrete Structures
Design codes for different types of structure provide detailed recommendations on
how to apply the tying force method. Among the three methods presented in the last
section, specific guidance is often necessary on application of the tying method. This
section presents an overview of the tying method application to precast reinforced
concrete structures.
Figure 2-7: Vertical and Horizontal tying in a structure (NIST, 2007)
Figure 2-7 shows a scheme of providing ties. Internal ties should be available at each
floor and roof level approximately at right angles and they should be continuous, and
at each end they should be anchored to peripheral ties. In the British Standard BS
8110-1:1997, the internal ties should be capable of resisting a tensile force equal to
the greater value of the two following relationships (in kN/m width):
36
Equation 2-2
Equation 2-3
Whichever is larger.
Where:
is dead load on floor (in kN/m2)
is the imposed (live load) on floor (in kN/m2)
is the greater of the distances between centres of columns, frames or walls
supporting any two adjacent floor spans in the direction of the considered tie
is the lesser of these two values: (20 + 4n0) or 60 kN/m; where n0 is the number of
storeys in the structure.
The value of 60 kN was chosen based on an estimation of the equilibrium state that a
tie connecting two horizontal members on top of the lost column should be able to
provide with regard to the floor area applying load to the tie.
But as will be seen in this study (Chapter 4), for precast concrete floor slabs, this
value is subject to many other local factors at the connection zone. Also it will be
shown (in Chapter 4) that solely specifying the tie force does not necessarily
guarantee robustness of the structure, because in the case of the connection between
PCFSs it is mainly the elongating capacity of the tie bar that provides this
characteristic for the structure in catenary action.
2.3 Research on Structural Tying System of PCFS
There have been many experimental and numerical investigations regarding the
robustness of steel and concrete frames. But most of the research in the field of
progressive collapse focuses on the connections between the main structural
elements i.e. columns and beams and they usually consider the flooring system as an
integrated structural element which does not fall apart (Shi et al., 2010), (Zolghadr
37
Jahromi et al., 2013), (Vlassis et al., 2008), (Izzuddin et al., 2008), (Sasani, 2008).
This assumption not only dismisses the intrinsic segmental nature of the precast
concrete flooring system but also neglects the effect of the debris from such floors on
the lower levels which itself causes extra loading on the remaining structure. This
phenomenon has more importance as it has not been considered in the current
building regulations (Izzuddin et al., 2008).
Despite questioning the effectiveness of tying resistance for PCFS against
progressive collapse (CPNI, 2011), there is a scarcity of experimental and numerical
(finite element) research on connections between precast concrete floor units and
how they contribute to the robustness of this type of structure. Most of the research
on PCFS in steel frame concentrate on the composite behaviour of the PCFS on the
steel beam, considering the shear stud (Lam et al., 1999), (Lam et al., 2000), (Lam
and Nip, 2002), (Fu and Lam, 2006), (Hegger et al., 2009) and not the ties
connecting the floor slabs especially in the case of loss of a column. The only major
study on the tying system connecting the PCFS with special attention to the bond-
slip phenomenon of the rebar inside concrete was conducted by Engström (1992).
2.3.1 Bending Tests on Tie Connections
Bending tests on the connection of PCFS have been conducted by Engström (1992),
Rosenthal (1978), and Gustavsson (1974). The scope of these studies has been to
obtain the adequate floor integrity by means of connection between the concrete
floor slabs, deformation capacity and anchorage capacity of the tie connection
respectively. In all of the mentioned experiments the PCFS units rested on three
beams and the middle one (under the connection) was raised in order to apply
bendingtotheslabs’connection(Figure 2-8). Although the boundary conditions of
these experiments were not a faithful representation of the real PCFS flooring system
(because there was no consideration of axial restraint), their results shed some light
on the behaviour of tie connections between PCFS, including:
Smooth tie bars may enable more elongation in the connections, but the bond
stress provided by them did not give sufficient anchorage between concrete
and steel bars, in comparison to ribbed bars.
38
Increasing the tie bar dimension increased the bending resistance of the
connection, although a balance should be struck between concrete and steel
tensile strength.
Due to pure bending action in these tests, under sagging moments, the tie bar
served better if placed at the bottom of the section and under hogging
moment at the top. Placing the tie bar at mid-height of the section would
provide some resistance under bending in both directions.
Slabs were always separated first at one of the transverse joint interfaces.
Once the transverse joint was cracked, the tensile strength was carried by the
tie bar only.
The contribution of the tie bar in the transverse joint was negligible.
If the tie bar was sufficiently long (usually defined as 75ϕ) the end hooks
were not strained.
Smooth tie bars, even with end hooks, did not provide sufficient bond with
concrete.
However the lack of consideration of axial restraints and the different boundary
condition of the other far ends of the slabs from the real structure condition in the
aforementioned experiments, neglected the effect of arching action prior to the
catenary stage. Also as the slabs can move freely on both far ends (from the
connection, in the lack of axial restraint) the catenary action behaviour of these tests
may not illustrate behaviour close to what may happen in real floor slabs; as
adequate tension was not applied on the tie connecting the slab units.
39
Figure 2-8: Lifting tests on the connection between PCFSs (Engström, 1992)
In the bending tests of Engström (1992) the type, dimension and position of the tie
bars were studied. The present study will broaden the scope of investigation by
including the following additional parameters: grouting concrete strength, span and
depth of the slabs, and tie bar length. Also the effect of yield stress and ultimate
strain of the tie bars will be individually studied.
2.3.2 Formulation of Tie Behaviour in Bending
Engström (1992) seems to be the only one to have made an attempt to formulate the
requirement on tie connection between PCFS in the case of column loss. It was
assumed that after column loss, slabs are suspended by ties from both ends and that
the elongations (w) of all of the ties were the same at all times (Figure 2-9).
40
Figure 2-9: Pure suspension mode of action in a precast floor after loss of an interior column
(Engström, 1992)
For this formulation, the following parameters were defined:
Q: total load of each floor element applied at the centre of the element;
N: tensile force of the tie bar;
a: maximum vertical displacement of the connection
aqz: vertical displacement of the driving force (Q),
w: elongation (displacement) of the tie bar
l: length of the slab
For one slab, the deformed geometry yields:
(
)
Equation 2-4
Neglecting the quadratic terms of w, the vertical displacement of the driving force
can be written as:
√
Equation 2-5
41
The equilibrium of moments gives:
Equation 2-6
The ultimate strength and elongation of the ties are Nmax and wmax respectively.
Therefore, the maximum vertical resistance (Qmax) for the connection is defined as:
√
Equation 2-7
The above procedure suffers from the following drawbacks:
While the slabs are assumed to be completely suspended, the expected
behaviour from the side tie bars would be dowel action rather than catenary
(capturing the tensile force).
Even if the two sides of the slabs were provided with vertical restraint so that
the ties would only carry tensile forces, the assumption that all three would
have equal elongations should have been substantiated.
The above arrangement of ties neglects slab connections with walls and
beams e.g. at the edge of the structure.
In the case of loss of a column, as long as the slabs have adequate axial, vertical, and
rotational restraint at both far ends from the lost column, the connection between the
PCFSs can undergo a large vertical displacement (h in Figure 2-10). Hence it is the
tie bar’s elongation (e) localized to the connection zone that may provide the
required integrity of the floor slabs, provided that there is enough bond between the
tie bar and surrounding concrete.
42
Figure 2-10: Resistance mechanism of tie connection between PCFS in the case of lost column
The current tie connection regulation recommends a tie force derived based on
equilibrium of slab forces in the catenary action stage (as shown in section 2.1.1);
however, it neglects the fact that it is the ductility of the tie bar which dictates the
extent of the catenary action development (Figure 2-10). The present study will
demonstrate that a tie connection designed based on current regulations may fail.
Importantly, this study will show how to achieve robustness of PCFS system.
Based on the conducted literature survey presented above, the necessity of an
investigation on effectiveness of the tie connection is apparent. Such investigation
should particularly concentrate on parameters that affect the behaviour of the tie
connection in the catenary action stage because it is considered the dominant
mechanism under the column loss scenario (Elliot, 2002).
2.4 Construction Technology of PCFS
To understand the behaviour of the PCFS better, it is necessary to examine the
different components of this structural element. This can be achieved by considering
the design and manufacturing process of precast concrete members.
Tie Bar
PCFS Axial and Vertical Restraints
h
L
43
2.4.1 Design of PCFS
PCFSs are designed mostly as simply supported, one-way spanning units. The main
failure modes for PCFSs are (Elliot, 2002):
Flexural capacity
Shear capacity
Other design considerations are:
Deflections limit
Bearing capacity
Handling restrictions (usually imposed by manufacturer)
2.4.1.1 Flexural capacity
The flexural capacity of a PCFS is checked in serviceability limit state (SLS) and
ultimate limit state (ULS) designs. When using design factors of 1.35 for permanent
and 1.5 for variable load, the SLS design check is usually the critical loading
condition (Elliot, 2002).
2.4.1.1.1 Serviceability limit state (SLS)
The serviceability limit state of flexure is calculated based on lesser of the following
two relationships (Equation 2-8 and Equation 2-9):
( √ )
Equation 2-8
Equation 2-9
Where: Zb and Zt are the section modulus to the bottom and top fibre respectively. fcu
is the cube compressive strength of concrete. And fbc and ftc are the maximum fibre
stress in the bottom and top of the section respectively, and defined as:
44
(
)
Equation 2-10
(
)
Equation 2-11
in which
The final prestressing force (Pf) is the product of the effective prestress in the tendon
after all losses (fpe) and the cross sectional area of the prestressing strands (Aps):
e: eccentricity of the prestressing strand
and A is the total cross sectional area of the precast member
2.4.1.1.2 Ultimate limit state of flexure
The ULS flexure resistance can be calculated based on the following relationship
(Elliot, 2002):
Equation 2-12
in which:
Mur: is the slab ultimate bending moment resistance
fpb: is the design tensile stress in the tendons
Aps: is the total cross sectional area of the tendons per unit area of slab
d: is the effective depth of the precast concrete cross section
and X is the depth of concrete in compression calculated by equating the tensile force
in the tendons to compressive force of the concrete block
45
2.4.1.2 Shear capacity of PCFS
Unlike the bending resistance, shear capacity is only considered at the ultimate state.
The shear capacity is calculated for a cracked (Vco) and uncracked (Vco) section
separately. Obviously the shear capacity of the uncracked section is more than that of
cracked, as the whole section contributes to resisting the shear forces.
The shear capacity of uncracked PCFS is given by the following relationship:
√
Equation 2-13
where
√
fcp is the compressive stress at centre axis resulting from prestress after all losses
y’ :is the distance from section centroid to total area (A) centroid
bv :is the web width
The shear capacity in the cracked region of concrete is given by:
(
)
Equation 2-14
in which vc is a factor obtained from BS8110, Part 1, Table 3.9
and fpu is the ultimate strength of the prestressing strands.
Most PCFSs are manufactured with predefined standard specifications, and their
load carrying capacities are reported by the manufacturing companies, giving load-
span tables. An example is shown in Table 2-3.
46
Table 2-3: Load/Span Table (Bison, 2012)
Overall
structural
depth
Spans indicated below allow for characteristic service load (live load)
plus self-weight plus 1.5 kN/m2 for finishes
Characteristic service loads (kN/m2)
0.75 1.5 2 2.5 3 4 5 10 15
Effective span (m)
150 7.5 7.5 7.5 7.3 6.8 6.4 5.6 5 3.2
200 8.25 8.25 8.1 7.9 7.7 7.4 6.9 5.8 4.6
250 10.4 9.9 9.7 9.4 9.2 8.8 8.1 6.9 5.3
300 11.7 11.2 10.9 10.6 10.4 9.9 9.5 7.8 6.8
350 14.5 14 13.7 13.5 13.2 12.7 12.3 10.7 8.8
400 16 15.5 15.2 14.9 14.6 14.1 13.7 11.9 10
450 17.1 16.5 16.2 15.9 15.6 15.1 14.6 12.7 10.7
2.4.2 PCFS Manufacturing
The manufacturing process starts with cleaning the casting bed (Figure 2-11). The
casting bed should have a smooth surface which is oiled to provide detachable
surface with concrete. For concrete curing purposes there are heating pipes under the
metal surface of the casting bed which itself lies on a concrete base and insulation
material. The length of the casting bed depends on several parameters such as
utilisation of the casting bed, production flexibility, available space in the factory,
and strand patterns. The common length for the beds is 120m (Spiroll, 2014).
Figure 2-11: Installed Casting Beds (Spiroll, 2014)
The next step is positioning and pulling of the prestressing strands. This process is
done, by some manufacturers, with the same machine that cleans the bed. The
47
strands are tensioned to the desirable stress and anchored at the other end of the
casting bed.
The main stage of the manufacturing process is performed with a machine called
“extruder”.Extrudershavedifferentcharacteristics in termsofspeed,heightof the
slab they produce, number of hollow cores, slab width, and concrete compaction
technology. The concretemix is usually fed into themachine from the “concrete
hopper” and transferred to the nozzles thatmove the extruder forward with their
injection force (Figure 2-12).
Figure 2-12: Extruder Components (Elematic, 2014)
Most manufacturers use a concrete mixture with a rather low water to cement ratio.
This dry mixture with intense concrete compaction allows the concrete mixture to
plasticise during a short time and form and mould while the extruder passes on the
casting bed. After the concrete is cured, slabs are ready to be sawed into the required
lengths.
Machines designed for sawing the PCFS use diamond blades with different
diameters depending on the height of the slab. Based on the type of the cut required,
machines with suitable angle of saw are chosen. Cuts may be longitudinal or
transverse and each requires the corresponding saw. There are also saws that can be
adjusted to any angle between 0 to 90 degrees (Figure 2-13).
48
Figure 2-13: Different types of cut on PCFS (Spiroll, 2014)
Most of the transverse cuts are performed while the slab is on the casting bed,
providing slabs segments which can be moved to the storage area. This allows the
casting beds to have a faster turnaround. Slabs are lifted usually from their side
grooves (Figure 2-14). At the stock yard (storage area) other type of saws may be
used to give the slabs the required shape and size.
Figure 2-14: Slab lifting from its side grooves (Ultra-Span, 2012)
Longitudinal Cut
Transverse Cut
Inclined Cut
49
2.5 Providing Tying Resistance in Precast Concrete Floor
Slabs
This research focuses on the behaviour of precast concrete floor slabs supported by a
steel frame. It is necessary to understand details of the structural components and
connections that are used to achieve sufficient tying resistance required for structural
robustness according to current construction methods.
2.5.1 Precast Concrete Floor Slabs
Precast concrete floor slabs are prestressed units and are constructed in two main
categories: 1) solid elements (planks) or 2) with longitudinal hollow cores (HC). The
units usually have 1200 mm width and can be up to 10 m long, with different depths
(Way et al., 2007).
2.5.1.1 Hollowcore floor units
The majority of the manufacturers produce units with depths ranging from 150 to
450 mm and a nominal width of 1200 mm. High tensile prestressing strands or wires
are used as the reinforcement in hollowcore floor slabs, and there is no shear
reinforcement in them.
Figure 2-15: Hollowcore unit profile on steel structure (Hanson, 2014)
PCFS
Tie Bar
Grouting
Shear Key
50
The edges of the hollowcore units are profiled (shear key) such that it is possible to
grout the joint between two adjacent units to provide enough shear resistance
between them. The reinforcement for providing the tying resistance is located in this
grouted or concreted joint between the floor slabs (Figure 2-15).
2.5.1.2 Solid precast floor units
This kind of floor slab is used usually with structural in-situ concrete topping, and
their depth ranges from 75 mm to 100 mm. The prestressing reinforcement of solid
precast floor units is the same as for the hollowcore units. As there are no
hollowcores in this type of units their thickness is usually less than that of those with
hollowcores.
2.5.2 Connections of precast concrete floor slabs
Apart from the tie force that was discussed in section 2.2.1.1, other regulations
observe the placement of the tie bar in between the slabs for those types of structures
that require tying. Tie bars are normally placed in between the units and in the
hollowcores. For the latter, the top flange of the PCFS is removed and the core is
filled with in situ concrete (Figure 2-16).
Figure 2-16: Placement of tie bar in hollowcores (CCIP-030)
Placement of tie bars between units depends on the position of the grouting keys on
side of the PCFS, as shown in Figure 2-17. Other arrangements of tie bar are
51
possible for connection of PCFS to supporting walls and beams, but they are out of
the scope of the present study.
(a) (b)
Figure 2-17: Placement of tie bar in between units (CCIP-030)
2.6 Bond-Slip
As explained in the preceding section, in precast concrete floor systems, tying
resistance is provided by the reinforcement between the floor units. It is therefore
important that this means of resistance is reliably quantified.
The tying resistance critically depends on the bond-slip behaviour between the
reinforcement and the concrete. This behaviour is complex due to the nature of
concrete and other factors such as: randomness of the size, shape and texture of the
aggregates, and chemical and physical adhesion between the reinforcement and
concrete. Many research studies have been devoted to this subject; examples
including (Naaman and Najm, 1991), (Lahnert et al., 1986), (Edwards and
Yannopoulos, 1979), (Huang et al., 1996), (Engström et al., 1998), (Mazzarolo et
al., 2012).
Depending on the length of the reinforcement, there are two generic modes of bond-
slip behaviour: tie bar pull-out (anchorage failure) in the case of short tie bar and tie
bar yield in the case of long tie bar. There have been pull-out tests conducted on the
52
tie bar embedded in the grouting between the PCFSs (Regan, 1975), (Holmgren,
1975). The findings of these experiments can be helpful in the sense that when the tie
bar is embedded in the concrete and then tensioned, its strain is localised to the
section of the bar protruding from the concrete face, unlike the free steel bar along
which strain can be distributed more. To have more elongation of the tie bar
embedded in concrete, the use of smooth steel bars is suggested, but on the other
hand this type of bar would not provide enough bond with the surrounding concrete.
Hence in the present study the bond condition between steel and concrete is derived
with the assumption that grooved bars are embedded in concrete.
Similar tests by Engström et al. (1998) and Salo et al. (1984) focus on embedment
length in order to prevent anchorage failure of the tie bar in the connection between
PCFSs. Engström suggests that when the tie bar is grouted in the cores of the PCFS,
the embedment length should be between 0.5 m to 1 m. And (Salo et al., 1984)
suggest anchorage length of less than 60ϕ (where ϕ is the diameter of the tie bar) is
susceptible to failure. Later CCIP-030 regulated the anchorage length to be 75ϕ
(Whittle and Taylor, 2009). These recommendations will be checked in this study
through comparison with FE simulations (section 3.2.3) and it will be shown that the
BS regulations regarding tie bar length satisfy the required bond conditions between
reinforcement and surrounding concrete.
The two types of bond-slip behaviour are summarised in the CEB-FIP Model Code
2010 (CEB-FIP 2010) and are shown in Figure 2-18. For type I bond-slip
relationship the following relationships are provided:
Equation 2-15
Equation 2-16
Equation 2-17
Equation 2-18
Where:
53
τisthebondstress
: is the bond strength (dependent on concrete strength)
: is the residual bond stress (dependent on concrete strength)
s: the slip of steel bar in concrete in millimetres
α, : are the coefficients that can be found based on the concrete and bond
conditions in the tables below:
Table 2-4: Type I (tie bar pull-out) bond-slip relationship parameters (CEB-FIP 2010)
1 mm
3 mm
Clear rib spacing
3
α 0.4
√
Figure 2-18: Generic bond-slip relationships (CEB-FIP 2010)
For type II bond-slip relationship, the parameters in Table 2-5 are used.
54
Table 2-5: Parameters for Type II (tie bar yield) bond-slip relationship
1 mm
3 mm
Clear rib spacing
2
where fcm is the concrete cylindrical compressive strength.
It is assumed that the shear bond stresses over the area made up of the perimeter of
the tie bar multiplied by its length, should at least be equal to the product of steel
yield stress and its cross sectional area. By this method the minimum length that
provides the necessary bond stress which results in the yielding of the tie bar can be
determined (Mazzarolo et al., 2012):
Where:
fy: steel reinforcement yield stress
As: rebar cross sectional area
p: the perimeter of the steel reinforcement
lmin: is the length that if the reinforcement is less than, it does not yield
To apply and evaluate the bond-slip relationship in the connection between PCFSs
the tests conducted on long embedment of reinforcement in concrete are studied in
more detail in Chapter 3.
55
2.7 Summary and Objectives of Research
This chapter presented a brief review on robustness regulations in British Standard
and the background from which these rules have originated. It was shown that, in
particular, the tying method regulations are derived from the assumption that ties
connecting the slabs are able to develop catenary action. Then at catenary stage
where all the structural members are supposed to be in equilibrium, force values
required in ties to balance typical structural loads are recommended to be sufficient
to provide adequate robustness. The present study examines the initial assumption of
the British Standard about deformation capacity of ties, and casts doubt on available
tie ductility with current rules since they focus merely on the tie force.
It was seen that although the capability of current tying regulations has been
questioned (CPNI, 2011) yet the literature lacks substantial study into the behaviour
of ties and design parameters that may affect that. The only study concentrating on
PCFS and tie connection has been conducted by Engström (1992). It was seen that
since the tests were conducted without axial restraint, ties were not fully tensioned
and consequently the results could not represent precise tie behaviour of a real
structure. The assumptions made in derivation of an analytical relationship by
Engström (1992) for predicting the tie connection behaviour, also weaken the
resemblance of the analysed model compared to real connection behaviour. Hence
the necessity of research into the ductility of tie bars and the parameters affecting
that behaviour was established. This leads to the main objectives of this research as:
- To understand the load carrying mechanism for robust precast concrete floor
construction.
- Assessment of the adequacy, or lack thereof, of the current structural
robustness specification in achieving robust precast concrete floor
construction.
- To investigate methods of changing tie bar properties for enhancing
robustness of precast concrete floor construction.
For the tie bar to be able to facilitate its elongation to the fullest extent, it is
necessary to have enough bond with the concrete in which it is embedded. This
56
research will include some investigation on how to provide sufficient bond to allow
the tie bar to achieve its full tying force and elongation capacity.
57
Chapter 3. Validation of Numerical Modelling
The results reported in this thesis have been obtained by numerical simulations using
the finite element program DIANA (TNO-DIANA, 2010). This software package
was chosen for its ability to handle complex concrete behaviour. This chapter
presents examples of validating the numerical model. In analysis of robustness of
precast concrete floor slabs, the two important requirements for accurate modelling
are large deflection behaviour and bond slip relationship for the tie bar. This chapter
will present validation examples for these two situations: the experiments of Su et
al. (2009) numerical models for dealing with large deflection behaviour, including
both arching and catenary action in reinforced concrete members. And for
validating modelling of bond slip behaviour, some experiments by Engström et al.
(1998) were modelled.
3.1 Choosing DIANA FE Package
Initially in the present study, the commercial FE package ABAQUS was used to
model concrete structural elements in large deflections. However, as concrete has
proved to be a very sensitive material to be modelled in geometric nonlinearity
problems, the author was not able to obtain results that agreed with the experimental
results, especially for concrete members in catenary action stage. This deficiency has
also been observed by other researchers (Lee, 2009), (Garden, 1997). Hence the
general finite element software DIANA was used instead.
The main shortcomings of ABAQUS include:
1) Concrete ismodelled inABAQUS either by “smeared crackingmodel” or
“damagedplasticitymodel”.Thelatter assumes that concrete is an isotropic
material even after cracking (cracks happening in orthogonal directions with
respect to each other) which is far from the reality of concrete behaviour.
However, if using the smeared cracking model, ABAQUS would not allow
theuseofsmearedcrackingmodelin“ABAQUSExplicit”whichwouldbe
58
necessary for numerical simulation stability. DIANA, on the other hand,
provides more suitable options as explained in sections 3.2.1 and 3.2.2.
2) The suitable modelling method for reinforcement in ABAQUS for the
presentstudywouldbethe“embedded”optionsincethesteelbarshouldbe
modelled as an explicit material from concrete (the other option only assumes
higher stiffness for the concrete element which has the rebar). This option
assumes full bonding of reinforcement and the surrounding concrete. This
assumption would not be appropriate because it is necessary in the present
study to consider bond-slip phenomenon between the steel bar and concrete.
In contrast, DIANA allows the bond-slip function for reinforcement to be
modelled explicitly in concrete.
3.2 Modelling concrete in DIANA
There are two main material modelling methods available in DIANA: a) smeared
cracking and b) discrete cracking. In the smeared cracking approach, when the
tensile strength of concrete is breached in an integration point, it is assumed that
concrete is cracked and can carry no more tension at that point. On the other hand, to
model the cracks discretely specific interface elements are used which dictated the
knowledge of exact crack location in advance. The discrete cracking approach may
represent the discontinuity in the material better, but invokes some numerical
difficulties when used with other interface elements (TNO-DIANA, 2010). As in the
present simulation the effect of bond-slip needs to be taken into account, which is
modelled with interface elements available in the software for this purpose, the
smeared cracking approach is chosen.
The chosen smeared cracking approach (Multi-directional fixed crack model)
introduces several nonlinear plasticity models for both compressive and tensile
regimes of concrete. The compressive plasticity models include:
Tresca & Von-Mises
Mohr-Coulomb
Drucker-Prager
59
And the tensile regime modelling consists of:
Brittle cracking
Linear tension softening
Multi-Linear tension softening
Moelands tension softening
Hordijk tension softening
3.2.1 Compressive Behaviour
3.2.1.1 Tresca
Tresca yield criterion is based on maximum shear stress. In principal stress space
(σ1>σ2>σ3), shear stress is defined as the difference between maximum and
minimum stresses:
| | | | | | Equation 3-1
Where σ1, σ2, σ3 are the principal stresses in a multi-axial stress state, and σy is the
yield strength.
3.2.1.2 Von-Mises
The Von-Mises criterion is a smoother approximation of the Tresca yield function
(Figure 3-1).
√ [
] Equation 3-2
Both Tresca and Von-Mises are usually used for ductile material such as steel.
Figure 3-1: Tresca and Mohr-Coulomb yield criteria (TNO-DIANA, 2010)
60
3.2.1.3 Mohr-Coulomb
Mohr-Coulomb criterion is an extension of Tresca, but the yield function is mostly
dependent on the pressure. In a multi-axial principal stress state its function is
expressed as:
Equation 3-3
Where:
φ: is the internal frictional angle (usually ≈ 30o for concrete (TNO-DIANA, 2010))
c: is the cohesion:
Mohr-Coulomb yield function is usually used for concrete and other brittle materials.
This yield criterion shows more compatibility with test results over other
compressive behaviour options.
3.2.1.4 Drucker-Prager
Drucker-Prager is a smooth approximation of the Mohr-Coulomb (Figure 3-2). This
criterion in terms of principal stresses reads:
Equation 3-4
Where:
√
√
In the Drucker-Prager criterion the internal frictional angle is recommended to be
taken as 10o (TNO-DIANA, 2010).
This model is used mostly for modelling the plasticity of soil and masonry materials.
61
Figure 3-2: Mohr-Coulomb and Drucker-Prager yield criteria (TNO-DIANA, 2010)
3.2.2 Tensile Behaviour
3.2.2.1 Brittle Cracking
In this cracking criterion after the tensile stress in a point reaches the tensile strength
of the concrete it is assumed that the stress suddenly drops to zero and the element
carries no more tension at that point (Figure 3-3).
Figure 3-3: Brittle Cracking Behaviour (TNO-DIANA, 2010)
This model does not reveal compatible results with the experimental results in
catenary action of the concrete structural members, as it sometimes causes
discontinuity, and also does not capture the tension stiffening effect of reinforced
concrete.
62
3.2.2.2 Linear Tension Softening
This model assumes the tensile stress in a cracked element decreases in a linear
fashion (Figure 3-4).
Figure 3-4: Linear tension softening (TNO-DIANA, 2010)
The line of tension softening is defined by the fracture energy of the concrete (Gf).
The fracture energy is defined as the energy required to form a tensile crack of unit
area, and was calculated based on the CEB-FIP Model Code 2010 regulations. This
tension model showed acceptable results when compared to experimental results of
(Engström, 1992) and (Su et al., 2009) and was used as the tension softening model
in the simulations presented in this chapter.
Parameter h is the crack band width calculated by DIANA, which depends on the
element size (Table 3-1):
Table 3-1: Crack band width calculation (TNO-DIANA, 2010)
Element Type Crack band width (h)
Linear 2D √
Quadratic 2D √
3D √
Where A and V are the element area and volume respectively.
By assuming gradual decrease in tensile strength after the concrete has cracked, the
linear tension softening model, unlike the brittle cracking softening model, enables
tension stiffening of reinforced concrete to be modelled in a numerically stable
manner. Since it is the reinforcement that carries tensile forces during the catenary
action stage which is the main concern of this research, it will be shown that
63
assuming linear stiffening of concrete yields numerical simulation results in good
agreement with the test results (section 3.3.4).
3.2.2.3 Moelands Tension Softening
The Moelands tension softening relationship is a modification of the linear tension
softening and it is advised to be used in cases if the linear tension softening causes
convergence problems (Figure 3-5).
Figure 3-5: Moelands Tension Softening (TNO-DIANA, 2010)
The normalized relationship for the Moelands curve is defined as:
{
(
)
Equation 3-5
3.2.2.4 Hordijk Tension Softening
Hordijk tension softening model is defined by the following relationship
(Figure 3-6):
{
( (
)
) (
)
Equation 3-6
64
Figure 3-6: Hordijk tension softening (TNO-DIANA, 2010)
As with the linear tension stiffening model, the Hordijk and Moelands tension
softening models also give gradual reduction of tensile strength of concrete in a
cracked element, which helps with numerical stability. However, with a given
fracture energy, the linear tension softening model assumes a steeper reduction in
concrete tensile strength immediately after cracking. Hence, it gives concrete a lower
ultimate cracking strain ( ).Thisaffectsthe“fractureenergy/bandwidth”ratio
(
area under the curve of tension softening models).
Figure 3-7: Comparison of tension softening models
As shown in Figure 3-7 a higher ultimate cracking strain, resulting from nonlinear
tension softening models (Moelands and Hordijk), leads to a higher tension
stiffening effect and consequently failure of the structure occurs at a higher vertical
displacement.
-500
-400
-300
-200
-100
0
100
200
300
0 0.05 0.1 0.15 0.2 0.25 0.3
Ho
rizo
nta
l R
eact
ion
V
erti
cal
Lo
ad
(kN
)
Vertical Displacement (m)
Test A1
Linear A1
Moelands A1
Hordijk A1
65
3.2.3 Bond-Slip
There are interface elements available in the material model bank of DIANA for
modelling the bond-slip behaviour of reinforcement in concrete. These include:
Dörr Model (Figure 3-8):
Figure 3-8: Cubic function of Dörr model (TNO-DIANA, 2010)
This function is defined by:
{ ( (
) (
)
(
)
)
Equation 3-7
Noakowski Model (Figure 3-9):
Figure 3-9: Power Law of Noakowski (TNO-DIANA, 2010)
66
Which is defined by:
{
Equation 3-8
Multi-linear Model
As it can be seen in Figure 3-8 and Figure 3-9, the two predefined relationships for
bond-slip behaviour are for modelling local bond-slip effect and do not consider the
“pull-out”or“yieldandpull-out”failuremodesofthethisphenomenon.Hencethe
bond-slip relationship used for modelling the tests in this study is the Multi-linear
Model which allows the user to define the relationship between the bonds stress and
slip. As mentioned in the literature review chapter this relationship is calculated
based on the CEB-FIP Model Code 2010.
3.2.4 Concrete Elastic Material Properties
3.2.4.1 Tensile Strength
The CEB-FIP Model Code introduces a relationship for calculating the tensile
strength based on compressive strength:
(
)
Equation 3-9
Where:
fctm: mean axial tensile strength
fcrko,m= 1.40 MPa
fck: Compressive strength
fcko=10 MPa
67
3.2.4.2 Elastic Modulus
The elastic modulus of concrete in the CEB-FIP Model Code 2010 can be calculated
using the following relationship:
Equation 3-10
Where:
Eci: is the concrete elastic modulus
Ec0 = 2.15 ×104
fck: Compressive strength
Δf = 8 MPa
3.3 Axially Restrained Beams (Su et al., 2009)
Although the subject matter of this thesis is reinforced concrete slabs, the main focus
of this research is development of catenary action of axially restrained member. For
this behaviour, the experimental work of Su et al. (2009) is the most relevant for the
purpose of validation, even though the subject matter of their research is reinforced
concrete beam. In particular, the work of Su et al. (2009) was in the context of
structural robustness under column removal scenarios, which is closely related to the
present research.
Su et al. (2009) referred to some axially restrained slab tests by Guice and Rhomberg
(1988). However, this study was mainly focused on slab capacity under compressive
membrane action, which is not suitable for the purpose of progressive collapse
investigation where the slab deflection is rather high. Therefore, the tests of Guice
and Rhomberg (1988) were not used in the validation study of this research.
68
3.3.1 Test Subassemblies & Setup
As shown in Figure 3-10 each test subassembly consisted of two doubly reinforced
concrete beams connected with a central column stub and two short columns at the
two far ends. The centre column represented the lost column and had a 250 mm
square base for all the specimens, but the edge columns were with enlarged sizes for
ease of being anchored into the test setup.
Figure 3-10: Test Subassembly and Reinforcement Layout (Su et al 2009)
The test specimens constructed were varied in their reinforcement (A-series), beam
geometry (B-series), and the loading speed (C-series). The A-series specimens varied
in flexural reinforcement ratio (from 2ϕ12 to 3ϕ14), with the cross section
dimensions being the same (150 mm wide (b), 300 mm deep (h), span length (ln)
1225 mm (span to depth ratio of 4.08). In the B-series, the cross sections of all the
specimens were the same as for the A-series, but their span to depth ratio (ln/h) was
variable, with the range being from 6.58 to 9.08. The C-series tests examined the
effects of loading rate. This series of tests was chosen to examine the influence of
dynamic loading caused by rapid column removal. Since in this study, it was
assumed that loading acts in a static manner, only the A and B-series of the tests
were modelled.
The concrete cube strength fcu for the specimens in this experiment varied from 23.3
to 39 MPa. The yield strength for the steel reinforcement varied from 290 to 340
69
MPa, and the reinforcement diameter varied between 12 and 14 mm. Table 3-2 lists
the main parameters of the tests simulated in this research.
Table 3-2: Specimen Properties
Test
b × h
(mm)
ln
(mm)
fcu
(MPa)
Longitudinal
Reinforcement
Ties fctm
(MPa)
Ec
MPa
Top Bottom
A1 150 x
300
1225 32.3 2ϕ12 2ϕ12 ϕ8
@100
3.05 34214.23
A2 150 x
300
1225 35.3 3ϕ12 3ϕ12 ϕ8
@80
3.24 35042.98
A3 150 x
300
1225 39.0 3ϕ14 3ϕ14 ϕ8
@80
3.46 36013.98
B1 150 x
300
1975 23.3 3ϕ14 3ϕ14 ϕ8
@100
2.45 31449.95
B2 150 x
300
2725 24.1 3ϕ14 3ϕ14 ϕ8
@120
2.51 31715.64
B3 150 x
300
2725 26.9 3ϕ14 2ϕ14 ϕ8
@120
2.67 32455.72
The load was applied through the centre column. During the test, the vertical
displacement and load at the centre column stub were measured by a built-in load
cell and a displacement transducer. The horizontal and the vertical reaction forces at
the beam ends were measured at the support columns; however the exact locations of
measurement was not specified in the test report (Su et al., 2009). Figure 3-11 shows
the test setup. For the A- and B- test series, the load was applied with displacement
control at a constant rate of 5 mm/min, except for initial state and during the pauses
for inspection. In the C test series, the displacement rate was 0.2, 2, and 20 mm/s.
70
Figure 3-11: Test Specimen and Schematic Illustration of Test Setup (Su et al, 2009)
The two end columns were restrained axially, vertically and rotationally to simulate
the possible restraints applied to the concrete beam ends by the rest of the structure
in real building. To obtain the horizontal and rotational stiffness of the beam ends,
the corresponding displacement of the side columns were measured, but the exact
locations of measurements were not given in the original publication (Su et al 2009).
The reported values for horizontal and rotational stiffness of the connection between
the end column and the steel socket are 1000’000 kN/m and 17,500 kN-m/rad
respectively.
Using these reported stiffness values caused the FE model to predict higher
compressive arching action, and correspondingly higher axial reaction forces than
the test results; meaning that the used stiffness values were higher than what have
been applied on the test specimens. Through a correspondence between the author
and the researchers of the experiments (Miratashi, 2011), it was established that the
actual boundary stiffness may vary from what was reported. The researchers
suggested values of 1/10th
of the reported ones, stating that such values were
obtained from more precise measurements of their similar ongoing tests using the
same apparatus. Applying the modified values again overestimated the arching
action and corresponding axial reaction force at the beam ends. Hence to find the
71
appropriate restraint stiffness values, the author had to conduct an extensive set of
numerical simulations until one set of restraint stiffness values gave simulation
results consistently being in good agreement with the experimental results for all the
tests. These values were 1500 kN/m and 75 kN-m/rad for the axial and rotational
stiffness respectively.
3.3.2 Typical Beam Behaviour to Reach Catenary Action
Model A3 is used to show different stages of behaviour of the beams. Figure 3-12
shows the applied load on this beam and the horizontal reaction force in the supports.
Initially, steel acted in an elastic manner (up to Point B), and because the beam
deflection was very small, there was very little axial reaction force in the beam (up to
Point A). At larger deflections, the lower edge of the concrete beam at the supports
bore against the supports, thus developing compressive arching action.
The development of compressive arching action continued until the maximum at C
and D. Afterwards, the maximum compressive stress in concrete was reached and the
compressive membrane action force decreased. This was accompanied by a
reduction in the applied load that can be resisted by the beam (from C to E in
Figure 3-12). When the beam deflection was about ½ of the beam depth,
compressive arching action diminished and the beam quickly underwent large
deflections and activated catenary action (point E). Catenary action was stable and
the applied load on the beam increased until failure of the beam due to reinforcement
fracture (point F).
72
Figure 3-12: Behaviour of Stages of model A3
3.3.3 Finite Element Model
Due to the symmetry in both the test specimen geometry and loading, the finite
element model simulated only a quarter of the beam. For the sake of simplicity and
computational efficiency, the side columns were not included in the model; instead,
their effects were represented by a rotational stiffness of 75 kN-m/rad and an axial
stiffness of 1.5E3 kN/m. The downward displacement in the tests was modelled as a
prescribed displacement on the model beam end (corresponding to the actual beam
centre column stub). The test beam was doubly reinforced with stirrups spaced at
distances of 80-120 mm along the beam.
For the concrete beam, twenty noded solid brick quadratic elements (CHX60) were
used. This element type is consistent with the selected method of modelling the
reinforcement in DIANA. Both the longitudinal and transverse reinforcements were
modelled by the embedded reinforcement option available in DIANA.
-400
-300
-200
-100
0
100
200
300
0 0.05 0.1 0.15 0.2 0.25
Ho
rizo
nta
l R
eact
ion
V
erti
cal
Lo
ad
(kN
)
Vertical Displacement (m)
Test A3
A
BC
D
E F
73
Table 3-3: FE models divisions along 3D axis for mesh sensitivity study
Model Division Along
x-axis
Division Along
y-axis
Division Along
z-axis
A 1 1 8
B 1 2 16
C 1 2 24
D 2 4 32
E 3 8 32
F 3 8 48
G 4 10 56
H 4 10 64
The mesh size chosen for the model was based on the results of a mesh sensitivity
study using eight different element sizes (Table 3-3). Figure 3-13 shows the results
of mesh sensitivity study. Based on this result, mesh E can be used.
Figure 3-13: Results of mesh sensitivity study
0
50
100
150
200
250
A B C D E F G H
Ver
tica
l L
oa
d (
kN
)
Models
74
Figure 3-14: Finite element model
The prescribed displacement was applied on a node in the middle of the beam, and
the mentioned node was constrained vertically with adjacent nodes within a distance
of 125 mm to represent the middle column stub of the actual test set up. The far end
surface of the model is restrained totally in the vertical (y) direction and rotational
and axial stiffness have been provided by spring elements attached to the nodes of
this surface. Figure 3-14 shows a typical finite element model. The solution method
chosen for this model was the tangential regular Newton method which revealed
better results than the other solution methods.
3.3.4 Comparison between Simulation and Test Results
Figure 3-15 and Figure 3-16 show comparison between the simulation and test
results for the A and B-series of tests respectively, showing the applied load- and
axial reaction-vertical displacement relationships. In all cases, the observed arching
action and catenary action stages were closely followed by the numerical simulation
results.Furthermorethetests’failurewasdefinedbasedontheruptureofthetensile
reinforcement in the bottom of the section in the beams close to the centre column
stub. This is shown with the terminating points of the diagrams.
Prescribed
displacement
75
(a)
(b)
(c)
Figure 3-15: Comparison of experiment and FE results for A-series
Due to lower span to depth ratio of the A-series specimens, the development of
arching action was more than the B-series. This resulted in higher applied load to the
structure and less vertical displacement. In models A3 and B1, as the tensile
-500
-400
-300
-200
-100
0
100
200
300
0 0.05 0.1 0.15 0.2 0.25
Ho
riz
on
tal R
ea
cti
on
Verti
ca
l L
oa
d
(kN
)
Vertical Displacement (m)
Test A1
FE A1
-400
-300
-200
-100
0
100
200
300
0 0.05 0.1 0.15 0.2 0.25
Ho
riz
on
tal R
ea
cti
on
Verti
ca
l L
oa
d
(kN
)
Vertical Displacement (m)
TestA2
FE A2
-400
-300
-200
-100
0
100
200
300
0 0.05 0.1 0.15 0.2 0.25
Ho
riz
on
tal R
ea
cti
on
Verti
ca
l L
oa
d
(kN
)
Vertical Displacement (m)
Test A3
FE A3
76
reinforcement in the bottom of the section had a higher cross sectional area, concrete
reached its compressive strength and there was a sudden drop of the applied load.
In B-series models due to higher span/depth ratio the tensile reinforcement enabled
the models to develop more into the catenary stage and hence in them the final
applied load was higher than the load due to arching action. The axial reaction force
of all the models depict that after arching action (where the maximum axial force
occurred) the decrease in the axial force is a representation of onset of the catenary
action.
77
(a)
(b)
(c)
Figure 3-16: Comparison of experiment and FE results B-series
The presented results manifest the capability of the finite element modelling for
capturing the concrete cracking, crushing and its interaction with the reinforcement
inside. It is shown that by proper plasticity model of concrete and its cracking, the
FE model is able to predict the behaviour of the specimens in catenary action stage.
-300
-250
-200
-150
-100
-50
0
50
100
150
200
0 0.1 0.2 0.3 0.4 0.5
Ho
riz
on
tal R
ea
cti
on
Verti
ca
l L
oa
d
(kN
)
Vertical Displacement (m)
FE B1
Test B1
-250
-200
-150
-100
-50
0
50
100
150
200
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Ho
riz
on
tal R
ea
cti
on
Verti
ca
l L
oa
d
(kN
)
Vertical Displacement (m)
Test B2
FE B2
-250
-200
-150
-100
-50
0
50
100
150
0 0.1 0.2 0.3 0.4 0.5
Ho
rizo
nta
l R
eact
ion
V
erti
cal
Lo
ad
(kN
)
Vertical Displacement (m)
Test B3
FE B3
78
Also the plasticity model used for reinforcement was able to simulate the rupture of
the reinforcement close to the test results.
3.4 Verification of Bond-Slip Modelling
In the tests of Su et al. (2009), the longitudinal reinforcement was throughout the
length of the beam. Therefore, there was no reinforcement pull-out. However,
reinforcement pull-out may occur when using tie bar if the tie bar length is not
sufficiently long. It is important that the tie-bar bond-slip behaviour is accurately
modelled. For this purpose, the experiments conducted by Engström et al., 1998
were used. These experiments focused on the global bond-slip behaviour of the steel
tie-bar. Some of the test results used here, related to same series of experiments, are
reported in (Huang et al., 1996).
3.4.1 Test Specimens and Variables
Two types of specimens (shown in Figure 3-17) were modelled: I) concrete cubes
(400mm) with the steel rebar being placed in the centroid; II) concrete cuboids of
cross section 400 × 400 mm and the length of 400 or 500 mm with the steel rebar
being placed in the middle of the side to check the effect of the concrete cover on the
bond-slip behaviour. This type of specimen was provided with an extended nose-like
part to help the support of specimen in the testing machine and to balance the
eccentricity of the applied load. Both high strength and normal strength concretes
were considered for both specimen types.
79
Series-I Series-II
Figure 3-17: Test specimens of type I and II (Engström et al., 1998)
The embedment length of the steel rebar in concrete varied between 90 to 250 mm
for specimens made with high strength concrete, and between 150 to 500 mm for
normalstrengthconcrete.Thesteel rebar’snominalyieldstresswasreported tobe
500 MPa but the actual value from the tensile test was recorded as 569 MPa. The
steel rebar diameter was 16 mm in all specimens.
The concrete cover to the reinforcement in specimens of type (I) was 192 mm
(=12ϕ). The concrete cylindrical compressive strength was reported to be about 30
MPa for normal strength type and 110 MPa for high strength concrete. The fracture
energy for normal concrete ranged from 110 N/m to 165 N/m for the high strength
type. The tests were carried out after 40 days of concrete casting. Table 3-4 lists the
testspecimens’characteristics.
80
Table 3-4: Characteristics of the test specimen (Engström et al., 1998)
Test Types of
specimen
Bar
Location
Concrete
compressive
Strength
(MPa)
Concrete
Cover
(mm)
Embedment
Length
(mm)
N290 I centroid 30.6 192 290
N500 I centroid 26.8 192 500
H170 I centroid 101.5 192 170
N290m-16 II Mid-edge 28.6 16 290
H170m-16 II Mid-edge 110.9 16 170
The loading was applied by at the rate 0.1 mm/min at the end of the rebar protruding
out of the concrete block (the active end of the rebar). Slip was measured at the end
of the steel rebar relative to the nearest concrete.
3.4.2 Finite Element Model
For modelling bond-slip, DIANA introduces the reinforcement by interface elements
which have normal and tangential stiffness defined by user. The behaviour of the
interface element in the normal direction is assumed to be linear, but the behaviour
in the tangential direction (between shear stress and strain) can be considered as
nonlinear. There are two predefined material models for the relationship between
shear stress and strain and the user is provided with an option to define this
relationship. The relationship in CEB-FIP Model Code 2010 (section 2.6) was used
herein.
Table 3-5: Calculated concrete material properties (based on CEB-FIP Model Code 2010)
Model fc
(MPa)
ft
(MPa)
Ec
(MPa)
τmax
(MPa)
τy
(MPa)
τf
(MPa)
τy,f
(MPa)
lmin
(mm)
H170 101.5 6.21 46550.8 45.67 34.25 18.27 9.13 49.83
N290 30.6 2.41 31213.73 13.77 10.32 5.5 2.75 165.28
N500 26.8 2.13 29864.14 12.06 9.04 4.82 2.41 188.72
N290m-16 28.6 2.26 30518.3 12.87 9.6525 5.148 2.574 176.84
H170m-16 110.9 6.62 47945.6 49.90 37.42 19.96 9.981 45.60
The tensile strength and elastic modulus of concrete were calculated using
Equation 3-9 and Equation 3-10. However, for the elastic modulus, the CEB-FIP
Model Code advises that when the actual compressive strength of concrete at the age
of 28 days is known, the term Δf must be dropped from the relationship which is the
81
case for this experiment; and in the tensile strength relationship Δf must be deducted
from fcm (CEB-FIP, 2010). Since the steel rebar used in all of the test specimens is
the same, the slip values for the bond-slip diagram are constant (as slip is related to
the rib spacing of rebar). The calculated values for bond stress based on CEB-FIP
Model Code 2010 are shown in Table 3-5.
Figure 3-18: Concrete block FE model for simulation of bond-slip tests
Specimen Type I
Specimen Type II
Load
Rebar
82
The concrete model used in the finite element model to simulate these tests is
smeared cracking with Mohr-Coulomb plasticity and linear tension softening
behaviour. For better results and time efficiency, the 8-noded solid elements
(HX24L)wereusedfortheconcreteandtheinterfaceelements(“barinsolid”type,
HX30IF) were chosen to simulate the bond-slip interface between the concrete and
steel rebar. The reinforcement was modelled using the 2-noded truss element
(L6TRU) compatible with the solid element of concrete and bond-slip interface
element. The element size was 50 mm.
As reported in the experiment, the concrete block was supported in the axial
direction (in the same direction as loading) on the loading face by steel plates
covering 10 cm of the concrete surface on each side of the reinforcement in type I. In
the finite element model the other end of the specimen was constrained in the four
corner points to avoid any trivial movement. For type II, the nose-shape part of the
concrete block is restrained in all directions and the bottom of the concrete block
close to the reinforcement is restrained in the parallel direction to the reinforcement (
Figure 3-18).
3.4.3 Comparison between Simulation and Test Results
Figure 3-19 compares the load-slip curves. The simulation results are very close to
the test results. For test specimen type I, in the load-slip diagram after the maximum
load is reached, there is a plateau which marks yielding of the steel rebar, and this is
accurately captured in the numerical simulation. Afterwards, the applied load
decreases, reflecting pull-out failure or rupture of the reinforcement; again the
numerical simulation results follow the test results closely.
83
(a)
(b)
(c)
Figure 3-19: Comparison of experiments with FE results
For specimens type II, since the concrete cover is not enough to provide the
reinforcement with adequate bond, the reinforcement is not yielded before being
pulled-out (Figure 3-20). Here after the maximum load is reached, the bond stress
declines which marks the failure of the model.
0
20
40
60
80
100
120
140
0 2 4 6 8 10 12 14 16
Axia
l L
oa
d (
kN
)
Slip (mm)
FE
N290b
0
20
40
60
80
100
120
140
0 5 10 15 20
Axia
l L
oa
d (
kN
)
Slip (mm)
Experiment N500
FE
0
20
40
60
80
100
120
140
0 2 4 6 8 10 12 14 16
Axia
l L
oa
d (
kN
)
Slip (mm)
FE
Experiment H170
84
(a)
(b)
Figure 3-20: Comparison of experiment and FE results, Type II
Figure 3-21 showsthedeformedshapeofthemodel’smesh:
Figure 3-21: Bond-Slip specimens deformed mesh, Types I and II
0
20
40
60
80
100
120
0 2 4 6 8 10 12 14 16
Axia
l L
oa
d (
kN
)
Slip (mm)
Experiment N290m-16
FE
0
20
40
60
80
100
120
140
0 2 4 6 8 10 12 14 16
Axia
l L
oa
d (
kN
)
Slip (mm)
FE
H170m-16
85
3.5 Summary
In this chapter the capability of the commercial finite element software DIANA was
evaluated against analysis of concrete in large deformation in order to capture the
arching and catenary action behaviour of reinforced concrete structural members. It
was shown that the most suitable material model for concrete was the Mohr-
Coulomb plasticity in compression regime along with the tension softening in
tension. The reinforcement was modelled as embedded rebar in concrete with Von-
Mises perfect plastic behaviour. The suitable mesh for concrete was determined in a
sensitivity study. The FE model was able to predict the failure of the Su, et al. (2009)
experiments with good agreement. The model was able to depict catenary and
arching actions.
The material models of concrete and steel, and the bond-slip interface of DIANA
proved to be a powerful tool in predicting the pull-out tests of Engström et al.,
(1998). The bond-slip relationship was calculated based on the relationships
provided by CEB-FIP Model Code 2010.
86
Chapter 4. Parametric study of 2D Restrained
Slab
This chapter presents the results of an extensive parametric study of design
parameters on the behaviour and load carrying capacity of 2-D concrete floor slabs.
The objectives of this parametric study are:
(1) To investigate the effects of different design parameters with a view of how to
improve slab resistance;
(2) To provide a comprehensive database of results for the development and
validation of an analytical method that may be used as a design tool.
Figure 4-1 shows the simulated structure. It represents the accidental loading
condition of losing the centre column/beam support of a two-span precast panel
structure. The parameters considered here that affect the behaviour and load carrying
capacityoftheslabaretakenfromtheconnectionsandtheslab,including:tiebar’s
length, diameter, position, yield stress; slabs’ height and length; concrete grouting
compressive strength; and the stiffness of the constraint by which the slabs are
connected to the rest of the structure.
Figure 4-1: Two-dimensional representation of the slabs with pinned BC
Precast Concrete Slabs
Steel Tie Bar
Lost Column
87
The results will provide insight into whether or not the tying mechanism is able to
provide sufficient load carrying capacity under the accidental loading condition.
Suggestions will be made based on the condition of the connection between the two
slabs in order to enhance its performance in the case of an accidental action.
4.1 Illustrative behaviour
4.1.1 Slab with total axial restraint
To illustrate the complete range of the behaviour of the structural system, this section
presents the simulation results of two precast concrete floor slabs which have lost
their supporting column at the centre, shown in
Figure 4-1. A point load (as a prescribed displacement) was applied at the centre of
the two slabs to simulate the applied floor loads. Herein it was assumed that the
connection to the slabs at their far ends was capable of providing enough rotational
capacity and that they will not rupture.
For this model, the slabs are typical of those used in residential floor system with the
following dimensions: length 5 m, depth 265 mm and width 1.2 m. The
reinforcement diameter is 16 mm to provide the required tying resistance according
to British Standard (BS8110-1, 2007). The concrete is modelled using smeared
cracking with Mohr-Coulomb plasticity and Linear Tension Softening, and the tie
bar in the slabs is modelled using Von-Mises plasticity. Table 4-1 lists the material
properties used in this model:
88
Table 4-1: Material Properties used in FE Model
Concrete
Ec 33842.32 MPa
ν 0.2
fc 39 MPa
ft 2.97 MPa
Gf 0.0778 N.mm/mm2
Steel (Tie Bar)
Es 210000 MPa
ν 0.3
fy 500 MPa
As explained in Chapter 3, the general finite element software DIANA was adopted
in this study and concrete was modelled in two dimensions using the 8-noded
quadrilateral plane stress elements (CQ16M) which were able to predict concrete
cracks and the following catenary action properly. To model the bond-slip behaviour
between the tie bar and the surrounding concrete, the 2-noded line interface elements
of L8IF and for the tie bar the 3-noded truss elements of CL6TR were used.
The element sizes were chosen based on a sensitivity study, as explained in Chapter
3. Figure 4-2 shows the finite element mesh of the half of the structure with the
applied boundary condition. The mesh around the tie bar in the connection zone,
between the slabs, has been chosen to be finer than the other parts of the structure.
Figure 4-2: Mesh view of half of the model (left slab)
Concrete Slab
Steel Tie Bar
89
Figure 4-3-a shows the load-deflection results of the structure of the node on the top
of the connection. The structural behaviour goes through a number of stages before
failure due to the fracture of the tie bar. At the start of loading on the structure, the
behaviour is linear elastic while concrete is contributing to tensile resistance. This
stage terminates when the concrete at the bottom of the connection between the two
slabs reaches its tensile strength. This can be confirmed by the plot of the axial stress
in the bottom of the section (Figure 4-3-b). After the concrete has reached its tensile
strength and has developed tensile cracks, the applied load drops before increasing
again due to arching action in concrete.
90
(a)
(b)
(c)
(d)
Figure 4-3: FE Results for the 2D slab Model
-15
-10
-5
0
5
10
15
20
25
30
35
0 100 200 300 400 500
Load
(kN
)
Vertical Displacement (mm)
Accidental Action Load
FE Pinned BC
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
0 100 200 300 400 500Axi
al S
tre
ss in
Mid
dle
Bo
tto
m E
lem
en
t (M
Pa)
Vertical Displacement (mm)
-35
-30
-25
-20
-15
-10
-5
0
5
0 100 200 300 400 500
Axi
al S
tre
ss in
Mid
dle
To
p E
lem
en
t (M
Pa)
Vertical Displacement (mm)
-1200
-1000
-800
-600
-400
-200
0
200
400
0 100 200 300 400 500
Axi
al R
eac
tio
n F
orc
e (k
N)
Vertical Displacement (mm)
Tie Bar Strength
Tie Bar Axial Reaction Force
91
In arching action mechanism, the axial stress in the middle top element of the
concrete develops a negative, compressive, value (Figure 4-3-c) and at the same time
the two pinned constraints at both far ends of the slabs are pushed away from each
other which causes the increase of negative axial reaction force of the end restraints
of the slabs, as shown in Figure 4-3-d.
This behaviour continues until the vertical displacement of the slabs’ connection
reaches a point where the slab tips will cease pressing against each other and start to
move away from one another. As it can be seen in Figure 4-3-c and Figure 4-3-d, the
maximum negative values of these two diagrams happen at the same vertical
displacement, which shows by splitting the concrete slabs’tipsthevalueofreaction
force starts to decrease too. After this stage, the compressive force is released and
the structure develops catenary action in order to sustain the applied load.
During the catenary action stage, the only load carrying part of the model is the steel
tie bar and this makes the axial reaction force of the restraints to be equal to the yield
force of the tie bar (Figure 4-3-d), which at this stage has completely yielded. In fact,
as shown in Figure 4-4, the tie bar has yielded in bending, long before the
development of catenary action. Based on the input stress-strain curve, the tie bar
fractures when its maximum strain has reached 20% and rupture of the tie bar marks
the completion of the numerical model (Figure 4-4-a). Figure 4-4-b shows the tie bar
stress – structural deflection relationship.
92
(a)
(b)
Figure 4-4: Axial tie bar stress in the connection
The horizontal reaction force on the pinned boundary condition depicts that, at the
very early stages of loading, while slabs’connectionhasalinearelasticbehaviour;
there is no considerable amount of force exerted to the restraints (Figure 4-3-d). But
as the tensile strength of concrete is breached and by the onset of the arching action,
compressive forces are developed in the pinned restraints and this continues until the
arching action is transferred to the catenary action due to separation of the two
concrete slabs. From this point on, the value of the tying force is equal to the product
of the yield stress of the steel and the cross sectional area of the tie bar.
The variation of axial stress of the concrete element near the connection region
between the slabs is a good representation of the behaviour of the slabs, as it can
show the onset and termination of arching action, the occurrence of final crack, and
whether the concrete in the compressive region reaches its strength or is cracked
before that. This can be shown with the plots of the axial stress in the region of
connection between the two slabs with the crack propagation at the main stages of
the behaviour.
0
0.05
0.1
0.15
0.2
0.25
0 100 200 300 400 500
Stra
in
Vertical Displacement (mm)
0
100
200
300
400
500
600
0 100 200 300 400 500
Axi
al S
tre
ss M
idd
le T
ie B
ar E
lem
en
t (M
Pa)
Vertical Displacement (mm)
93
a b
c d
Figure 4-5: Axial Stress and Crack Pattern of Slabs' Connection
Figure 4-5-a, shows the concrete stress and crack state just after the linear elastic
limit of the connection. As it can be seen at this stage the axial stresses on top and
bottom of the section are almost the same, however the top fibre of concrete has
94
negative compressive values and the concrete in the bottom fibre has just passed the
tensile strength of the concrete.
After this stage because of the crack opening in the bottom of the section, the arching
action starts which leads to increased stress in the top fibre of the concrete in the
connection zone. Simultaneously with this phenomenon, the crack propagation
happens from the tensile zone in the bottom of the section (Figure 4-5-b). At the time
when the arching action has caused its maximum compressive effect in the concrete,
the vertical displacement of the connection reaches a value that the two slabs start
moving away from one another.
While the slabs move away from each other, cracks take over the whole height of the
connection, and as can be seen in Figure 4-5–c the top fibre of concrete in the
connection zone reaches its tensile strength and the final crack of the concrete
section occurs. From this point cracks have reached the top fibre of the connection
and concrete in this region does not carry any load, and it will be the tie bar that
provides the resistive load and helps to develop the catenary action (Figure 4-5-d).
4.1.2 Slabs with Elastic Axial Restraint
In realistic structures, the slabs will be connected to other structural members, such
as beams and walls. These components will provide flexible, rather than rigid,
support to the slabs. This section illustrates how the slab behaves if the axial
restraints are elastic. As shown in Figure 4-6, it is assumed that the two far ends of
the slabs are connected via a horizontal spring to fixed points and the slab ends are
vertically fixed.
Figure 4-6: Schematic 2D Slabs with Elastic Axial Restraints
The axial restraint stiffness to the slabs may be estimated as follows: The stiffness
comes from the remaining columns acting as one fix-ended beam of span 2L under a
point load at its centre (Figure 4-7, Figure 4-8).
95
Figure 4-7: Columns acting as fixed beams
The stiffness of a beam under a point (Figure 4-8) is given in Equation 4-1:
Figure 4-8: Calculation of stiffness for a fixed beam under a point load
Equation 4-1
Using typical column sizes in steel structures with PCFS (Fu and Lam, 2006), (Fu et
al., 2008), a restraint stiffness value of about 50 kN/mm was obtained. Due to infill
walls, the real stiffness may be much higher (Sasani, 2008). In the case of assessing
robustness of one or two slabs of a floor losing their vertical supports, the axial
Lost Support
L
L
PCFS
Remaining Columns
F
F
2L
96
restraint stiffness to these slabs can be orders of magnitude higher if the effect of in-
plane stiffness of the rest of the floor slabs is considered. So in the author’s
parametric study, the axial restraint value ranges from 100 to 10000 kN/m.
Comparing the load-deflection results in Figure 4-3 and Figure 4-10, it can be seen
that the slabs with rigid and flexible axial restraints undergo similar stages of
behaviour. However, during the initial stage, the development of compressive (arch)
membrane action is reduced as the supports become more flexible. Before full
development of catenary action, slabs with more flexible axial restraint (lower
Boundary Condition (BC) stiffness) experience less catenary force at the same slab
deflection, hence the slab resistance to the applied load is lower. Therefore, at the
same elongation of the tie bar (hence similar bar force and tensile stress in concrete),
the vertical displacement of the slab with lower BC stiffness is higher, as shown in
Figure 4-9, where L=L1=L2). This increased vertical deflection results in postponing
total concrete cracking, as shown in the load-deflection curves in Figure 4-10.
Figure 4-9: Effect of Horizontal Displacement on Vertical Deflection
Once the concrete slab has totally cracked through the thickness, the concrete stress
drops to zero (Figure 4-10-b) and the tension force is that of the tie bar (Figure 4-10-
c). Once the tie bar has reached its maximum resistance (yield force), the horizontal
displacement of the slab remains constant. At this stage, since the slab resistance
comes from the tie bar tensile force, acting on the slab vertical deflection, the load –
Δ
Δ
h2
h1 L1
L2
L
97
vertical deflection relationship of the slabs coincide for all the different BC stiffness
values.
(a)
(b)
(c)
Figure 4-10: Behaviour of slabs with elastic axial restraint
As shown in Figure 4-10(a), in some of the models analysed, the maximum load
carrying capacity of structure before total cracking of concrete is higher than the
maximum resistance after total cracking when the tie bar provides all the tensile
-15000
-10000
-5000
0
5000
10000
15000
20000
25000
30000
35000
40000
0 100 200 300 400 500
Load
(N
)
Vertical Displacement (mm)
Accidental Action LoadBC Stiffness 50kN/mmBC Stiffness 100 kN/mmBC Stiffness 1000 kN/mmBC Stiffness 10 000 kN/mm
-30
-25
-20
-15
-10
-5
0
5
0 100 200 300 400 500
Axi
al S
tre
ss in
Co
ncr
ete
Mid
dle
To
p
Ele
me
nt
(MP
a)
Vertical Displacement (mm)
BC Stiffness 50 kN/mm
BC Stiffness 100 kN/mm
BC Stiffness 1000 kN/mm
BC Stiffness 10000 kN/mm
-1200
-1000
-800
-600
-400
-200
0
200
400
0 100 200 300 400 500
Axi
al R
eac
tio
n F
orc
e (k
N)
Vertical Displacement (mm)
BC Stiffness 50 kN/mm
BC Stiffness 100 kN/mm
BC Stiffness 1000 kN/mm
BC Stiffness 10000 kN/mm
98
resistance. Since concrete cracking is brittle, using the strength of the slabs before
total cracking of the concrete for providing robustness of the system, is not
recommended.
4.1.3 Slabs with Partial Tie Bar
In the previous systems, the tie bars are through the entire length of the slabs. It is
possible for slabs with partial tie bars (tie bar length < total slab span) to develop
catenary action, provided the slab portions without tie bar are not cracked and the tie
bar length offers enough bond between the tie bar and the surrounding concrete for it
not to slip through.
The results of such a case are shown in Figure 4-11, with a tie bar length of 2.5 m in
each slab. It can be seen that even after the final cracking of the connection between
the slabs, the load-displacement diagram of the slabs shows the development of
catenary action. Figure 4-12 shows that in this case, the concrete in the unreinforced
region is not cracked in tension.
Figure 4-11: Partial Tie Bar, development of Catenary Action
99
Figure 4-12: Crack Pattern, Partial Tie Bar
However, such a load carrying mechanism would be sensitive to the tie bar size. If
the tie bar size is large, total concrete cracking can happen in the unreinforced region
of the concrete grouting between the slabs or infill of the hollowcores, and the
structure model would not be able to carry any load (Figure 4-13).
Figure 4-13: Crack Pattern, Partial Tie Bar, showing concrete cracking in the unreinforced
zone of grouting
To demonstrate this effect, a model with tie bar diameter of 64mm (Figure 4-14) was
simulated. The slab possesses no strength due to negligible development of catenary
action.
100
Figure 4-14: Partial Tie Bar, large tie bar size resulting in total crack of concrete grouting in the
unreinforced zone
To further confirm this, Figure 4-15 compares the tensile stress development in the
unreinforced region of the grouting between slabs in the above two cases. It can be
seen that in the case of the smaller tie bar, the tensile stress in concrete is lower than
the tensile strength. But for the slab with large tie bar diameter, the tensile stress
breaches the tensile strength and the concrete stress diminishes after cracking. In the
model with smaller tie bar, the force obtained by the product of the axial stress, if
added for all the elements in height of the slab, and multiplied by the cross sectional
area of the slab is the same as the force derived by the product of the steel cross
sectional area and its yield stress.
Figure 4-15: Axial Stress in Unreinforced Region, a) no crack through in the unreinforced
region (small tie bar); b) Crack through in the unreinforced region (large tie bar)
-20
-10
0
10
20
30
40
50
60
0 100 200 300 400 500
Load
(kN
)
Vertical Displacement (mm)
101
Based on the above results, it can be concluded that it is possible for damaged slabs
(simulating internal support removal) to resist the accidental loads through the
development of catenary action. However, the provision of suitable tie bar is a key
factor. In the following section, an extensive set of parametric studies will be carried
out.
4.2 Accidental Load Calculation
The following calculations were performed to check whether or not the damaged
slabs can resist the accidental load in order to provide resistance against progressive
collapse:
gk = qk = 3.8 kN/m2
Length of each slab L = 5 m
The accidental load on the slabs with the above permanent and variable loads is:
Accidental Loading =
This uniformly distributed load (UDL) gives a total load of 61.2 kN on the two slabs
with a total length of 10 m and a width of 1.2 m. In the numerical model the total
load was applied as a point load at the centre of the two slabs. In order to give the
same maximum bending moment in the slabs, the applied point load should be
halved (30.6 kN).
Comparing the simulation result of ultimate load carrying capacity of 21 kN (in
catenary action) in Figure 4-3-a and Figure 4-10-a for the slabs with the required
accidental limit load, it appears that the slabs are not able to resist the accidental
load. Figure 4-3-a shows that the slabs can resist a force equal to the accidental load
under compressive arching action. However, it is not advisable to use compressive
arching action for structural robustness because compressive arching action is very
sensitive to axial restraint and has a brittle failure mode. Similarly Figure 4-10-a
shows that the model with the lowest BC stiffness provides a resistance just more
than the accidental load before final cracking of concrete.
102
Appendix 1 presents further simulation results for slabs that have been designed
according to the current British Standard regulations (BS8110-1, 2007) and
presented in the manufacturer’s load-span table (Table 2-3) . The results again show
that with the current tying force requirement, it is not possible to provide resistance
against the design accidental loading on the structure in a reliable manner by
developing catenary action. The current tying force requirement neglects the
important factor of deformation capacity of the structure.
In the next section, a parametric study will be conducted to examine how different
design parameters may affect the development of catenary action in slabs, with the
principal aim to identify methods that can increase catenary action resistance.
4.3 Parametric Study
For determining the methods of enabling damaged precast concrete floor slabs
(simulating removal of the centre support) to resist progressive collapse, the effect of
each of the following parameters on tie bar elongation is investigated:
Precast concrete slab’s height
Precast concrete slab’s span
Tie bar length
Tie bar position
Tie bar diameter
Tie bar yield Stress
Grouting concrete strength
Ultimate strain of the steel tie bar
Figure 4-16 shows the basic geometric and material properties of the slabs. When
carrying out the parametric studies, the values of the parameter which is investigated
are changed while other values are kept constant. As changing each of the above
parameters may lead to a change in the concrete cracking trend, for each of the
parameters, a new mesh sensitivity study was conducted to prevent the convergence
problems inherent in the smeared cracking model of concrete. The presented results
are the outcome of about five thousand simulations.
103
Figure 4-16: Slab's reference case dimensions
4.3.1 Height of the Slab
The range for the slabs height (150 mm- 450mm) is basedon themanufacturer’s
product listings (Bison, 2012). In all cases, the cover to the tie bar is 40 mm.
Figure 4-17 shows cross-sectional dimensions of the slabs.
Tie Bar Diameter = 16 mm
Tie Bar Height = 45 mm
5 m
265 mm
104
Figure 4-17: Hollowcore Sections (Bison, 2012)
The variation of the maximum vertical load, vertical displacement and strain of the
tie bar in the connections are shown in Figure 4-18. With this tie bar diameter, slabs
with heights larger than 350 mm are not able to develop catenary action, and the tie
bar reaches its rupture strain before the cracks reach to the top of the concrete
section. In this situation the structure is relying on the compressive forces in the
concrete only. This could cause an abrupt failure under accidental loading, and hence
is not a reliable mechanism for enhancement against progressive collapse. On the
other hand, as the slab height decreases the tie bar in the connection has the
opportunity of providing the catenary action since the concrete of the connection is
cracked totally before the tie bar is ruptured.
105
(a)
(b)
(c)
(d)
Figure 4-18: Effects of varying Slab Height on Vertical Load and Axial Reaction Force
0
5
10
15
20
25
30
35
40
45
50
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
sh150
sh200
sh250
sh300
sh350
-300
-200
-100
0
100
200
300
400
0 100 200 300 400 500 600
Axia
l R
eact
ion
Fo
rce
(kN
)
Vertical Displacement (mm)
150
200
250
300
350
-6
-5
-4
-3
-2
-1
0
1
2
3
4
0 100 200 300 400 500 600
Axia
l C
on
crete
Str
ess
(MP
a)
Vertical Displacement (mm)
150
200
250
300
350
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 100 200 300 400 500 600
Tie
Str
ain
Vertical Displacement (mm)
150
200
250
300
350
106
As shown in Figure 4-19, the catenary action resistance is the catenary action force
multiplied by the distance between them (=slab vertical deflection h + distance from
the slab centre to the position of catenary action force d); therefore, increasing the
slab thickness increases the distance between the catenary action force and the centre
of the cross-section, thus increasing the vertical load when the vertical displacement
of the slabs is the same as shown in Figure 4-18-a. It is clear from Figure 4-18-c that
the concrete is totally cracked for all slab thicknesses so final failure of the slabs is
governed by reinforcement rupture. By increasing the slab thickness, the strain in the
tie bar also increases at the same vertical displacement of the slabs as shown in
Figure 4-18-d. This results in earlier rupture of the tie bar.
Figure 4-19: Force diagram
As a summary, as the applied loads in the case of thick slabs are likely to be higher,
but the catenary action resistance does not increase by the same proportion when the
slab thickness is increased. As the slab thickness increases, catenary action ceases to
become an effective means of preventing progressive collapse. Hence it is
L
h
F
𝑃
F d
107
recommended that a balance should be struck between the compressive arching
action of concrete and tensile force exerted from steel tie bar.
4.3.2 Slab Span
Precast concrete floor slabs are able to cover structure spans of as long as about 10
m. In this study the effect of changing the slabs’span is investigated for the range of
spans from 3 to 10 m. As by changing the slab thickness, changing the slab span
changes the ultimate limit state resistance of the undamaged slabs and hence alters
the required accidental load.
Taking the slab span/depth ratio as the controlling parameter, then the effect of
increasing slab span is similar to that of reducing the slab depth. As shown in
Figure 4-20, as the slab span decreases, the vertical load in the slab increases.
However, the reinforcement fractures at lower slab deflections. Also, the ratio of the
slab catenary action resistance to the slab ULS resistance decreases to be much less
than that which is required to resist the accidental load, indicating that it is not
effective to use catenary action to control progressive collapse in short slabs.
Figure 4-20: Variation of slab span affecting the connection response
4.3.3 Tie Bar Length
Among the components present in the connection of the precast concrete floor slabs,
it is only the tie bar that provides reliable tensile resistance and ductility to enable the
damaged structure to undergo catenary action and hence the structure may be able to
0
5
10
15
20
25
30
35
40
45
50
0 100 200 300 400 500 600 700
Lo
ad
(k
N)
Vertical Displacement (mm)
3 m
4 m
5 m
6 m
7 m
8 m
9 m
108
survive a local damage before failure. Since tie bars may not be provided over the
entire span of the slabs, it is important to understand how the tie bar length affects
development of catenary action.
The length of the tie bar affects not only the required elongation for catenary action,
but also the bond-slip behaviour of the tie bar in the surrounding concrete. As the tie
bar in the connection is always in tension in the catenary action, if the length is too
short it could be pulled out of the concrete while steel is still elastic; or it could reach
its yield stress and then be pulled out. In either of these cases the anchorage failure
causes the connection to have a brittle collapse.
The desirable failure mode is when the steel yields and then ruptures while it still has
enough bond with concrete that hinders the tie bar from slipping through. This is
satisfied if enough length of the tie bar is provided and if the bond between the
reinforcement and the concrete is in good condition. The good bond condition is
defined in CEB-FIP Model code 2010 and is assumed throughout this study.
Regulations, based on BS EN 1992-1-1, for the length of the bar tying the
hollowcore floor slabs state that if the yield load of the straight tie bars between units
is more than 30 kN, tie bars should use a minimum anchorage length of 100ϕ,
otherwise it can be taken as 75 ϕ.
Furthermore, as explained in section 4.1.3, the tie bar should be long enough so that
the unreinforced portion of the connections between the slabs or hollowcore infill
does not crack through after the slab has reached its catenary action resistance. In
this parametric study, the length of the tie bar is varied from the full length of the
slabs (5 m) to the minimum length that allows the tie bar to yield. This minimum
length is calculated based on the yield stress of the tie bar and the bond stress that it
develops with the surrounding concrete (Mazzarolo et al., 2012), which is about 100
mm for the present model. Tie bar lengths lower than this minimum will have either
the yield and pull-out failure, or just the pull-out failure mode.
The comparison of the load-displacement relationships of the different slabs with
different anchorage lengths in Figure 4-21-a shows that for long tie bar lengths
(those by which rupture of the tie bar can be achieved) up to a length of 290 mm, the
catenary action force is successfully developed and there is only a very small change
109
of the maximum load and displacement in the catenary stage, The reduction being
about 3% when the tie bar length changes from the full slab length of 5 m to 290
mm. This fact shows that the strain on the tie bar is localized to the region near the
slabs’connection.
However, as shown in Figure 4-21-b, if the tie bar length is further shortened,
catenary action cannot be fully developed and the slab resistance is reduced. This is
because before final cracking of concrete in the connection between the slabs, the tie
bar has slipped and the bond is not enough to hold the tie bar to allow it to develop to
its ultimate stress and sufficient elongation.
(a)
(b)
Figure 4-21: Variation of Tie Bar Length on the development of catenary action
The trend of the load-displacement diagram of the models with short tie bar length
shows that as the tie bar length is decreased the amount of the applied load on the
structure is decreased too. This phenomenon can be explained by the amount of force
exerted to the connection by the tie bar. As the tie bar length is decreased the region
0
5
10
15
20
25
30
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
5000 mm
1000 mm
500 mm
290 mm
0
5
10
15
20
25
30
0 50 100 150 200 250 300 350
Lo
ad
(k
N)
Vertical Displacement (mm)
250 mm
200 mm
150 mm
100 mm
110
in which the tie bar applies its load into the concrete is less and hence the reaction
force from the connection to each of the slabs will be less.
In summary, the length of the tie bar recommended in BS EN 1992-1-1 is enough to
enable full catenary action to develop.
4.3.4 Tie Bar Height
Results of the investigations in the previous sections show that catenary action can
develop in the damaged slabs, but the slab load carrying capacity during the catenary
action stage may not be sufficient to sustain the accidental loads by changing the
parameters investigated. Since catenary action is primarily dependent on the
reinforcement provided, this and the next two sub-sections investigate how tie bar
parameters may be changed to enable the slab to sustain the accidental load.
BS EN 1168 sets out regulations of the tie bar positioning with regard to the grouting
keyscastedontheslabs’longitudinal profile. According to this code, if the slabs are
placed close to each other in such a way that their grouting keys are at the bottom of
the section, the tie bar should be placed in the space between half height of the
section and centre of the grouting key (Figure 4-22-a). This regulation holds for the
case when the tie bar in the hollowcore slabs too.
(a) (b)
Figure 4-22: Placing the Tie Bar between Hollowcore Units (CCIP-030)
On the other hand, if the grouting keys are on top of the section when placing the
hollowcore floor slabs adjacent to each other, the tie bar can be placed in a space
between the centre of the grouting keys and 40 mm from the bottom face of the
111
connection (Figure 4-22-b). This is to provide enough concrete cover for the tie bar
so that the concrete is not crushed when the bar is in sagging moment.
The dimensions of the grouting keys are different in different concrete units but their
centre distance from the top or bottom faces of the concrete slab vary in a range of
30 to 50 mm. Hence the tie bar position range chosen for this study starts from 30
mm from bottom of the connection section and goes as high as 30 mm from the top
face of the slab.
Figure 4-23 compares the load-deflection relationships. There is some difference in
y-intercept of the load-deflection curve during the catenary action stage. However, it
can be seen that as the tie bar is moved up, the maximum deflection (hence the
maximum load that can be resisted by the slab) increases and then decreases. This
may be explained by how the tie bar is mobilising the surrounding concrete.
Considering the equilibrium of a slab, as the tie bar is moved up and the distance
between the centre line of the slab height and the tying force is decreased, it is
expected that the vertical applied load carried by the slab decreases. This is because
the distance between the centre line of the slab height and the tying force is
decreased, similar to the case where the height of the slabs was varied
(section 4.3.1). This explains the change in the trend of the load-deflection curves.
However, when the tie bar takes different positions in the height of the slab, different
concrete covers are achieved and hence the tie bar will possess different post-peak
bond-slip behaviours (Engström et al., 1998), (Torre-Casanova et al., 2013).
Furthermore, the additional factor of concrete cracking at the bottom of the slabs also
affects the effect of tie bar position as the cracked concrete gives different bond
behaviour to the tie bar than uncracked concrete.
When the tie bar is positioned at the top of the section, it does not contribute to the
tensile stresses of the section until the whole section is cracked. Therefore, catenary
action does not develop until complete separation of the slabs. However, at the same
time, since the tie bar experiences no tensile strain during bending of the slabs, it has
higher strain available during the catenary action stage to enable the slab to sustain
lateral deflections and high loads.
112
Figure 4-23: Effects of tie bar height (measured from bottom of slab) on catenary action
development
Thus, moving the tie bar position has three effects on the slab performance: concrete
cover to tie bar for bond behaviour, change in distance between centre of slab and
position of tensile force and strain. As the tie bar moves up towards mid-height of
the slab, the beneficial effects of increased concrete cover and reduced tie bar strain
dominate and enable the slab to undergo large deflections and hence to sustain
higher loads. Afterwards, the detrimental effects of reduced concrete cover and
reduced distance between the tensile force in the tie bar and the slab centre take over
and the slab resistance decreases when the tie bar moves from the centre of the slab
towards the top. These effects are shown in Figure 4-23. To enable the slab to
achieve the highest load carrying capacity in catenary action, it is preferable to place
the tie bar in the centre of the slab.
4.3.5 Tie Bar Diameter
Figure 4-24 compares the effects of changing the tie bar diameter from 10 mm to 30
mm, showing the slab-deflection curves and the axial force (Figure 4-24-b) and tie
bar strain (Figure 4-24-c) results. In all cases, concrete was totally cracked through
-5
0
5
10
15
20
25
30
35
0 100 200 300 400 500 600 700 800
Lo
ad
(k
N)
Vertical Displacement (mm)
Tie Height 30
Tie Height 70
Tie Height 90
Tie Height 130
Tie Height 170
Tie Height 210
113
(as shown by the sharp reduction in tension load in Figure 4-24-b) and the ultimate
failure was due to tie bar fracture (Figure 4-24-c).
(a)
(b)
(c)
Figure 4-24: Effects of changing tie bar diameter
Increasing the tie bar size not only increases the tensile force during the catenary
action stage, it also increases the maximum slab deflection at failure. This increase in
-10
0
10
20
30
40
50
60
70
80
90
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Diameter 10 mm
Diameter 14 mm
Diameter 18 mm
Diameter 22 mm
Diameter 30 mm
-200
-100
0
100
200
300
400
500
0 100 200 300 400 500 600Axia
l R
eact
ion
Fo
rce
(kN
)
Vertical Displacement (mm)
D=10 mm
D=14 mm
D=18 mm
D=22 mm
D=30 mm
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 100 200 300 400 500 600
Str
ain
Vertical Displacement (mm)
D=10 mm
D=14 mm
D=18 mm
D=22 mm
D=30 mm
114
displacement is a result of increased elongation of the tie bar since as the tie bar
diameter is increased; a longer length of the tie bar is activated as the bond surface is
increased. These two beneficial effects are exhibited as the higher load-deflection
slope and higher displacement in Figure 4-24-a as the tie bar diameter increases.
4.3.6 Tie Bar Yield Stress
Figure 4-25 compares slab load-displacement relationships for changing the tie bar
yield stress from 200 to 600 MPa. Increasing the tie bar yield stress has the clear
benefit of increasing the catenary action force and hence the slab resistance.
However, since the tie bar geometry is unchanged, the slab ultimate deflection does
not benefit from significant change.
Figure 4-25: Effects of changing tie bar yield stress
Hence it is recommended that to increase the load carrying capacity of a connection,
a tie bar with higher yield stress can be used, although the displacement of catenary
action is not changed.
0
5
10
15
20
25
30
35
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
200 MPa
300 MPa
400 MPa
500 MPa
600 MPa
115
4.3.7 Grouting Concrete Strength
After aligning the precast concrete floor slabs on their supporting beams closely to
each other, and placing the tying reinforcements in the gaps between the floor units;
the space between the slabs is filled with grouting concrete. Tie bars are also placed
in the hollowcores of the slabs and again the length of the hollowcores encasing the
tie bar is filled with grout.
Figure 4-26: Effect of concrete strength on load-displacement of slabs' connection (stresses in
MPa)
Grouting concrete not only contributes to the bending capacity of the connection, but
also affects the bond-slip behaviour of the tie bar as bond stress relationships are
based on the compressive strength of the concrete (CEB-FIP Model Code 2010).
Stronger concrete provides more tensile strength in cracking which is expected to
provide higher load carrying capacity of the connection. However, once the slab is in
catenary action stage after the grout has completely cracked through, since the
tensile resistance comes from the tie bar reinforcement, the grout strength is not
expected to have any effect on the slab load-deflection behaviour. This is
demonstrated in Figure 4-26. This figure indicates that total crack through the grout
is delayed by using higher strength grout, but in the catenary action stage slab
behaviour is hardly changed. As a conclusion, using high strength grout is not
0
5
10
15
20
25
30
35
40
45
50
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Dispalcement (mm)
fc = 30
fc = 50
fc = 70
fc = 90
116
effective in increasing the slab catenary action resistance for controlling progressive
collapse.
4.3.8 Ultimate Strain of the Steel Tie Bar
During the catenary action stage, the yield stress of the tie bar determines the slope
of the load-deflection curve while the maximum deflection of the slab is controlled
by the elongation of the tie bar. It is expected that increasing the tie bar ultimate
tensile strain would increase the slab maximum deflection and hence the slab load
carrying capacity. Figure 4-27 compares the slab load-deflection relationships for
different tie bar maximum strains from 5% to 40%.
The results confirm this expectation. The tie bars have developed complete yield
stress and their reaching the maximum tensile strains determines the failure of the
slabs. Since the yield stress of the tie bar is not changed, the slab load-deflection
behaviour before total concrete cracking is unchanged. Also since the tensile load in
the tie bar is the same, the different load-deflection curves coincide until tie bar
fracture.
Figure 4-27: Effects of changing Tie Bar Ultimate Strain
Comparing the effects of changing the tie bar ultimate strain with changing other
parameters, it is clear that using ductile tie bars with high elongation is the most
0
5
10
15
20
25
30
35
0 100 200 300 400 500 600 700
Lo
ad
(k
N)
Vertical Dispalcement (mm)
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
117
effective method of increasing the catenary action resistance of damaged slabs for
controlling progressive collapse.
4.3.9 Summary
This chapter presented a two dimensional model of the slabs and the longitudinal
connection between them. It has been assumed that the supporting column in
between the two slabs and in the middle of the longitudinal tying connection has
been lost due to an accidental action on the structure, hence the behaviour of the
connection and the parameters that affect it have been identified.
Different stages of behaviour were identified and it was shown that due to the brittle
nature of concrete, in the case of a column loss, it was the catenary action provided
by the tie bar that delivered a reliable resisting mechanism against the progressive
collapse and not the arching action of concrete.
The accidental limit load was calculated for a typical residential building and it was
shown that the designed connection based on the current regulations was not able to
resist the load limit.
Different parameters in the connection zone were examined and their effect on the
catenary action was concluded as follows:
Slab height: as the slab height increases it was shown that the tie bar ruptures
sooner, hence the use of thinnest possible slab is recommended
Slab Span: although this parameter did not show any effect on the elongation
of the tie bar, it was shown that the slab span affects the equilibrium equation
of slabs under catenary action. The shorter the slab, the higher the load
carrying capacity, but at the same time the tie bar is strained in earlier stages
of loading and this decreases the vertical displacement
Tie bar length: as long as the length required for capturing enough bond
between concrete and tie bar is provided, the tie bar length does not affect the
resistance or displacement of the connection
118
Tie bar height: it was shown that due to different factors such as concrete
cover, distance between the forces on the slab, and straining of tie bar, the
optimum location for the tie bar was the mid-height of slab
Tie bar diameter: increasing the tie bar diameter, boosted both resistance and
the elongation of the tie bar where the latter results in more vertical
displacement as well
Tensile ultimate strain of tie bar: directly affected the elongation and ductility
of tie bar in catenary action, resulting in both higher resistance and vertical
displacement
Tie bar yield stress: directly affecting the tying force, increase in tie bar yield
stress increases the load carrying capacity of the connection; but the
elongation of the tie bar is unchanged
Compressive strength of grouting concrete: increase in this parameter delays
the cracking of concrete, but does not affect the catenary action as the tie bar
is main load bearing unit in this stage
Based on the conducted parametric study, recommendation for each of the
parameters with the regard to their effect on the catenary action was made and the
effective parameters (tie bar diameter, height, ultimate tensile strain, and the slab
height) were identified in order to develop an analytical relationship to predict the
connection behaviour in catenary action in the next chapter.
119
Chapter 5. 2D Slab Analytical Load-Displacement
Relationship
This chapter develops and assesses the accuracy of an analytical model to predict the
behaviour of 2D slabs. The assessment is made against the results of FE simulations
presented in the previous chapter. The analytical model should give the load carrying
capacity of the structure until fracture and should be able to deal with flexible end
restraints. The results of the previous chapter will be used to help make some
assumptions.
It was seen in Chapter 4 that tie bars in thick slabs ruptured before the slabs
developed catenary action. Under the accidental loading that the thick slabs are
required to resist, relying on compressive arching action of the concrete slabs may
lead to a sudden failure of the structure. This is against the spirit of design rules that
require the structure to provide a ductile failure mode. Hence the derivation in this
chapter is focused on the catenary action.
5.1 Development of the Analytical Relationship
5.1.1 Axially Restrained Slabs
Figure 5-1 shows the free body diagram of half of the 2-D slab model. It is assumed
that the slab is in pure catenary action with no bending resistance. The catenary force
is F and the externally applied load is P. The catenary action force is provided by the
tying resistance of the steel tie bar.
120
At catenary stage, the tie bar is yielded with a constant force equal to its yield
resistance (=steel yield stress multiplied by the cross sectional area of the tie bar).
Assuming that the far ends of the slabs are pinned, the equilibrium equation of the
forces applied on the slab can be written as (Equation 5-1):
)(2
dhFLP
Equation 5-1
Where:
P: the vertical force applied on the connection between the two slabs, representing
the accidental load, half of which is carried by each slab
L: length of each slab
F: tying force which is equal to fy,steel Asteel
h: vertical displacement of the slab
d: the distance of the tie bar to the centre of the slab
Initial Position of the Slab
Figure 5-1: Free body diagram of one slab
L
h
F 𝑃
F
L + e
d
𝑃
Initial Position of the Tie Bar
121
From Equation 5-1 the vertical force applied on the slabs’ connection (P as a
function of vertical displacement h) can be written in terms of the slab vertical
displacement (Equation 5-2):
Equation 5-2
Figure 5-2 compares the analytical equation (Equation 5-2) with the finite element
simulation. It is seen that the analytical solution provides close agreement with the
numerical result for the catenary action stage. Further comparisons are in Appendix
2.
Figure 5-2: Comparison of FE model with analytical relationship, slab height 265 mm, width
1200 mm, span 5 m, tie bar height 45mm, diameter 16 mm, steel yield stress 500 MPa
5.1.2 Elastic Axially Restrained Slabs
In chapter Chapter 4, it was shown that the behaviour of support conditions did not
affect behaviour of the slabs during the catenary action stage. This section will
present the analytical derivations to prove this. Figure 5-3 shows the same structure
-15
-10
-5
0
5
10
15
20
25
30
35
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Analytical Relationship
FE Pinned BC
122
as in Figure 5-1, except that the end of the slab has moved (Δ). This horizontal
displacement can be calculated by dividing the catenary action force (tying force) by
the axial stiffness of the support.
Figure 5-3: Slab deflection with axial displacement
Using the free body diagram shown in Figure 5-3, the equilibrium relationship can
be written as:
dhFLP
2
Equation 5-3
Giving:
L
dhFP
2
Equation 5-4
Where:
P: is the vertical force
F: tying force
h: vertical displacement of the connection between the slabs
Δ
h+d L+e
L
F
P/2 P/2
123
d: vertical distance between the centre line of the slab height and tying force
L: slab length
And Δ: the horizontal movement of slab end due to flexibility of the support.
e: is the elongation of the tie bar.
As the support horizontal displacement (Δ) is negligible compared to the length of
the slab (L), the support stiffness has almost no effect on the equilibrium of the
system in catenary stage. Figure 5-4 shows typical comparison between Equation 5-4
with numerical simulation result for a case with elastic axial restraint. The agreement
is quite good. More comprehensive comparisons are provided in Appendix 2.
Figure 5-4: Comparison of the FE model with elastic BC with the analytical relationship, slab
span: 7 m
5.2 Maximum Slab Displacement
In order to obtain the maximum resistance of the slab in catenary action, it is
necessary to be able to estimate the maximum slab vertical displacement (h). It is
assumed that the maximum slab displacement is reached when the maximum strain
in the tie bar has reached its rupture strain (εult). At this stage, the strain in the tie bar
is distributed along a certain length of the tie bar (Lp). Figure 5-5 shows a schematic
distribution of strain at the connection between the slabs. As explained in
section 2.6, strain is localized to the connection zone.
0
5
10
15
20
25
0 100 200 300 400 500 600
Loa
d (
kN
)
Vertical Displacement (mm)
Analytical
Prediction
Elastic BC L=7 m
124
Figure 5-5: Strain distribution at the connection between PCFS
Figure 5-6 shows the same tie bar as in Figure 5-5, but in a typical tie bar strain
diagram of an FE model. If the strain distribution (area under the strain curve: Sd),
and the strained length of the tie bar (Lp) are known, then the total tie bar elongation
(e in Figure 5-1 and Figure 5-3) can be calculated, from which the maximum
displacement h of the slab can be calculated.
Figure 5-6: Strain distribution along the tie bar in the FE model
The strain distribution (Sd) has been calculated based on the trapezium rule along the
tie bar (Figure 5-6). The region between each pair of the integration points along the
PCFS
Tie Bar
Strain Distribution
of Tie Bar at
Connection (Sd)
Lp
Sd εult
Lp
125
tie bar is assumed as a trapezium, and the summation of these areas gave the strain
distribution of the FE models. For prediction of the strain distribution, it proved to be
a good estimation to consider the two parallel sides of the trapezium as εult and
.
Hence the strain distribution over the penetration length of strain (Lp) at the ultimate
strain of the tie bar (εult) can be written as:
→
Equation 5-5
On theotherhand the rupture strainof tiebarcanbewritten in termsof thebar’s
total elongation (e), and the strain penetration length (Lp), which with Equation 5-5
gives:
Equation 5-6
By referring to Figure 5-3 the following quadratic equation can be estimated for the
verticaldisplacementoftheslab(neglectingthesquareofe,andΔ):
Equation 5-7
Having all the parameters calculated, Equation 5-7 is solved for the vertical
displacement h. Hence the maximum force applied on the connection is derived with
Equation 5-4, which is then compared with the required accidental limit force on the
connection between the slabs.
Among the parameters studied that affect the elongation of the tie bars, the following
parameters have noticeable effects: slab height, tie bar position and diameter, and
ultimate strain of the tie bar material.
126
5.2.1 Slab Height
Figure 5-7 shows the strain distribution along the tie bar length with slab height
variation. A curve of best fit is also shown in the figure.
Figure 5-7: Variation of tie bar strain distribution with slab height
The relationship for strain distribution (Sd,S.H) and slab height (SH) can be written as:
Equation 5-8
The above trend can be explained as follows: Considering two models with different
slab heights at the same vertical displacement at a stage before plasticity of the tie
bar, and assuming the second model has a bigger height, it can be seen that
(Figure 5-1 and Equation 5-1) due to equilibrium of the forces applied to each slab,
the vertical force on the second model is bigger (P2 > P1); as the distance of the
influence line of the tie bar force from the centre line of the slab is more in this slab
(d2 > d1).
As the resisting force, provided by the tie bar, is the same in both cases, the
excessive force in the second model is carried by more straining of the tie bar, hence
ε2 > ε1. Expressing the tie bar strain in terms of the ratio of the elongation of the tie
bar (e) over the length of the tie bar elongated (Lp).
Sd = 3E-10(SH)4 - 5E-07(SH)3 + 0.0003(SH)2 - 0.0947(SH)
+ 25.327
0
2
4
6
8
10
12
14
16
18
20
0 100 200 300 400 500
Str
ain
Dis
trib
uti
on
(m
m)
Slab Height (mm)
127
1
11
pL
e
And 2
22
pL
e
And noticing that the lengths and vertical displacements of the two slabs are the
same, it is apparent that the elongation of the two tie bars is equal and they can be
approximated by Pythagoras theorem (neglecting the quadratic term of e):
L
hee
2
2
21
This yields that:
1
1
2
212
pp L
e
L
e
Knowing that e1 = e2, gives: Lp1 > Lp2, hence the slab with lower height has higher
strain distribution along its tie bar length.
5.2.2 Tie Bar Diameter
Figure 5-8 shows that the tie bar strain distribution length increases with increasing
tie bar diameter. When the tie bar diameter increases, the tie force is also increased
according to the square of its radius. In the meantime, the bond force between the tie
bar and the concrete increases linearly (determined by the interface area). Therefore,
a longer tie bar strain distribution length is necessary to distribute the tie bar force.
128
Figure 5-8: Effect of tie bar diameter on strain distribution length
The estimated linear relationship between strain distribution (Sd,T.D) and the tie bar
diameter (D) is:
Equation 5-9
5.2.3 Tie Bar Position
Figure 5-9 shows how the tie bar strain distribution changes with tie bar position.
Due to the number of factors affecting the strain distribution, when the tie bar
position changes in the height of the slab; the variation of this parameter cannot be
explained with the geometry relationships solely. Herein, the concrete cover to tie
bar, concrete cracking, and end rotation of the slab affect the strain distribution of the
tie bar.
Sd = 0.3988(D) + 7.9593
0
5
10
15
20
25
0 5 10 15 20 25 30 35
Str
ain
Dis
trib
uti
on
(m
m)
Tie Bar Diameter (mm)
129
(a)
(b)
Figure 5-9: Variation of strain distribution with tie bar position
The quadratic relationship between the strain distribution (Sd,T.H) and the tie bar
height (TH) is written as:
Equation 5-10
If the tie bar is placed below the mid-height of the section, the effect of the tie bar
position is similar to that of increasing the slab height as explained in section 5.2.1.
Thus, when the tie bar moves up towards the mid-height, the effect is similar to
reducing the slab thickness in section 5.2.1, which leads to an increased tie bar strain
distribution length. Also as the tie bar moves up, the concrete cover is more, hence it
is able to deploy more bond stress with the surrounding concrete (Engström et al.,
1998), (Torre-Casanova et al., 2013).
Sd = 0.0035(TH)2 - 0.298(TH) + 19.468
0
5
10
15
20
25
30
35
40
45
0 20 40 60 80 100 120 140
Str
ain
Dis
trib
uti
on
(m
m)
Tie Bar Height (mm)
Sd = -0.1132(TH) + 50.645
0
5
10
15
20
25
30
35
40
45
0 50 100 150 200 250
Str
ain
Dis
trib
uti
on
(m
m)
Tie Bar Height (mm)
130
When the tie bar moves above the mid-height, the strain distribution analogy for the
slab height holds but this time it starts to lose its concrete cover which leads to less
bond stress. Also when the tie bar is above the mid-height, the tie bar is in tension at
a later stage compared to the case when it is places below the mid-height due to later
concrete cracking under a sagging moment. Hence the tie bar has less opportunity of
developingtensionalongthetiebar’slength.
5.2.4 Ultimate Strain of the Steel Tie Bar
Figure 5-10 shows that the tie bar strain distribution length increases almost linearly
with increase in the ultimate strain of the tie bar. The increased tie bar ultimate strain
gives opportunity for the tie bar stress to be dissipated along a longer length of the
tie bar.
Figure 5-10: Effect of tie bar ultimate strain on connection behaviour
The linear relationship for strain distribution (Sd,ε) variation with respect to ultimate
strain of the tie bar is:
Equation 5-11
y = 76.42x + 0.5736
0
5
10
15
20
25
30
35
0 0.1 0.2 0.3 0.4 0.5
Str
ain
Dis
trib
uti
on
(m
m)
Ult. Strain of Tie Bar
131
Figure 5-7 to Figure 5-10 give the best fitting equations to the various trends to
enable the tie bar strain distribution length to be calculated as functions of the
various design parameters. To obtain the overall value, the average of the four values
is taken.
5.3 Validation of the Analytical Prediction for the
Maximum Catenary Action Resistance
Using the approximate expressions between the tie bar strain distribution and the
various design parameters (slab thickness, tie bar diameter, tie bar position and tie
bar ultimate strain), the maximum tie bar elongation can be calculated. Equation 5-7
then gives the maximum slab displacement which can be used in Equation 5-4 to
give the ultimate resistance of the slab in catenary action. The analytical calculations
are presented here for the base case.
Table 5-1: Calculation of strain distribution for the base case
Design Parameter Value Sd
Tie Bar height 45 mm 13.14 (Equation 5-8)
Diameter 16 mm 14.34 (Equation 5-9)
Slab Height 265 mm 13.47 (Equation 5-10)
Ultimate Strain 0.2 15.85 (Equation 5-11)
Average Sd: ≈14
The vertical displacement can be written as (by Equation 5-6 and Equation 5-7):
√ √ (
)
And using the Equation 5-4 the resistance of the base case slab is shown in
Figure 5-11, where good agreement between the calculated vertical displacement and
slab resistance can be observed:
132
Figure 5-11: Comparison of FE base case with the analytical calculation
For prediction of the resistive force (P) of other models, where more than one
parameter is changed, the strain distribution calculated (based on Equation 5-8 to
Equation 5-11) is normalised with respect to the average strain distribution of the
base case (Equation 5-12):
{ √ (
) }
Equation 5-12
Where:
Atie: is the cross sectional area of the tie bar
d: is the distance between the mid-height of slab to the tie bar position
Sd,S.H.: is obtained from the strain distribution relationship for slab height
(Equation 5-10)
Sd,T.D.: is obtained from the strain distribution relationship for tie diameter
(Equation 5-9)
0
5
10
15
20
25
30
0 100 200 300 400 500
Load
(kN
)
Vertical Displacement (mm)
133
Sd,T.H.: is obtained from the strain distribution relationship for tie height (position,
Equation 5-8)
Sd,ε: is obtained from the strain distribution relationship for ultimate tie bar strain
(Equation 5-11)
The value obtained from the above equation can be checked against the accidental
load (Pacc) required on two damaged slabs to see if the connection between the slabs
has enough robustness to withstand this loading condition (Equation 5-13).
Equation 5-13
To obtain a suitable tie bar for a given accidental load, a trial and error procedure for
Atie should be followed until Equation 5-12 gives a force equal or greater than Pacc in
Equation 5-13.
5.3.1 Comparison of FE and Analytical Results
The results of the analytical calculations for all the numerical simulation models in
chapter Chapter 4 are shown in Table 5-2 to Table 5-5. The overall agreement
between the analytical and numerical results is very good.
Table 5-2: Comparison of FE and analytical results, Tie Bar height
Parameter Displacement (mm) Load (N)
Tie Height
(mm)
Calc. FE Abs.
Difference
Calc. FE Abs.
Difference
30 440 416 24 21837 20400 1437
70 476 467 9 21528 21000 528
90 546 518 28 23659 23500 159
130 752 730 22 30351 31200 849
170 667 642 25 25336 24000 1336
210 617 610 7 21717 24700 2983
134
Table 5-3: Comparison of FE and analytical results, Tie Bar Ult. Tensile Strain
Parameter Displacement (mm) Load (N)
Tie Ult.
Strain
Calc. FE Abs.
Difference
Calc. FE Abs.
Difference
5 227 167 60 12660 6350 6310
10 310 311 1 16018 22800 6782
15 376 369 7 18649 17400 1249
20 432 421 11 20885 19400 1485
25 481 472 9 22865 22000 865
30 525 519 6 24660 24700 40
35 567 569 2 26314 27700 1386
40 605 593 12 27857 29100 1243
Table 5-4: Comparison of FE and analytical results, Slab Height
Parameter Displacement (mm) Load (N)
Slab Height
(mm)
Calc. FE Abs.
Difference
Calc. FE Abs.
Difference
150 475 481 6 20330 19900 430
200 453 457 4 20455 19700 755
250 436 428 8 20768 20200 568
300 421 413 8 21180 20000 1180
350 407 385 22 21614 20300 1314
135
Table 5-5: Comparison of FE and analytical results, Tie Bar Diameter
Parameter Displacement (mm) Load (N)
Tie
Diameter
(mm)
Calc. FE Abs.
Difference
Calc. FE Abs.
Difference
10 394 384 10 7565 8220 655
12 407 415 8 11187 11500 313
14 419 413 6 15614 15100 514
16 432 422 10 20885 19600 1285
18 443 449 6 27038 27000 38
20 455 462 7 34107 34700 593
22 466 480 14 42128 43700 1572
28 498 498 0 72230 63200 8630
30 509 515 6 84381 76700 7681
The absolute values of difference between FE and analytical displacement of all
models have average and standard deviation of 12 mm and 11.7 mm respectively.
And the absolute values of difference between the analytical and FE results of
resistance have the average and standard deviation of 1.8 kN and 2.3 kN
respectively.
5.4 Summary
This chapter has presented the derivation and validation of an analytical model to
calculate the load-displacement relationship of the slab in catenary action, based on
observations of the finite element simulation results presented in Chapter 4. Under
catenary action, the slab is assumed to possess no bending resistance and deforms in
a straight line. The maximum slab displacement is related to the tie bar elongation,
which is based on a trapezium strain distribution along a tie bar strain distribution
length Lp. This strain distribution length has been found to be dependent on the slab
height, the tie bar diameter, position and ultimate tensile strain. Approximate
equations have been proposed to calculate the tie bar strain distribution length as
functions of these parameters. Using the proposed relationships, the maximum slab
displacement and slab resistance have been calculated for all the numerical models
in Chapter 4. Good agreement was found between the analytical and numerical
136
results. The proposed method may be used in design estimation of the potential of
the damaged slabs, after removal of the central support, to resist the accidental load.
137
Chapter 6. Three-dimensional behaviour of PCFS
with column removal
This chapter presents the results of an extensive parametric study on a three
dimensional finite element model of the precast concrete floor slabs. This model
consistsofeightPCFS’ssegmentstied together, representing a full span of a typical
flooring system. The results of the FE analysis are used to assess the current building
code regulations with regard to the effectiveness of tying PCFS’s to achieve
robustness, and recommendations are made based on the presented parametric study.
Three case scenarios are considered herein (Figure 6-1):
1. Loss of an edge column
2. Loss of a centre column
3. Loss of a corner column
Figure 6-1: Plan of a floor system
The parametric study focuses on the parameters that were found effective in Chapter
4. These include: the tie bar diameter and ultimate strain, and the slab height. Based
on the findings of the Chapter 4 the tie bar is placed in the mid-height of the slab in
Corner Column
Edge Column
Centre Column
Longitudinal Tie
1.2 m
Transverse Tie
PCFS
138
all 3D models as this was shown to be the optimum position. Also the effect of
transverse reinforcement on the performance of the connection between PCFS is
studied (Figure 6-1).
Figure 6-2: Modelled slab resting on steel structure
In steel framed structures, although the primary system for structural robustness is
through tying of the principal members at the beam-column joints, they have proved
to lack sufficient rotational capacity for development of catenary action (Byfield and
Paramasivam, 2007). This research will investigate whether ties connecting the
PCFS can resist the accidental loading without relying on the beam-column ties. In
these simulations, one steel member is assumed to have lost load carrying capacity
and other steel members are considered intact (Figure 6-2) so as to be able to provide
the PCFS with appropriate support condition.
6.1 Finite Element Model
The finite element model is depicted in Figure 6-3-a including the boundary
conditions. It is assumed that the surrounding structural elements do not fail to
provide the required vertical restraint to the slab segment in question. The same axial
restraint is applied to the 3-D model as discussed in Chapter 4. Figure 6-3-c shows
the slab layout and positions of the tie-bars.
Stable Steel Beam
Stable Steel Column
PCFS
Lost Column
Beam-Column
Joint
139
(a)
(b)
(c)
Figure 6-3: Floor arrangement and three dimensional finite element model representation
Due to symmetry, only half of the floor was simulated. Loading of the model is
applied via prescribed displacement. The nodes on the edge beam have been tied
together to have a displacement with linear proportionality so that they represent the
beam underneath the slabs on top of the lost column. The concrete material model is
the smeared cracking model with Mohr-Coulomb plasticity in compression and
Edge
1st Hollowcore Tie
4@1200 mm
5000 mm
140
linear tension softening in tension, as used in Chapter 3. The material model for
reinforcement is the Von-Mises plasticity model.
In order to model the grouted hollowcores which contain the tie bars, the middle
hollowcore of each model has been filled to the tie bar length which is necessary to
capture the cracking length of the slab as explained in Chapter 4 (Figure 6-3-b). The
failure of the floor slab is marked with rupture of the 1st tie in the hollowcore
(Figure 6-3-c).
The base case of the parametric study is the same as the one for the 2D model
in Chapter 4: slab height 265 mm, slab width 1200 mm, tie bar diameter 16 mm, and
span length 5 m.
Table 6-1 lists the material properties used in the numerical models.
Table 6-1: Material property of concrete and steel used in the FE model
Concrete
Ec 33842.32 MPa
ν 0.2
fc 39 MPa
ft 2.97 MPa
Gf 0.0778 N.mm/mm2
Steel (Tie Bar)
Es 210000 MPa
ν 0.3
fy 500 MPa
6.2 Effects of Loss of an Edge Column
If the structure suffers the loss of an edge column on the side of the structure, it may
be possible for the structure to provide an alternative load path, through the
development of catenary action (Figure 6-4).
141
Figure 6-4: Possible structural behaviour after loss of an edge column, (Baldridge and Humay,
2003)
6.2.1 Slab Height
The slab heights in this study give the precast slab span/height ratio ranging from 33
– 14. Typically, the slab span/height ratio for PCFS is 30 (SpanRight, 2013). The
effect of slab height on load carrying capacity of the floor system is shown in
Figure 6-5-a. It can be seen that similar to the trend of the 2D case in Chapter 4, as
the slab height increases, the load carried by the floor system increases, but this leads
to higher tie bar strain at the same vertical deflection. Figure 6-5-(b) and (c) show the
tie bar strain development in the connection at the first and the second tie bars. In all
cases, failure of the slab was due to tie bar fracture when the tie bar strain reached
the assumed tie bar fracture strain of 20%. Figure 6-5-d shows that the concrete in
the connection is thoroughly cracked and the force in the catenary action is only
provided by the tie bar.
142
(a)
(b)
(c)
(d)
Figure 6-5: Slab Height effect on the connection behaviour
0
100
200
300
400
500
600
0 200 400 600 800 1000 1200 1400 1600
Lo
ad
(k
N)
Vertical Displacement (mm)
Slab Height 150
Slab Height 200
Slab Height 250
Slab Height 300
Slab Height 350
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 200 400 600 800 1000
Ed
ge T
ie B
ar S
tra
in
Vertical Displacement (mm)
Slab Height 150
Slab Height 200
Slab Height 250
Slab Height 300
Slab Height 350
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 200 400 600 800 1000 1200 1400 1600
1st
Ho
llo
wco
re T
ie S
tra
in
Vertical Displacement (mm)
Slab Height 150
Slab Height 200
Slab Height 250
Slab Height 300
Slab Height 350
-14
-12
-10
-8
-6
-4
-2
0
2
4
0 200 400 600 800 1000 1200 1400 1600
Co
ncr
ete
Axia
l S
tress
(M
Pa
)
Vertical Displacement (mm)
Slab Height 150
Slab Height 200
Slab Height 250
Slab Height 300
Slab Height 350
143
As expected, when the slab height increases, the first peak load in the slab load-
deflection relationship (Figure 6-5-a) increases as a result of increased bending
moment capacity of the slabs. However, since the reason for using deep slabs is to
resist increased design loads, the residual load carrying capacity of the slab system
after column removal may not be sufficient to resist the accidental loading. Below a
procedure for estimation of required floor resistance for accidental loading is
proposed:
(1). The slab bending moment capacity per unit width can be calculated as follows
(Elliot, 2002):
Equation 6-1
In which:
Mur: is the slab bending moment resistance
fpb: is the design tensile stress in tendons
Aps: total cross sectional area of the tendons per unit area of slab
d: the effective depth of the precast concrete cross section
and X is the depth of neutral axis calculated by equating the tensile force in the
tendons to compressive force of the concrete block
Based on the manufacturers product listing (CFS, 2013) and (O'Reily, 2013), the
prestressingstrands in thehollowcores slabs thinner than300mmare10φ10mm
andforslabsthickerthan300mmare10φ12.5mm.Theultimatetensilestressof
strands is taken as 1500 MPa and design stress is about 80% of that. It is usually
assumed that 70% of the design stress is lost due to construction deficiencies. Based
on this information, BS 8110 part1 table 4.4 provides the coefficients for calculation
of force provided by the prestressing strands and the location of the neutral axis (X).
(2). The ultimate limit state (ULS) load carrying capacity of the undamaged floor
system, under uniformly distributed loading, can be calculated as follows:
144
Equation 6-2
Assuming a ratio of accidental load/ULS load of about 0.3, the accidental load (ALS)
is:
Equation 6-3
Based on which the maximum bending moment per unit slab width in the damaged
floor system is calculated. This is done by considering a plate with three fixed edges
and a free edge (as the edge column is lost in this case). (Moody, 1990) provides
relationships for calculation of the moment of different loading situations of plates
with different boundary conditions.
Figure 6-6: Plate structure with 3 fixed and 1 free edges (Moody, 1990)
As in this case the ratio a/b (Figure 6-6) is almost equal to one (a being the one slab
span, and b being four times of slab widths), the moment for concrete due to uniform
distributed loading at { is calculated by:
145
Equation 6-4
Where:
P: is the uniform distributed load (ALS in here)
(3). The moment due to the uniformly distributed load calculated in the previous
stage, is equated to the moment that a point load in the middle of the edge of the
slabs (at { ) would produce. This is done due to limitation of displacement
control analysis when using DIANA.
Equation 6-5
Where p is the point load.
Table 6-2 shows the ULS undamaged slab resistance, the required equivalent point
load resistance for the damaged accidental situation, the slab first peak resistance,
and the slab final resistance under catenary action.
Table 6-2: Accidental load based on the ultimate bending resistance of the slabs
slab height
(mm) Mur (kN.m)
ULS
(kN/m2)
“1span”
w
Point
Load
(kN) “2
spans”
First
peak
resistance
(kN)
Max.
Load in
Catenary
(kN)
150 75.76 20.20 148.07 55 180
200 133.36 35.56 210.93 125 130
250 193.88 51.70 276.97 224 115
300 351.32 93.68 448.79 344 90
350 446.32 119.01 552.46 513 50
146
The results in Table 6-2 suggest that only the thinnest slab (150mm) can develop
sufficient slab resistance in catenary action to resist the accidental load. For greater
slab heights, the required floor resistance under accidental loading is proportionally
increased as the slab height increases, but the floor resistance under catenary action
does not increase because this resistance is limited by the amount of reinforcement.
Although tie bars are useful to prevent accidental detachment of one precast unit
from others, the conclusion of this study casts doubt on the effectiveness of using tie
bars to develop an alternative load carrying mechanism to control progressive
collapse of the global structure.
6.2.2 Tie Bar Diameter
Figure 6-7 shows the slab load-deflection relationships for different tie bar
diameters. Because slab dimensions are unchanged, the bending resistance of the
floor system, as evidenced by the first peak, does not change. Increasing the tie bar
diameter increases the load carrying capacity of the floor system during the catenary
action stage. However, the results again suggest that the development of catenary
action would not be sufficient to resist the applied loads under accidental load
condition.
147
(a)
(b)
(c)
Figure 6-7: Effect of tie bar diameter on floor behaviour
Nevertheless, the results in Figure 6-7(a) clearly show very useful increase in floor
resistance under catenary action when increasing the tie bar diameter. It is thus
possible to use increasing the tie bar diameter as a possible method of boosting floor
resistance under accidental loading. Figure 6-7(b) and Figure 6-7(c) again show that
0
50
100
150
200
250
300
0 200 400 600 800 1000 1200 1400
Lo
ad
(k
N)
Vertical Displacement (mm)
Tie Diameter 12
Tie Diameter 16
Tie Diameter 20
Tie Diameter 24
Tie Diameter 30
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 200 400 600 800 1000
Ed
ge T
ie B
ar S
tra
in
Vertical Displacement (mm)
Tie Diameter 12
Tie Diameter 16
Tie Diameter 20
Tie Diameter 24
Tie Diameter 30
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 200 400 600 800 1000 1200 1400
1st
Ho
llo
wco
re T
ie S
tra
in
Vertical Displacement (mm)
Tie Diameter 12
Tie Diameter 16
Tie Diameter 20
Tie Diameter 24
Tie Diameter 30
148
failure of the slabs was caused by fracture of the tie bars which reached the fracture
strain of 20%.
6.2.3 Tie Bar Ultimate Tensile Strain
Catenary action develops as a result of large deflection of the floor system. In the
previous sections, floor failure was due to fracture of the tie bars. Therefore, it is
expected that if the tie bar deformation capacity (ultimate tensile strain) is changed,
it would have direct influence on the development of catenary action. Figure 6-8
shows results of changing the tie bar ultimate tension strain, ranging from 5%
(typical for reinforcement) to an artificially high value of 40%. Again, failure of the
floor system was due to the tie bars reaching their respective ultimate tensile strains.
However, the results in Figure 6-8(a) indicate that if the tie bar ultimate tensile strain
could be increased, it would be a very effective method of increasing the floor
resistance under catenary action. For comparison, the final floor resistance at 40% tie
bar ultimate tensile strain is more than double that at 20% tie bar ultimate tensile
strain. In contrast, the final floor resistance at 5% tie bar ultimate tensile strain is
about 60% of that at 20% strain. Whilst it may not be possible to increase the
ultimate tensile strain of tie bars, the results of this investigation clearly suggest that
for improved robustness of the floor system (which is the main aim of providing the
tie bars), tie bars with high ductility (hot rolled) instead of low ductility (high
strength, cold formed) should be used.
149
(a)
(b)
(c)
Figure 6-8: Effect of changing tie bar ultimate tensile strain
6.2.4 Transverse Tie Bars
At present in the construction practice, according to the EuroCode, only longitudinal
ties are introduced in order to provide robustness between the PCFSs. This section
investigates whether it would be effective to enhance the floor resistance under
accidental loading by introducing transverse tie bars between the precast floor units.
0
50
100
150
200
250
300
0 500 1000 1500 2000
Lo
ad
(k
N)
Vertical Displacement (mm)
Tie Ult. Strain 0.05
Tie Ult. Strain 0.1
Tie Ult. Strain 0.15
Tie Ult. Strain 0.2
Tie Ult. Strain 0.25
Tie Ult. Strain 0.3
Tie Ult. Strain 0.35
Tie Ult. Strain 0.4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 200 400 600 800 1000 1200
Ed
ge
Tie
Ba
r S
tra
in
Vertical Displacement (mm)
Tie Ult. Strain 0.05
Tie Ult. Strain 0.1
Tie Ult. Strain 0.15
Tie Ult. Strain 0.2
Tie Ult. Strain 0.25
Tie Ult. Strain 0.3
Tie Ult. Strain 0.35
Tie Ult. Strain 0.4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 500 1000 1500 2000
1st
Ho
llo
wco
re T
ie S
tra
in
Vertical Displacement (mm)
Tie Ult. Strain 0.05
Tie Ult. Strain 0.1
Tie Ult. Strain 0.15
Tie Ult. Strain 0.2
Tie Ult. Strain 0.25
Tie Ult. Strain 0.3
Tie Ult. Strain 0.35
Tie Ult. Strain 0.4
150
Figure 6-9 shows the arrangement of the transverse tie bar connecting the precast
units in the x-direction.
Tie bar in top of the section Tie bar in the middle hieght
Model 1 Model 2
Model 3 Model 4
Model 5 Model 6
Figure 6-9: Arrangement of transverse ties
Table 6-3 presents the characteristics of the transverse tie bars such as their length,
and height position. Tie bar diameters is the same as base case (16 mm).
Transverses Tie
Longitudinal Tie
151
Table 6-3: Position (from bottom of the section) and lengths of the transverse tie bars
Model Position (mm) Length (mm)
1 132 600
2 220 600
3 132 600
4 220 600
5 132 4800
6 220 4800
The results in Figure 6-10 show that introducing transverse tie bars have some
beneficial effects increasing the load carrying capacity of the floor structure,
especially when the tie bars are placed above the mid-height of the slab section
(Models 1,3,5 compared to Models 2, 4, 6).
Figure 6-10: Effects of introducing transverse tie bars
However, since the added tie bars are not connected to any of the support structure,
their contribution to the floor resistance is through increased slab bending resistance,
rather than catenary action. This is confirmed by observing the maximum strains in
the longitudinal and transverse tie bars (Figure 6-11 (b) and (c)). While the
longitudinal tie bars have reached their ultimate tensile strain, the transverse tie bars
are at low levels of strain. What this investigation has demonstrated is the limited
0
50
100
150
200
250
300
0 200 400 600 800 1000 1200
Lo
ad
(k
N)
Vertical Displacement (mm)
Base Case
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
152
effectiveness of introducing transverse tie bars if these tie bars are not anchored to
the edge supports.
(a)
(b)
(c)
Figure 6-11: Strain in main longitudinal tie (b) and two of the transverse ties (c)
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 200 400 600 800 1000 1200
Axia
l S
tra
in
Vertical Displacement (mm)
1st Hollowcore Tie
-0.001
0
0.001
0.002
0.003
0.004
0.005
0.006
0 200 400 600 800 1000 1200
Axia
l S
tra
in
Vertical Displacement (mm)
Trans. Tie A-1
Trans. Tie A-2
1st Hollowcore Tie
Trans. Tie A-1
Trans. Tie A-2
153
Figure 6-11 shows that the strain progress in the transverse tie bars is much lower
compared to the longitudinal ties. This, in general, holds true for the other models
with different tie arrangements.
6.2.5 Procedure of Improving Robustness of Precast Floor Systems
Based on the results of this section, the following hierarchy (benefit from high to
low) of improving the robustness of precast floor system can be suggested:
- Use the thinnest slab depth;
- Use tie bars with the highest ductility;
- Increase tie bar diameter;
- Introduce transverse tie bar.
6.3 Loss of a Centre Column
In this section the loss of a central column is considered. Figure 6-12 shows the
simulation case and the finite element mesh with boundary condition applied on it.
The parameters studied here are the tie bar strain, and diameter and the slab height.
Also the effect of adding transverse ties of the slabs to the adjacent floor panel is
studied. This model has symmetry boundary conditions in the plane of the floor in
two perpendicular directions; hence a quarter of the model is simulated here.
154
(a)
(b)
Figure 6-12: Loss of centre column in a floor system
6.3.1 Slab Height
Figure 6-13 shows the effect of the slab height on the behaviour of the connection
between PCSFs when a central column is lost.
Lost Centre Column Modelled Slab
155
(a)
(b)
(c)
Figure 6-13: effect of slab height when central column is lost
It is seen that the load bearing capacity of the slabs is increased by the increase in
their height in the arching action stage, but as the catenary behaviour is mostly
dependent on the tie bar, this area of the connection behaviour is not affected
significantly.
0
500
1000
1500
2000
2500
3000
0 500 1000 1500 2000
Lo
ad
(k
N)
Vertical Displacement (mm)
Slab Height 150
Slab Height 200
Slab Height 250
Slab Height 300
Slab Height 350
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 100 200 300 400 500 600 700 800
Ed
ge T
ie B
ar S
tra
in
Vertical Displacement (mm)
Slab Height 150
Slab Height 200
Slab Height 250
Slab Height 300
Slab Height 350
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 500 1000 1500 2000
1st
Ho
llo
wco
re T
ie S
tra
in
Vertical Displacement (mm)
Slab Height 150
Slab Height 200
Slab Height 250
Slab Height 300
Slab Height 350
156
Based on the calculated distributed accidental load in Table 6-2, and considering the
fact that the area of floor slabs in question is twice of the previous case the point load
which causes the same bending moment as the distributed load on the slabs is
calculated based on the relationships in (Timoshenko, 1959). Here a point load in a
centre of a slab with four fixed edges is considered:
Figure 6-14: Plate structure with point load at centre (Timoshenko, 1959)
The moment in the centre of the slab (at {
) due to uniform distributed load is
calculated by:
Equation 6-6
Where:
P: is the uniform distributed load
a: is the width of slabs
The equivalent point load is calculated by:
Equation 6-7
157
Table 6-4 shows the equivalent point of the accidental actions:
Table 6-4: Equivalent of accidental point load on the connection
slab height
(mm)
Accidental
Load (kN/m2)
Point Load
(kN) “2spans”
150 7.95 291.51
200 11.33 415.27
250 14.88 545.30
(265) 15.95 584.33
300 24.12 883.56
350 29.69 1087.68
In the case of centre column loss, it can be seen that all the slabs are able to capture
the accidental limit load within the arching action. However the arching action in a
concrete floor leaves the slabs susceptible to sudden brittle crushing failure of
concrete. Hence relying on arching action is not recommended.
6.3.2 Tie Bar Diameter
As in previous cases an increase in the tie bar diameter allows the connection to
carry more load, and again it is seen that the force that the connection undergoes in
the arching action stage is more than the calculated accidental load that the structure
should be able to carry.
158
(a)
(b)
(c)
Figure 6-15: Effect of tie bar diameter when central column is lost
0
200
400
600
800
1000
1200
1400
1600
0 200 400 600 800 1000 1200 1400 1600
Lo
ad
(k
N)
Vertical Displacement (mm)
Tie Diameter 10
Tie Diameter 14
Tie Diameter 18
Tie Diameter 24
Tie Diameter 30
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 200 400 600 800 1000
Ed
ge T
ie B
ar S
tra
in
Vertical Displacement (mm)
Tie Diameter 10
Tie Diameter 14
Tie Diameter 18
Tie Diameter 24
Tie Diameter 30
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 200 400 600 800 1000 1200 1400 1600
1st
Ho
llo
wco
re T
ie S
tra
in
Vertical Displacement (mm)
Tie Diameter 10
Tie Diameter 14
Tie Diameter 18
Tie Diameter 24
Tie Diameter 30
159
6.3.3 Tie Bar Ultimate Strain
Figure 6-16 shows the effect of tie bar ultimate tensile strain variation on the
connection behaviour in the case of central column lost. Similar to the previous cases
this parameter directly affects the catenary action of the connection.
Figure 6-16: effect of tie bar strain when central column is lost
6.3.4 Transverse Tie Bars
In the case of the central column removal, since the slabs in question can be
connected to adjacent slabs with tie bars in both horizontal directions; the failure
mode may change and depend on the failure of the last connecting tie in the
transverse direction.
Figure 6-17: Transverse tie bars in loss of a centre column
0
200
400
600
800
1000
1200
1400
0 500 1000 1500 2000 2500
Lo
ad
(k
N)
Vertical Displacement (mm)
Tie Ult. Strain 0.05
Tie Ult. Strain 0.1
Tie Ult. Strain 0.15
Tie Ult. Strain 0.2
Tie Ult. Strain 0.25
Tie Ult. Strain 0.3
Tie Ult. Strain 0.35
Tie Ult. Strain 0.4
0
200
400
600
800
1000
1200
1400
0 500 1000 1500 2000
Lo
ad
(k
N)
Vertical Displacement (mm)
Base Case
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
160
The results in this section show similar structural behaviour of the floor system as in
the case of edge column loss, with the floor system being able to develop catenary
action. However, since the total applied load under accidental loading is much higher
than in the case of edge column loss (increased floor area) but the slab resistance
under catenary action is similar to that with edge column loss, the floor resistance
under catenary action is very low compared with the first peak load, indicating that
using catenary action would not be effective in resisting the floor load under
accidental loading condition.
6.4 Loss of Corner Column
Figure 6-18 shows a deformed shape of a model with the loss of a corner column.
Since there is no axial restraint available, it is not possible to develop catenary
action. Furthermore, since there is little slab bending resistance, the floor resistance
is very low, as shown by the load-displacement relationship in Figure 6-19-a.
Although this study is brief, it suggests that using the current construction practice,
precast floor system has little resistance to accidental loading in the event of corner
column loss. The method of providing tie bars is not effective. The only means of
providing resistance under corner column loss by improved bending resistance of the
corner bay slab through continuous construction.
Figure 6-18: Deformed shape of the model with loss of a corner column
161
(a)
(b)
Figure 6-19: Load-displacement and tie bar strain diagram under corner column loss
6.5 Conclusions
This chapter has presented the results of an extensive set of parametric studies to
investigate the residual floor resistance after removal of an edge, an interior and a
corner column. The parameters investigated include those that were found to be able
to increase precast floor catenary action in Chapter 4. The main conclusions are:
(1) In the case of corner column removal, the existing means of providing tie
bars is not effective. The structure will have very little resistance due to a
lack of horizontal restraints to help develop catenary action and a lack of
bending resistance. Continuous slab construction in the corner bays should be
considered.
(2) In the case of interior column removal, the floor resistance under catenary
action is very low and is unlikely to resist the accidental loading.
0
10
20
30
40
50
60
0 200 400 600 800 1000 1200
Lo
ad
(k
N)
Vertical Displacement (mm)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0 200 400 600 800 1000 1200 1400
Str
ain
Vertical Displacement (mm)
162
(3) In the case of edge column loss, catenary action can develop in the tie bars to
provide some resistance to accidental loading. To further improve the
effectiveness of this method, the following (in order of decreasing benefit)
should be considered:
- Use the thinnest slab depth;
- Use tie bars with the highest ductility;
- Increase tie bar diameter;
- Introduce transverse tie bar.
163
Chapter 7. Conclusion and Further Studies
This thesis presented the results of a numerical investigation into the resistance of tie
connection between Precast Concrete Floor Slabs (PCFS) against progressive
collapse and methods of improving this resistance. The objectives of the research
were:
A reliable FE model to examine the factors affecting the behaviour of the
connection between two Precast Concrete Floor Slabs (PCFS) until fracture
was established;
Parameters that affect PCFS tie connection behaviour in catenary action were
identified;
A predictive analytical method to foretell the behaviour of the PCFS tie
connection, validated by the parametric study results, was developed;
Effectiveness of current building code regulations and practical construction
methods for providing robustness of PCFS tie connection in steel framed
buildings, was assessed;
Methods for improving robustness of precast concrete floors in steel frame
buildings were suggested;
Mechanisms of collapse, due to loss of a column in different locations, in
representative steel framed buildings with PCFS were studied, and
suggestions for each case were made.
The main conclusions of this research were:
7.1 Literature Review
A brief review of the incidents leading the British Standard to consider the
robustness regulations was given. The background theory of robustness regulations
with special interest to tying regulations was presented and it was shown that the
current tying method is derived by only considering the equilibrium of forces in
catenary action. Neglecting the ductility of the tie bar, as the main parameter for
development of catenary action, can make the tying method susceptible to failure.
164
It was shown that although the effectiveness of the tying method has already been
under scrutiny, there was a lack of suitable research on tying method and parameters
affecting the tie behaviour and ductility. The only major study (Engström, 1992) on
tying of PCFS was reviewed and it was shown that there were fundamental issues in
the tests’ boundary conditions such as lack of axial restraint needed for inducing
proper tension in the tie bar. The assumptions made for development of an analytical
relationship by Engström (1992) suffered from some drawbacks such as:
Assumption of full tension in ties while slabs are totally suspended
Assumption of equal elongation in ties at both ends of slab and the tie
between slabs
The model could have benefitted from more realistic boundary conditions
By a brief review on the construction technology of PCFS, it was seen that due to
prestressing of tendons in the PCFS, prestressed slabs can cover a rather large span
and also can carry higher load in comparison to ordinary concrete slabs. This leads to
higher accidental load being applied to the PCFS tie connection, while the current
tying regulations do not take this extra loading into account.
7.2 Finite Element Model & Validation
The commercial package TNO-DIANA was chosen in this study as it was considered
to be an effective tool in modelling concrete owing to its many concrete modelling
options in different situations.
To establish the validity of DIANA models, a series of tests by Su et al. (2009) on
two axially restrained concrete beam assemblies with the middle support column
removed were simulated. The simulation results were close to the test results for the
vertical load-displacement relationship, the axial reaction force-displacement
relationship and the different stages of structural behaviour of the assembly.
Based on the validation results and the sensitivity study results, the following
simulation methodology was adopted:
The chosen material model for concrete was the Mohr-Coulomb plasticity for
compression and linear tension softening for tensile regime. The
165
reinforcement was modelled with Von-Mises plasticity with perfect plastic
behaviour.
The element type chosen was the 20-noded solid brick quadratic element,
capable of capturing the bending of the model properly.
The mesh sensitivity study showed the element size of about 40 mm to yield
the best results along with the optimum analysis cost efficiency.
The boundary condition of the model should have appropriate axial restraint
in order to enable the model to depict arching and the following catenary
action.
The reinforcement was modelled using the embedded reinforcement option in
DIANA, which assumes a fully bonded condition of the steel rebar and
surrounding concrete.
7.3 Two-Dimensional Analysis of Slabs
To investigate whether the tie connection, designed according to current regulations,
between PCFS is able to resist the accidental loading after removal of the central
support, a two-dimensional model was developed. A series of simulations according
to manufacturer (Bison, 2012) load-span tables was performed. It was shown that the
connection’styingresistancewasnotcapableofdevelopingenoughcatenaryaction
in order to resist the accidental load limit.
To suggest methods for improvement of connection resistance, a series of
parametrical studies were carried out to evaluate the effects of changing the
following design parameters: tie bar diameter, position, length, yield stress and
ultimate strain; the slab’s height, length; and the compressive strength of the
grouting concrete in between the slabs which encases the tie bar. In this parametric
investigation, it was assumed the slabs were vertically supported and had various
degrees of axial restraint. Furthermore, it was shown that the axial restraint has only
a minor effect as long as the restraint has a capacity greater than the tensile strength
of the tie bars.
To improve the connection resistance to accidental loading, the following methods
may be used:
166
Slab height: as the slab height increased it was shown that the tie bar ruptured
sooner; hence the use of thinnest possible slab is recommended.
Tie bar height: it was shown that due to different factors such as concrete
cover, distance between the forces on the slab, and straining of the tie bar, the
optimum location for the tie bar was at the mid-height of the slab.
Tie bar diameter: increasing the tie bar diameter boosted both resistance and
the elongation of the tie bar where the latter resulted in more vertical
displacement as well.
Tensile ultimate strain of the tie bar: directly affected the elongation and
ductility of the tie bar in catenary action, resulting in both higher resistance
and vertical displacement.
The tie bar yield stress: directly affected the tying force. Increase in the tie
bar yield stress increased the load carrying capacity of the connection, but the
elongation of the tie bar is unchanged.
7.4 Predictive Analytical Relationship of the 2D Model
Based on the parametric study results, a predictive analytical relationship has been
developed for the catenary action stage of the PCFS tie connection. In this
development, it was assumed that the tie bars could develop their full tensile strength
and slab failure was due to the tie bar fracture. The slabs were assumed to be cracked
through at the slab-slab connection interface and each slab underwent rigid
movements. Vertical displacement of the PCFS was purely due to elongation of the
tie bar. In order to obtain the total tie bar elongation, strain distributions along the tie
bar, from the numerical simulation models, were used to develop a series of
regression equations to relate the tie bar elongation as a function of the four factors
as follows:
Strain distribution with slab height:
Equation 7-1
Strain distribution with tie bar diameter:
167
Equation 7-2
Strain distribution and tie bar height:
Equation 7-3
Strain distribution and ultimate tensile strain:
Equation 7-4
The resulting load-vertical displacement of the PCFS is as follows:
{ √ (
) }
Equation 7-5
Based on comparison with all the models in the parametric study in Chapter 5, the
analytical method was shown to produce reasonably close results with the simulation
results for the load-vertical displacement relationship, the maximum displacement
and the maximum resistance of the PCFS tie connection.
7.5 Three-Dimensional Simulation of the Floor System
To assess whether the conclusions of the 2D simulations were applicable to the
more realistic 3-D structure, and to investigate the feasibility of using transverse tie
bars to improve the PCFS resistance to accidental loading, a number of scenarios of
realistic 3-D arrangement were simulated. These scenarios included corner column
removal, edge column removal and interior column removal.
The simulation results suggest that the transverse ties were not effective in boosting
the resistance of the structure under catenary action.
168
When a corner column of the structure was lost the development of catenary action
was not possible, due to a lack of axial restraint. The only method of resisting the
accidental loading in this case may be providing the slabs with enough bending
resistance.
In the case of an edge column removal, the connection had the chance to develop
catenary action, but it was shown that the current design regulations did not provide
the connection with enough resistance to withstand the accidental load limit. In the
case of an internal column removal, the connection developed catenary action; but
since the load bearing surface related to the connection was increased, it was less
likely for the catenary action resistance to capture the accidental loading.
7.6 Limitations of the Current Study
Due to difficulty of overcoming numerical divergence when using load control in
DIANA simulations, displacement control was employed in this study. This means
that only point loading was considered. Nevertheless, because the conclusions of this
research were based on relative performance of one PCFS tie connection under one
point load, it is considered that the conclusions of this research are generally valid
when the slabs are under other loading conditions.
7.7 Future Studies
For completion of understanding of the PCFS tie connection behaviour, the
following recommendation may be made:
Experiments on the PCFS connection with realistic boundary conditions are
required:
The current literature regarding the tying of PCFS would benefit from
experimental data that considers other structural elements in the floor.
Although it was shown that theaxialstiffnessoftheslabs’restraintsdid not
affect the catenary action significantly; this cannot be true if no axial restraint
are considered for the slabs as in Engström (1992), since in this case, the
effect of arching and catenary actions are neglected.
Dynamic effect of column removal can be considered:
169
In the current study the behaviour of the PCFS tie connection was studied
considering that the load on the connection was applied in a static fashion.
Although depending on the nature of the accidental action this may be the
case, the column removal may happen in an abrupt manner which could add
dynamic aspects to the situation.
The effect of connections of the transverse beams on supporting the slabs
may be taken into account:
The focus of the current study was on the tie bar between the PCFS, while
under any longitudinal tie connection lies a beam, which itself is connected to
other structural members at its far end. If this connection is designed to
capture bending moments, it could have some effect on supporting the slabs
in question.
Considering discrete cracking for the connection between the PCFSs may
enhance the accuracy of the FE analysis:
Experiments (Engström, 1992) have shown that in the case of a column loss
and large displacement of the connection between slabs, the interface
between the slabs and grouting concrete detaches and forms the first crack of
the connection. Although the adopted concrete material model was able to
predict the location of the cracks properly, applying the discrete crack
approach that depicts the actual discretisation of the mesh in concrete would
allow more realistic study of the strain distribution along the tie bar.
Modelling the discrete crack in the finite element software DIANA is via the
interface elements. In the current version of DIANA the combination of two
interface elements leads to numerical divergence and since the bond-slip
behaviour of the tie bar was modelled with interface elements as well, the
smeared cracking approach was chosen over the discrete cracking model.
170
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176
Appendix 1: Assessment of tie bars designed
according to British Standard regulations
Slabs with recommended span and load from a manufacturer (Bison, 2012) are
modelled. Based on the permanent and variable load in the load-span table
(Table 2-3), the accidental loading and tie bar diameter are determined for a 5-storey
building. It is shown that, due to high bending resistance of PCFS, the tie bar is not
capable of providing robustness. Hence it is recommended that the tie bar be
designed considering the ductility of the tie bar (as addressed in section 5.3).
Since the variable load in Table 2-3 includes concrete topping and selfweight of the
PCFS, the imposed load is considered to be constant as in section 2.1.1. The tie force
is calculated according to Equation 2-2 (repeated below), and the tie bar diameter is
determined based on the required cross sectional area of two steel rebars at yield
stress in each slab of width 1.2 m. The accidental load as a point load is calculated as
explained in section 4.2.
Equation A1-1
Equation A1-2
Based on the calculation results using the two above equations, the following
simulations (Table A1-1) were performed.
177
Table A1-1: Simulations based on Bison (2012) load-span table
Slab
Height
(mm)
Slab
Length
(m)
Variable
Load
(kN/m2)
Imposed
Load
(kN/m2)
Tying
Force
(kN/m)
Tie
Diameter
in 2D
(mm)
Accidental
Load (kN)
Model
Name
250
6.9 10 3.8 101.57 28 71.48
SH250-
SL6.9
8 5 3.8 75.1 24 66.88
SH250-
SL8
8.8 4 3.8 73.22 24 70.05
SH250-
SL8.8
9 3.5 3.8 70.08 24 69.84
SH250-
SL9
9.9 1.5 3.8 55.96 20 68.9
SH250-
SL9.9
10.4 0.75 3.8 50.47 20 69.26
SH250-
SL10.4
300
6.8 15 3.8 136.36 32 84.04
SH300-
SL6.8
7.8 10 3.8 114.81 28 80.8
SH300-
SL7.8
9 7 3.8 103.68 28 82.44
SH300-
SL9
9.9 4 3.8 82.36 24 78.8
SH300-
SL9.9
10.9 2 3.8 67.43 24 78.04
SH300-
SL10.9
11.7 0.75 3.8 56.78 20 77.92
SH300-
SL11.7
Figure A1-1 compares the simulations’ load-deflection curves with the accidental
loads.
0
10
20
30
40
50
60
70
80
0 100 200 300 400 500 600 700
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
SH250-SL6.9
178
0
10
20
30
40
50
60
70
80
0 100 200 300 400 500 600 700
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
SH250-SL8
0
10
20
30
40
50
60
70
80
0 100 200 300 400 500 600 700
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
SH250-SL8.8
0
10
20
30
40
50
60
70
80
0 100 200 300 400 500 600 700
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
SH250-SL9
0
10
20
30
40
50
60
70
80
0 100 200 300 400 500 600 700
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
SH250-SL9.9
179
0
10
20
30
40
50
60
70
80
0 100 200 300 400 500 600 700 800
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
SH250-SL10.4
0
10
20
30
40
50
60
70
80
90
0 100 200 300 400 500 600 700
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
SH300-SL6.8
0
10
20
30
40
50
60
70
80
90
0 100 200 300 400 500 600 700
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
SH300-SL7.8
0
10
20
30
40
50
60
70
80
90
0 100 200 300 400 500 600 700 800
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
SH300-SL9
180
Figure A1-1: comparison between simulation load-deflection curves and accidental load, with
BS tie bar
The results in Figure A1-1 indicate that relying only on the tying force, as specified
in the current design method, is not able to provide robustness for PCFS.
Using Equation 5-12 and the procedure explained in section 5.3, improvement can be
made to the designs in Table A1-1 by changing the tie bar diameters. Table A1-2
shows required tie bar diameters for slabs of thickness 250mm and 300mm.
0
10
20
30
40
50
60
70
80
90
0 100 200 300 400 500 600 700
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
SH300-SL9.9
0
10
20
30
40
50
60
70
80
90
0 100 200 300 400 500 600 700 800
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
SH300-SL10.9
0
10
20
30
40
50
60
70
80
90
0 100 200 300 400 500 600 700 800
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
SH300-SL11.7
181
Table A1-2: Models with recommended tie diameters
Slab Height
(mm)
Model Name Tie
Diameter
in 2D
(mm)
Slab Height
(mm)
Model Name Tie
Diameter
in 2D
(mm)
250
SH250-SL6.9 32
300
SH300-SL6.8 36
SH250-SL8 32 SH300-SL7.8 36
SH250-SL8.8 34 SH300-SL9 38
SH250-SL9 34 SH300-SL9.9 38
SH250-SL9.9 34 SH300-SL10.9 38
SH250-SL10.4 36 SH300-SL11.7 38
The new simulation results are shown in Figure A1-2. They all confirm that the new
tie bar diameters are able to provide the slabs with sufficient resistance against the
accidental loads.
0
10
20
30
40
50
60
70
80
90
0 100 200 300 400 500 600 700 800
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
Rec-SH250-SL6.9a
0
10
20
30
40
50
60
70
80
0 100 200 300 400 500 600 700 800
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
Rec-SH250-SL8
182
0
10
20
30
40
50
60
70
80
0 100 200 300 400 500 600 700 800
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
Rec-SH250-SL8.8
0
10
20
30
40
50
60
70
80
0 100 200 300 400 500 600 700 800
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
Rec-SH250-SL9
0
10
20
30
40
50
60
70
80
0 200 400 600 800 1000
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
Rec-SH250-SL9.9
0
10
20
30
40
50
60
70
80
0 200 400 600 800 1000
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
Rec-SH250-SL10.4
183
0
10
20
30
40
50
60
70
80
90
100
0 100 200 300 400 500 600 700
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
Rec-SH300-SL6.8
0
10
20
30
40
50
60
70
80
90
100
0 100 200 300 400 500 600 700 800
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
Rec-SH300-SL7.8
0
10
20
30
40
50
60
70
80
90
100
0 100 200 300 400 500 600 700 800
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
Rec-SH300-SL9
0
10
20
30
40
50
60
70
80
90
100
0 200 400 600 800 1000
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
Rec-SH300-SL9.9
184
Figure A1-2: Comparison of simulations’ load-deflection curve with accidental load, with
recommended tie bar
It should be noted that other parameters (such as tie bar position and ultimate strain)
could have been changed in order to enhance the slab performance. Here, though,
only the tie bar diameter was changed to illustrate the practicality of the tie bar size
determined by this method.
Table A1-3 compares the tying forces for the simulated cases recommended by the
BS and tying force resulting from the tie bar diameter recommended in Table A1-2.
0
10
20
30
40
50
60
70
80
90
0 200 400 600 800 1000
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
Rec-SH300-SL10.9a
0
10
20
30
40
50
60
70
80
90
0 200 400 600 800 1000
Lo
ad
(k
N)
Vertical Displacement (mm)
Accidental Load
Rec-SH300-SL11.7
185
Table A1-3: Comparison of tie forces of the BS and recommended ties
Slab Height
(mm)
Model Name BS Tie
Force
(kN/m)
Recommended
Tie Force
(kN/m)
Percentage more
(Recommended/BS)
250
SH250-SL6.9 101.57 334.93 69.67
SH250-SL8 75.1 334.93 77.57
SH250-SL8.8 73.22 378.11 80.63
SH250-SL9 70.08 378.11 81.46
SH250-SL9.9 55.96 378.11 85.19
SH250-SL10.4 50.47 423.9 88.09
300
SH300-SL6.8 136.36 334.93 67.83
SH300-SL7.8 114.81 334.93 72.91
SH300-SL9 103.68 378.11 78.04
SH300-SL9.9 82.36 378.11 82.56
SH300-SL10.9 67.43 378.11 85.72
SH300-SL11.7 56.78 423.9 87.97
This comparison shows that the recommended tie force is 60-80% more than the tie
force calculated according to BS regulations. However, it should be noted that the
results of this study shows relying merely on the tie force does not necessarily
provide the PCFS connection with adequate robustness under accidental loading, and
other design parameters should be taken into account.
186
Appendix 2: Evaluation of Analytical Relation
To evaluate the relationship of the tying force of PCFS connection in catenary action
several models have been constructed in which the parameters in the tying force
relationship (Equation 5-4 presented in Chapter 5) has changed. Also these models
are verified against the relationship for prediction of the maximum displacement and
tying force in the catenary action stage (Equation 5-12 in Chapter 5). Table A2-1 and
Table A2-2 show the characteristics of the constructed models with their model
number which is used in the graphs below in order to specify the models.
187
Table A2-1: Models with tie bar of diameter 10 mm
Tie Bar
Diameter
(mm)
fy.steel
(MPa)
Slab
Height
(mm)
Slab
Length
(m)
Model
10
400
200
5 10-1
6 10-2
7 10-3
250
5 10-4
6 10-5
7 10-6
300
5 10-7
6 10-8
7 10-9
500
200
5 10-10
6 10-11
7 10-12
250
5 10-13
6 10-14
7 10-15
300
5 10-16
6 10-17
7 10-18
600
200
5 10-19
6 10-20
7 10-21
250
5 10-22
6 10-23
7 10-24
300
5 10-25
6 10-26
7 10-27
188
Table A2-2: Models with tie bar of diameter 20 mm
Tie Bar
Diameter
(mm)
fy.steel
(MPa)
Slab
Height
(mm)
Slab
Length
(m)
Model
20
400
200
5 20-1
6 20-2
7 20-3
250
5 20-4
6 20-5
7 20-6
300
5 20-7
6 20-8
7 20-9
500
200
5 20-10
6 20-11
7 20-12
250
5 20-13
6 20-14
7 20-15
300
5 20-16
6 20-17
7 20-18
600
200
5 20-19
6 20-20
7 20-21
250
5 20-22
6 20-23
7 20-24
300
5 20-25
6 20-26
7 20-27
189
0
2
4
6
8
10
12
14
16
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-1
0
2
4
6
8
10
12
14
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-2
0
2
4
6
8
10
12
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-3
190
-5
0
5
10
15
20
25
30
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-4
0
5
10
15
20
25
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-5
0
2
4
6
8
10
12
14
16
18
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-6
191
0
5
10
15
20
25
30
35
40
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-7
-5
0
5
10
15
20
25
30
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-8
0
5
10
15
20
25
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-9
192
0
2
4
6
8
10
12
14
16
18
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-10
0
2
4
6
8
10
12
14
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-11
0
2
4
6
8
10
12
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-12
0
5
10
15
20
25
30
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-13
193
0
2
4
6
8
10
12
14
16
18
20
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-14
0
2
4
6
8
10
12
14
16
18
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-15
0
5
10
15
20
25
30
35
40
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-16
0
5
10
15
20
25
30
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-17
194
0
5
10
15
20
25
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-18
0
2
4
6
8
10
12
14
16
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-19
0
2
4
6
8
10
12
14
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-20
0
2
4
6
8
10
12
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-21
195
0
5
10
15
20
25
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-22
0
2
4
6
8
10
12
14
16
18
20
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-23
0
2
4
6
8
10
12
14
16
18
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-24
0
5
10
15
20
25
30
35
40
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-25
196
0
5
10
15
20
25
30
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-26
0
5
10
15
20
25
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 10-27
197
0
5
10
15
20
25
30
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-1
0
5
10
15
20
25
30
35
0 100 200 300 400 500 600 700
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-2
0
5
10
15
20
25
30
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-3
0
5
10
15
20
25
30
35
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-4
198
0
5
10
15
20
25
30
0 100 200 300 400 500 600
Axis
Tit
le
Vertical Displacement (mm)
Model 20-5
0
5
10
15
20
25
30
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-6
0
5
10
15
20
25
30
35
40
45
50
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-7
0
5
10
15
20
25
30
35
40
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-8
199
0
5
10
15
20
25
30
35
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-9
0
5
10
15
20
25
30
35
40
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-10
0
5
10
15
20
25
30
35
40
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-11
0
5
10
15
20
25
30
35
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-12
200
0
5
10
15
20
25
30
35
40
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-13
0
5
10
15
20
25
30
35
40
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-14
0
5
10
15
20
25
30
35
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-15
0
5
10
15
20
25
30
35
40
45
50
0 100 200 300 400 500
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-16
201
0
5
10
15
20
25
30
35
40
45
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-17
0
5
10
15
20
25
30
35
40
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-18
0
5
10
15
20
25
30
35
40
45
50
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-19
0
5
10
15
20
25
30
35
40
0 100 200 300 400 500 600 700
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-21
202
0
10
20
30
40
50
60
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-22
0
5
10
15
20
25
30
35
40
45
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-23
0
5
10
15
20
25
30
35
40
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-24
0
5
10
15
20
25
30
35
40
45
50
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-26
203
0
5
10
15
20
25
30
35
40
45
0 100 200 300 400 500 600
Lo
ad
(k
N)
Vertical Displacement (mm)
Model 20-27