numerical modeling for the seismic response of …

Microsoft Word - Thesis-ALLreviewed9999.docCONCRETE TILT-UP BUILDINGS
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF APPLIED SCIENCE
(Civil Engineering)
(Vancouver)
Concrete tilt-up building is a prevalent construction technique used for industrial and
commercial applications in North America. This construction technique offers many
significant advantages over conventional cast-in-place construction. This includes the
reduction in construction time and the amount of formworks. Despite the large array of
buildings that has been constructed using such technique, the nonlinear behaviour of the
concrete tilt-up buildings is still not well understood.
The nonlinear behaviour of the concrete tilt-up building has been studied in this thesis. The
nonlinear response of the concrete tilt-up building is largely affected by the nonlinear
behaviour of the connectors between the panels and the slab, and between the panels. Past
researches have been conducted to experimentally examine the nonlinear behaviour of
the tilt-up panel connectors. The experimental results were used in this thesis to
develop an empirical numerical model capable of reproducing the force-deformation
response of the tilt-up connectors under combined axial and shear deformation. The
numerical model takes the shear strength and stiffness degradation into account after
axial cycles of inelastic deformation.
A finite-element software was developed specifically to study the nonlinear static and
dynamic behaviour of concrete tilt-up buildings. A typical tilt-up building designed in the
study of Olund (2009) was modeled. Incremental dynamic analysis was performed using
the developed finite element software to assess the seismic performance of the prototype
tilt-up building. The results of the incremental dynamic analysis provided valuable
information to understand the nonlinear behaviour of the concrete tilt-up buildings under
seismic load. Detailed parametric studies were carried out to examine the nonlinear
behaviour of tilt-up buildings. Parameters such as connector configurations; variation of
the roof stiffness and strength; and coefficient of friction between the panels and slab were
studied.
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1.2 Work needed ...................................................................................................................... 2
1.4 Thesis overview ................................................................................................................. 4
2 - LITERATURE REVIEW .............................................................. 6
2.1 Review of the current construction techniques and design approach ................................ 6
2.1.1 Roof diaphragm .......................................................................................................... 6
2.2 Review of previous experimental tests conducted on connectors ................................... 19
2.2.1 Experimental testing by Lemieux, Sexsmith and Weiler (1998).............................. 20
2.2.2 Experimental testing by Devine (2009) .................................................................... 23
2.3 Previous research on roof diaphragm .............................................................................. 26
2.4 Review of previous analytical models ............................................................................. 28
2.4.1 Olund (2009)............................................................................................................. 28
3 - NUMERICAL MODEL FOR THE EM5 CONNECTOR ....... 33
3.1 Summary of experimental tests used in this study........................................................... 33
3.2 General mechanical behaviour of the EM5 connector..................................................... 35
3.2.1 Monotonic and cyclic envelope ................................................................................ 35
3.2.2 Cyclic behaviour and damage................................................................................... 37
3.3.1 Mathematical rules for the uncoupled model............................................................ 39
3.3.2 Coefficients of the uncoupled model ........................................................................ 48
3.3.3 Model of coupling..................................................................................................... 48
3.3.5 Model of paired connectors ...................................................................................... 53
3.4 Validation of the model ................................................................................................... 56
3.4.1 Shear behaviour ........................................................................................................ 56
3.4.5 Sensitivity ................................................................................................................. 63
OF TILT-UP BUILDINGS................................................................ 64
4.4 Model Validation ............................................................................................................. 92
4.4.1 Hand calculation ....................................................................................................... 92
4.4.3 Comparison with another numerical model............................................................ 101
5 - NONLINEAR RESPONSE OF TILT-UP BUILDINGS ........ 106
5.1 Seismic performance evaluation of the prototype building ........................................... 107
5.1.1 Seismic hazard ........................................................................................................ 107
5.2.2 Connection configuration - Nonlinear dynamic analysis........................................ 119
5.2.3 Coefficient of friction ............................................................................................. 121
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6 - SUMMARY AND CONCLUSIONS ......................................... 125
6.1 Summary........................................................................................................................ 125
6.3 Conclusion ..................................................................................................................... 128
APPENDIX B. TIME-HISTORY ANALYSIS RESULTS .......... 135
APPENDIX C. MODEL CODE...................................................... 141
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Table 2.1 Connections shear strength (Modified from Lemieux et al., 1998)........................... 22
Table 2.2 Steel deck diaphragm configurations (Essa et al., 2003)........................................... 27
Table 2.3 Shear strength and stiffness in monotonic tests (Essa et al., 2003) ........................... 27
Table 2.4 Shear strength and stiffness in cyclic tests (Essa et al., 2003)................................... 28
Table 3.1 Summary of the experimental tests used to develop the EM5 connection model ..... 34
Table 3.2 Coefficients used for the uncoupled model ............................................................... 48
Table 4.1 Roof model parameters .............................................................................................. 91
Table 4.2 Peak rocking drift reduction .................................................................................... 103
Table 5.1 Summary of the selected ground motions................................................................ 107
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LIST OF FIGURES
Figure 2.1 Formwork layout before casting of concrete tilt-up panels (Devine, 2009)............... 8
Figure 2.2 EM1 joist seat (left - CAC Concrete handbook 2006 , right - Devine, 2009).......... 13
Figure 2.3 EM2 shear plate (left - CAC Concrete handbook 2006; right - Devine, 2009)........ 14
Figure 2.4 Use of EM2 shear plate to connect the diaphragm angle to the panels (CAC
Concrete handbook 2006 ) ......................................................................................................... 14
Figure 2.5 EM3 shear plate (CAC Concrete handbook 2006)................................................... 15
Figure 2.6 EM4 shear plate (CAC Concrete handbook 2006)................................................... 15
Figure 2.7 EM5 connection (left - CAC Concrete handbook 2006 , right - Devine, 2009) ...... 16
Figure 2.8 Slab to Panel Connection (Lemieux, Sexmith and Weiler, 1998)............................ 17
Figure 2.9 Free body diagram of tilt-up panels (Olund, 2009).................................................. 18
Figure 2.10 Shear test setup (Lemieux et al., 1998) .................................................................. 20
Figure 2.11 Load-displacement plots of shear tests (Lemieux et al., 1998) .............................. 21
Figure 2.12 Rendering of testing setup (Devine, 2009)............................................................ 24
Figure 2.13 Steel deck roof diaphragm testing setup (Essa et al., 2003)................................... 26
Figure 2.14 IDA drift results from 8 ground motion records (Olund 2009):............................. 30
Figure 3.1 Shear behaviour of the EM5 connector .................................................................... 35
Figure 3.2 Axial behaviour of the EM5 connector .................................................................... 36
Figure 3.3 Trilinear model for cyclic behaviour ........................................................................ 37
Figure 3.4 Unsymmetric connection behaviour......................................................................... 38
Figure 3.5 Force-deformation response regions of the EM5 connector model ......................... 40
Figure 3.6 Optimum coefficient rDisp for each cycle ............................................................... 43
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Figure 3.7 Model for the increase of the coefficient rForce when maximum displacement is
increased .................................................................................................................................... 46
Figure 3.8 Optimum backbone parameters F1, F2 VS Maximum vertical displacement for the
coupled tests............................................................................................................................... 49
Figure 3.9 Optimum degradation rate parameter Wref Vs Maximum vertical displacement ... 50
Figure 3.10 Horizontally dissipated energy at failure (left) and shear displacement at failure
(right) for all the coupled tests as a function of the maximum vertical displacement ............... 52
Figure 3.11 Backbones of the paired connector model.............................................................. 55
Figure 3.12 Comparison of the shear model with monotonic shear test results ....................... 56
Figure 3.13 Comparison of uncoupled shear model with experimental curves......................... 57
Figure 3.14 Comparison of axial model with monotonic test results ........................................ 58
Figure 3.15 Comparison of model with pure axial cyclic tests.................................................. 58
Figure 3.16 Comparison of axial model with seven uplift tests ................................................ 59
Figure 3.17 Comparison of model with coupled tests where the connection is held-up ........... 60
Figure 3.18 Comparison of model with coupled tests where the connection is brought down to
zero axial displacement.............................................................................................................. 61
Figure 3.19 Comparison of the proposed EM5 model with the model of Devine (2009) ......... 62
Figure 4.1 Simplified roof model............................................................................................... 66
Figure 4.4 Local degrees of freedom ......................................................................................... 70
Figure 4.5 The two rocking cases and their coordinates............................................................ 72
Figure 4.6 Inputs and outputs of the contact algorithm ............................................................. 74
Figure 4.7 Inputs and outputs of the uplift algorithm ................................................................ 76
Figure 4.8 Functioning of the uplift algorithm on a 2 panel example ....................................... 77
x
Figure 4.10 Functioning of the rocking algorithm..................................................................... 79
Figure 4.12 Modification of external force for equilibrium check ............................................ 82
Figure 4.13 Degrees of freedom defined at the middle of the panels ........................................ 83
Figure 4.14 Use of contact algorithm in dynamics .................................................................... 86
Figure 4.15 General layout of the typical building used for the analysis (Olund, 2009)........... 88
Figure 4.16 OpenSees 2D roof model........................................................................................ 90
Figure 4.20 Two panel model for hand calculation ................................................................... 96
Figure 4.21 Comparison of pushover behaviour of the model with F.Devine's model (2009) for
five configurations ................................................................................................................... 100
Figure 4.22 Perform 3D panel model ...................................................................................... 102
Figure 4.23 Comparison of the proposed model with the Perform model using time history
analysis with one ground motion record scaled at four amplitudes......................................... 104
Figure 5.1 Design spectrum and spectra of the selected ground motions with 5% damping .. 108
Figure 5.2 IDA results for the 10 selected ground motion records.......................................... 109
Figure 5.3 Pushover loading .................................................................................................... 112
Figure 5.4 Connection failure sequence................................................................................... 114
Figure 5.5 Pushover curves for configurations using two panel-slab connectors per panel.... 115
Figure 5.6 Pushover curves for configurations using three panel-slab connectors per panel.. 116
Figure 5.7 Pushover curves for configurations using four panel-slab connectors per panel ... 116
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Figure 5.8 Demand parameters for various connector configurations..................................... 119
Figure 5.9 Sliding and rocking demand for various values of the coefficient of friction µ .... 121
Figure 5.10 Demand parameters for various roof diaphragm characteristics .......................... 123
1
1.1 General description of tilt-up construction method
Tilt-up construction is a method widely used in North America to build warehouses, schools,
and offices. Inspired by timber construction, where the façade of a building was often
assembled on the ground before being tilted in place, the first tilt-up buildings appeared in the
1940's. The process is the following: a concrete slab is cast first and is then used as a formwork
to cast concrete panels. Once the panels are set, they are lifted and put at their final place using
a crane.
Compared to regular concrete construction methods, tilt-up requires a very small amount of
formwork. In addition, all wall elements can be cast at once. This makes tilt-up buildings
generally faster to build and more economical than cast-in-place buildings.
Tilt-up construction was primarily used for large, single story box-type warehouse structures
with few doors and small openings. Advances in the method allow nowadays for the
construction of multiple story buildings with more varied shapes and large openings. In this
way, a broad range of structures can now be conceived using tilt-up construction. Moreover,
tilt-up structures can be more functional and architecturally pleasing than in the past.
In tilt-up buildings, panels are used to resist both the vertical load from the roof and floor and
the lateral loading from wind and earthquakes. When it comes to using tilt-up panels as lateral
force resisting systems, panels can be divided into two categories: panels with large openings,
which behave like moment frames, and solid rectangular panels. The latter have a very
particular behaviour compared to regular concrete shear walls. Indeed, solid panels can rock
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and slide on their foundation, and their behaviour is greatly affected by the type and number of
connectors used to link them to the slab and to each other.
The specificity of concrete construction has only been taken into account recently in the
Canadian building code. Their design was first addressed in 1994 in the Design of Concrete
Structure Standards CAN/CSA A23.3-94. Currently, tilt-up buildings in Canada are designed
following the recommendations of Weiler (2006) in the Cement Associations of Canada
Concrete Design Handbook - Third Edition.
1.2 Work needed
The seismic behaviour of solid tilt-up panels systems is currently not well understood.
Displacement of panels is mainly due to two motions: sliding and rocking. The deformation of
the panel is relatively small as compared to the sliding and rocking motions. The modeling of
the sliding and rocking motions is highly nonlinear. Due to the presence of frictional and
contact forces, a minimum force is required to trigger panel rocking or sliding. These motions
are also responsible for a significant amount of energy dissipation during cyclic loading.
The current connection design procedure aims at preventing both sliding and overturning
failure. The capacity calculation for those two failure modes are made independently. The
necessary number of panel-slab connections is calculated first to resist the base shear of the
roof and the panels, without consideration of degradation due to axial connection damage
during rocking. Overturning capacity is then calculated by using the panel weight and the
strength of the panel-panel connectors, without taking into account the overturning capacity of
the panel-slab connections.
This procedure is made using static analysis. The ductility of the rocking motion is much
higher than the one of the sliding motion. Indeed, on the one hand, the sliding capacity is
limited by the panel-slab connection whose failure in shear occurs between 2 mm and 16 mm
depending on the connector embedded in the panel. On the other hand, the overturning
capacity is limited by the vertical displacement capacity of the panel-slab connection and by
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the shear capacity of the panel-panel connectors. The vertical displacement capacity of the
panel-slab connection is about 100 mm, and the shear capacity of the panel-panel connector is
above 33 mm. The ability for the system to deform inelastically is usually taken into account
by using a higher force reduction factor (Rd) for the overturning demand (typically 1.5) than
for the sliding demand (for which Rd=1.0 is usually used). However, this choice has not been
justified by detailed dynamic analysis.
The main obstacle facing dynamic analysis of tilt-up buildings using solid panels is the model
of the connections. The nonlinear response of the connections plays a key role in the seismic
behaviour of the building, and until very recently, limited experimental data was available to
understand their mechanical behaviour. Experimental tests have been conducted at the
University of British Columbia by Lemieux et al. (1998) to investigate the ductility and the
strength of the most common tilt-up connectors. Although these tests provided very valuable
information about the cyclic response of the connectors, experimental results were still missing
in order to understand the cyclic behaviour of connectors used in tilt-up buildings with solid
panels. In particular, no test was performed in the axial direction of the EM5 connector, this
direction being the one resisting overturning in the panel-slab connection.
A new test series on the connectors has been performed by Devine (2009) at the University of
British Columbia. This series includes uplift tests on the panel-slab connection, as well as
coupled tests, where the shear behaviour of panel-slab connectors previously damaged by uplift
was studied. It showed that uplift of the panel-slab connection can reduce the shear strength
significantly. In the work of Devine (2009), the experimental data was used to get the pushover
response of many connection configurations. However, dynamic effects as well as cyclic
behaviour were not investigated. Olund (2009) studied some potential dissipating energy
mechanisms using incremental dynamic analysis. Because of the lack of knowledge about the
connector’s behaviour, this study focused on buildings where the panel-slab connectors were
ignored.
As a result, in order to better understand the seismic behaviour of tilt-up building using rigid
panels, a nonlinear dynamic tilt-up model incorporating a realistic connection model is needed.
4
The research objectives of this study are to:
• Use the available experimental data to develop a numerical model of the EM5
connector. This model has to take the coupling between vertical and horizontal motions
into account.
• Incorporate this connection model in a building model to study the seismic response of
tilt-up structures with rigid panels.
• Use the developed finite element models to:
- Assess the current design procedure.
- Investigate alternative connection configurations.
- Conduct parameters study to confirm the sensitivity of important building
parameters (such as roof stiffness, coefficient of friction between the panels and
the foundation, coupling of vertical and horizontal connection behaviours) on
the structural design.
1.4 Thesis overview
The remaining chapters of this thesis will address the objectives listed above. Chapter 2 serves
as literature review of the current design procedure and the state-of-the-art analytical and
experimental work conducted today. Chapter 3 presents a numerical model for the EM5
connector. The hysteresis of the numerical model was calibrated using the experimental data
conducted at the University of British Columbia. Coupling between shear and axial behaviour
is included. Chapter 4 describes a numerical tilt-up building model that incorporates the
connector model presented in Chapter 3. A typical building designed in Olund (2009) is
described and implemented. The static and dynamic behaviour of the model were verified.
Chapter 5 summarizes the parameter studies of the prototype tilt-up model. Both nonlinear
5
static and dynamic analyses were conducted. Chapter 6 presents the conclusions and
recommendations resulting from this study.
6
This chapter first summarizes the construction techniques and design requirements of tilt-up
construction in Canada. Then, the main research studies performed on tilt-up construction
using rigid panels and steel roof diaphragms are presented. Previous research focused
primarily on connector response via experimental testing and on building response.
2.1 Review of the current construction techniques and design approach
Design approach and construction techniques for concrete tilt-up buildings were summarized in
Olund (2009) and Devine (2009). Design guidelines for wall panels and connectors are
provided by Weiler (2006) in the Concrete Design Handbook.
2.1.1 Roof diaphragm
Design approach and construction techniques for the roof diaphragm are summarized in Olund
(2009) and Devine (2009). Typically, two kinds of diaphragms are used in tilt-up buildings in
North America. In the United States, the most common is panelized plywood roof diaphragms
supported by glulam beams. In Canada, only corrugated cold-formed steel decks are used. The
steel decking is supported by steel beams and/or open web steel joists. A steel angle is added
along the perimeter of the deck to insures it is supported along the edges.
7
The design of steel diaphragms in Canada follows a document called “Design of Steel Deck
Diaphragms – 3rd Edition” (Canadian Sheet Steel Building Institute, 2006). Two methods are
described in this document: the Tri-Services method developed by S.B. Barnes and Associates
(1973), and the Steel Deck Institute (SDI) method (Luttrell, 2004).
The Tri-Services method used to be popular in Canada. However, it can be only applied to
steel decks having button punched and seam welded side-lap connections. Recent testing by
Tremblay et al., (2002) at the Ecole Polytechnique in Montreal showed that button punched
and welded deck connections have extremely limited ductility compared to nail and screw
fastened decks. This resulted in a change in design habits, nails and screws being much more
common now.
The SDI method provides design guidelines as well as equations to compute the stiffness, the
strength, and the period of the diaphragm. For seismic and wind design, the main requirement
is to provide sufficient capacity in order to avoid fastener failure and shear buckling of the
deck. The capacity is adjusted by choosing the adequate deck thickness and fastener type and
spacing. Lateral force demand on the diaphragm is computed using a beam model: the
maximum shear occurs at the ends of the diaphragm, and the maximum moment occurs at the
middle of the roof diaphragm (resulting in compressive and tension forces). Since the
maximum shear and tension always occur close to the perimeter of the roof diaphragm, the
perimeter of the diaphragm is usually constructed using thicker metal and the interior roof
diaphragm is usually constructed using thinner metal. For seismic design, the design forces are
typically reduced by a reduction factor which depends on the fastener type used. If screws are
used for side laps, and the deck is attached to beams and joists with pin connections, a
reduction factor of 1.95 (RdRo = 1.5 x 1.3) can be used.
2.1.2 Wall panels
Tilt-up panels are aimed at providing both vertical and lateral load resistance in a building.
Design of tilt-up panels for vertical and out-of-plane loading is governed by flexure, since tilt-
up panels are often very slender in the out-of-plane direction. Regarding the in-plane direction,
8
the behaviour is usually governed by sliding and rocking mechanisms, which are resisted by
friction, panel weight, and the connectors. This section summarizes the construction practices
and the current design practices for out-of-plane and in-plane loading of tilt-up panels.
i) Construction technique
Panel wall construction is described in Devine (2009). During tilt-up building construction, a
slab on grade is cast first. Once this slab is set, a chemical that prevents concrete bonding is
spread on the top: it will prevent the future panels to bond with the slab. Then formwork and
rebars are laid as shown on Figure 2.1.
Figure 2.1 Formwork layout before casting of concrete tilt-up panels (Devine, 2009)
Concrete tilt-up panels are then cast. Usually, the interior face of the panel is facing up, in
order to allow a more convenient placing of the connectors. Once panels are set, they are lifted
and put at their final place using a crane. Temporary bracings are needed to hold them in place
before the roof diaphragm is installed.
9
ii) Design tilt-up panels for vertical and out-of-plane loading
Since tilt-up panels are usually very slender in the out-of-plane direction, the design procedure
typically pays special attention to the out-of-plane loading due to both wind and earthquake
forces, as well as the eccentricity of the vertical load and the related P- effect. Design
guidelines are presented by Weiler (2006) in the Concrete Design Handbook.
Eq. 2.1 shows the equation used to check the flexural capacity of tilt-up panels.
br MM ≥ (2.1)
• bbf MM δ= is the factored moment including P- effect;
• btf
1
− =δ is the moment magnification factor to account for the P- effect;
• 0
2
)( 28
+++= tfwftf
f
M is the factored moment not including P- effect;
• Pwf is the factored panel weight above mid height;
• Ptf is the factored axial load at the top of the panel;
• e is the axial load eccentricity at the top of the panel;
• 0 is the initial deflection at panel mid height;
• Kb is the bending stiffness of the wall panel;
• l is the clear vertical span of the panel;
• wf is the factored lateral load.
2010 NBCC requires that a building should be designed to resist seven combinations of
different loading types, including dead, live, snow, wind and earthquake loads. For out-of-
plane flexure, the following two combinations have to be taken into account:
10
1.25 Dead Load + 1.4 Wind load + (0.5 Live Load or 0.5 Snow Load) (2.2)
1.0 Dead Load + 1.0 Earthquake Load + (0.5 Live Load or 0.5 Snow Load) (2.3)
Wind forces are calculated in accordance with 2010 NBCC Clause 4.1.7. In seismic areas, Eq.
2.3 governs. The earthquake forces on panels are calculated using Eq. 2.4 from 2010 NBCC
Clause 4.1.8.17 for Elements of structure:
ppEaap WSISFV )2.0(3.0= (2.4)
Where:
• Sa(0.2) is the 5% damped spectral response acceleration at a period of 0.2 seconds.
Value based on published climatic data within 2010 NBCC.
• Fa is the acceleration-based site coefficient that is adjusted based on the soil conditions
and Sa(0.2).
• n
h
h A 21+= is a height factor, where hx, hn are the height above the base level. For
out-of-plane forces, hx is taken as the center of mass of the panel.
• E
I is the importance factor. For normal importance structures, E
I equals 1.0.
W is the panel weight
Category 1 of 2010 NBCC Table 4.1.8.17 applies for the design of tilt-up panels for out-of-
plane bending:
• Cp is the component risk factor. Usually taken as 1.0.
• Ar is the dynamic amplification factor. For short period buildings with flexible walls,
this is equal to 1.0. If the natural frequency of the component is close to the
fundamental period of the building, this factor could increase to as much as 2.5
• Rp is the response factor associated with the ductility of the component. Its value is
normally 2.5 for design of reinforced tilt-up wall panels out-of-plane.
11
The slenderness of panels is also limited to 50 and 65 for panels having a single and double
reinforcing mat, respectively.
iii) Design of tilt-up panels for in-plane loading
In the in-plane direction, panels act as the lateral-force resisting system for the building.
Therefore, they take lateral loads from roof, floors and out-of-plane walls. As for in-plane
loading, 2010 NBCC Clause 4.1.7 is used to calculate wind load, and clause 4.1.8 is used for
earthquake load. According to this last clause, the design base shear of the building should be
computed using Eq. 2.5:
• V is the design base shear in kN.
• Fa,v is a site coefficient. It depends on the soil condition and the building period.
• Sa(T) is the spectral acceleration in unit of g, given by the uniform hazard spectrum at
the building’s fundamental period, T.
• Mv is a coefficient that takes the shear force due to higher modes into account. In tilt-up
buildings, higher modes are usually not considered, and this coefficient is chosen as 1.
• Ie is the importance factor. Since tilt-up construction can be used to build either low
occupancy warehouses or schools, this coefficients can vary between 0.8 and 1.3.
• W is the weight of the building in kN.
• Rd is the force reduction factor based on ductility.
• Ro is the force reduction factor based on overstrength.
This equation is valid as long as the building is either medium size and regular, or small size,
or located in an area of low seismicity. Tilt-up buildings usually fall in the "Regular structure"
category and medium sized, so this equation can be used most of the time. Otherwise, dynamic
12
analysis would be required. If Rd is higher than 1.5, the base shear does not have to be higher
than 2/3* Sa(0.2).
Eq. 2.6 is obtained from 2010 NBCC and can be used to assess the fundamental period of the
building.
05.0 na hT = (2.6)
Where hn is the height of the shear wall in meters.
This equation does not take into account the flexibility of the diaphragm and is thus giving a
period much lower than the one given by modal analysis. This is a conservative estimate as a
smaller period increases the design shear forces on the tilt-up building. 2010 NBCC allows
engineers to use a higher period (up to a period of 1.5 times the period calculated using Eq.
2.6) calculated by model analysis.
The weight of the building, W, includes the weight of the in-plane panels, the roof, half of the
out-of-plane panels, and 25% of the design snow load. The bottom of the out-of-plane panels is
restrained from sliding in the out-of-plane direction. This restraint is assumed to take half of
the forces applied on those panels, that is why only half of out-of-plane panel weight is
considered.
Panels without connectors have limited lateral capacity. This capacity is provided by contact
forces between the panels and the foundation slab. Connection technique and design are
described in the next section.
2.1.3 Connectors
Contrary to cast-in-place concrete buildings, tilt-up construction requires connectors to link all
the structural elements to each other. According to Lemieux et al. (1998), these connectors are
used for the transfer of:
• Gravity loads from beams and joists to the tilt-up panels.
• Shear forces from roof joists and steel deck to the tilt-up panels.
13
• Shear forces between tilt-up panels.
• Shear forces from tilt-up panels to the floor slab or foundation.
• Forces from mechanical equipment or architectural items to the tilt-up panels.
Weiler (2006) provides guidelines for the use of five standard connectors in the Concrete
Design Handbook. These five connectors are now the most common in Canada. These
connectors and their use are described below.
EM1 - Joist Seat
The EM1 is a joist seat connector made of a steel L89 x 89 x 6 x 300 mm angle which can be
embedded in concrete with two 15M Gr 400 rebar anchors. This connection is used as a seat
for the openweb steel joists. Figure 2.2 shows a drawing of this connector and a picture before
concrete casting.
Figure 2.2 EM1 joist seat (left - CAC Concrete handbook 2006 , right - Devine, 2009)
14
EM2 - Shear Plate
The EM2 connector is a shear plate connector made of a steel PL150 x 9.5 x 200 plate with
two 16mm studs welded. Figure 2.3 shows a drawing and a picture of the EM2 connection.
Figure 2.3 EM2 shear plate (left - CAC Concrete handbook 2006; right - Devine, 2009)
This connection is usually welded with an edge angle to be connected to the steel deck. Figure
2.4 shows the details of the EM2 connection welded with the steel angle.
Figure 2.4 Use of EM2 shear plate to connect the diaphragm angle to the panels (CAC
Concrete handbook 2006 )
EM3 - Shear Plate
The EM3 is a shear plate connector which is a stronger than the EM2 connection. It is made of
a PL 200 x 9.5 x 200 plate with four 16mm studs. Figure 2.5 shows the details of the EM3
connector.
Figure 2.5 EM3 shear plate (CAC Concrete handbook 2006)
It can be used for the same functions as the EM2 connection. Typical use includes attaching an
angle seat for an Open Web Steel Joists (OWSJ) or providing an anchorage in a panel in order
to attach this panel to the slab.
EM4 - Shear Plate
The EM4 is a shear plate connector which is even stronger than the EM3 connector. It is made
of a PL 225 x 9.5 x 460 mm plate with eight 16mm studs. Figure 2.6 shows the details of the
EM4 connector. The EM4 connector is typically used to attach girders of the steel decking
system.
16
EM5 - Edge angle
The EM5 is an edge angle connection, which is made of an L 38 x 38 x 6 x 200 mm steel angle
welded to a bent 20M rebar anchor, as shown in Figure 2.7.
Figure 2.7 EM5 connection (left - CAC Concrete handbook 2006 , right - Devine, 2009)
This connection is used to connect edges of concrete elements. Its long rebar provides
sufficiently deep and wide anchorage to avoid cone failure at the angle. It is most commonly
used in the panel-panel connection and panel-slab connection. Figure 2.8 shows how the EM5
connection can be used in a slab along with an EM2 or an EM3 connection in a panel in order
to make a panel-slab connection. The EM5 and EM3 connectors are welded together. A filler
bar is added in between to make the weld stronger.
17
Figure 2.8 Slab to Panel Connection (Lemieux, Sexmith and Weiler, 1998)
Connection layout design
The choice of the number and the configuration of connectors is based on static analysis. For
seismic design, roof shear and panel shear are computed using Eq. (2.5) where half of the mass
of the out-of-plane panels is included in the roof shear. The value of RdRo is chosen as 1.0.
Using a free body diagram shown in Figure 2.9, the forces in the connectors can be calculated.
18
Figure 2.9 Free body diagram of tilt-up panels (Olund, 2009)
The base shear capacity is calculated by adding the frictional force at the base of the panels and
the shear capacity of each of the panel-slab connectors. The number of connectors is chosen
based on the horizontal capacity needed to resist the base shear generated by the earthquake
force. It should be noted that during an earthquake shaking the shear capacity of the bottom
connectors might be damaged due to vertical motion during panel rocking. Such behaviour is
observed in tests from Devine (2009). However, this phenomenon has not been taken into
account in the current design procedure.
The overturning demand of the panel is calculated using Eq. 2.7.
od
wallsplabeinroof
• h is the panel height.
• Vroof is the seismic force due to the roof, the out-of-plane panels, and the design snow
load. Since there are usually one row of panel on each side of the building, only half of
this force is applied in the free body diagram.
19
• Vin-plane walls is the seismic force due to the in plane panels of the row.
• Rd and Ro are chosen as 1.5 and 1.3 for the EM5-EM5 connection.
The overturning capacity is computed using Eq. 2.8.
( ) bV N
bV bN
WWM downHold
Where:
• Nipp is the number of in-plane panels in the building, half of them being in the
considered row.
• Vp/p is the sum of the shear capacity of all the panel-panel connections between two
panels.
• VHold-down is the shear force applied by the out-of-plane panels to the in-plane panels
through the edge connectors.
This overturning capacity is adjusted by adding sufficient panel-panel connectors. It should be
noted that the overturning capacity of the panel-slab connectors is not taken into account.
2.2 Review of previous experimental tests conducted on connectors
Connectors play an important role in the seismic response of solid wall tilt-up panel buildings.
This section summarizes the state-of-the-art experimental tests that have been conducted on
tilt-up connectors.
2.2.1 Experimental testing by Lemieux, Sexsmith and Weiler (1998)
Lemieux, Sexsmith and Weiler (1998) investigated the strength and ductility of the common
tilt-up connectors described in Section 2.1.3. Those connectors were subjected to tension and
shear tests, under both monotonic and cyclic loading history. The results were then used to
recommend the shear and tension strength of those connectors in the Concrete Design
Handbook.
The tilt-up model developed in this thesis requires only the uplift behaviour of the EM5
connector, and the shear behaviour of the EM2~5 connectors. The former was not investigated
in this experimental program, so only the latter is presented here.
Figure 2.10. shows the experimental setup for the shear strength tests. Connections were
embedded in a 140 mm thick concrete panel with specified strength of 30 MPa. This value was
chosen because it is the minimum thickness recommended by the Concrete Design Handbook.
Those specimens were cast in a construction site, and then bolted to the strong floor.
Figure 2.10 Shear test setup (Lemieux et al., 1998)
For each connection type, one monotonic shear test and two cyclic tests were conducted. For
the cyclic tests, three cycles were performed at an increasing displacement target. Depending
on the tests, between six and eight displacement increments were applied. Figure 2.11 shows
21
the load-displacement plots of the all the shear tests. Test numbers are summarized in Table
2.1.
Figure 2.11 Load-displacement plots of shear tests (Lemieux et al., 1998)
22
Based on the results presented in Figure 2.11, the cyclic envelope appears to follow the
monotonic plot in the elastic range, but the monotonic envelop appears to overestimate the
cyclic envelop in the large deformation range. The EM5 connection appears to be the most
ductile connection. It can sustain a monotonic displacement of 15 mm, whereas monotonic
tests on EM2, EM3 and EM4 failed between 2 and 5 mm. For cyclic loading, the maximum
displacement of the EM2, EM3 and EM4 connectors is between 2 and 4 mm, whereas the EM5
fails at 7 mm. Strength results (in units of kN and kips in brackets) and failure modes of the
connectors are summarized in Table 2.1.
Table 2.1 Connections shear strength (Modified from Lemieux et al., 1998)
The failure of the embedded connectors often implies cone failure of the concrete. So the
thickness of a panel is an important parameter.
23
The average strength of the EM5 connector is higher than the one of the EM2 and EM3
connectors. This means that when those connectors are used in series (e.g. in a panel-slab
connection, as shown in Figure 2.8), the lower force capacity and ductility from the EM2 and
EM3 connector must be used in design.
Tension tests were performed on the EM5 connector, but not in the axial direction.
2.2.2 Experimental testing by Devine (2009)
Further experimental tests on the EM5 connection were carried out by Devine (2009) at the
University of British Columbia. There were two main objectives for this new test series. The
first was to determine the uplift behaviour of the EM5 connector, which is crucial to model the
panel rocking behaviour. The second was to study the coupling between the axial damage and
the shear response.
Figure 2.12 shows the experimental setup used by Devine (2009). Such setup allowed both
uplift and shear loading. The testing setup was made in a way that reproduces some of the
constraints that connections would have in a tilt-up building: it allowed controlled out-of-plane
movement, and rotation was restrained. The EM5 connectors were embedded in a piece of
concrete whose specified strength was 25MPa. Testing was conducted more than 100 days
after casting to reproduce field conditions.
24
Four types of tests were performed:
• Pure shear tests with both monotonic and cyclic displacement load history.
• Pure axial tests with both monotonic and cyclic displacement load history.
• Coupled tests, held up: three axial cycles were applied first, the connection was then
held up, and finally displacement cycles were applied in the shear direction with
increasing amplitude until failure.
• Coupled tests, brought down: same as the previous tests, except that the connection was
brought back to zero axial displacement before applying the shear load cycles.
This protocol for coupled tests made clearer how vertical displacement affects the shear
response of the connector than a test where the two directions are tested simultaneously.
Tests showed that the axial cyclic ductility of the EM5 connection can be easily improved by
changing the type of filler bar, which is an element used to strengthen the weld that attaches
the EM5 connector to another element. When the typical round filler bar was replaced by a
thicker square bar, the cyclic displacement capacity was increased from 75 mm to more than
100 mm.
25
Pure shear tests were performed in order to validate results from the previous testing program.
A different cyclic loading protocol was applied, using higher displacement increments. The
results are consistent with previous testing. In all three pure shear tests, failure occurred due to
the fracture of the EM5 rebar.
Eight coupled tests were performed. Four levels of vertical displacement (25 mm, 50 mm, 75
mm and 100 mm) were tried. For each level, two tests were conducted, one where the
connection is held up, one where the connection is brought back to zero vertical displacement.
The shear strength and stiffness were found to be greatly affected by the previous axial
damage. In particular, the strength during cyclic loading dropped from an average of 220 kN
for the undamaged connection to 140 kN for connections having undergone an uplift
displacement of 100 mm.
2.3 Previous research on roof diaphragm
The roof diaphragm plays an important role in the seismic behaviour of tilt-up buildings. The
base shear due to the roof diaphragm is greatly affected by its strength and stiffness.
Many experimental studies have focused on the static response of the roof diaphragm in the
past. A recent testing series at the Ecole Polytechnique in Montreal by Essa, Tremblay and
Rogers (2003) investigated the shear stiffness and strength of steel deck diaphragms for both
static and monotonic loadings. The aim of this testing series was to validate the design and
modeling recommendations from the Steel Deck Institute.
Figure 2.13 shows the experimental setup used in this study. Load was applied parallel to the
deck direction. The tested dimensions of the tested specimen were 6.10 m x 3.66 m.
Figure 2.13 Steel deck roof diaphragm testing setup (Essa et al., 2003)
Nine common deck configurations were tested. This includes two deck thicknesses, two deck
profiles, six kinds of deck-to-frame fasteners and three kinds of sided lap fasteners. For each
27
configuration, two specimens were tested: one with a monotonic loading and one with a cyclic
loading. Table 2.2 shows the characteristics of the specimen tested.
Table 2.2 Steel deck diaphragm configurations (Essa et al., 2003)
Strength and stiffness of the monotonic tests and cyclic tests are shown on Table 2.3 and Table
2.4, respectively.
Table 2.3 Shear strength and stiffness in monotonic tests (Essa et al., 2003)
28
Table 2.4 Shear strength and stiffness in cyclic tests (Essa et al., 2003)
The results show that the diaphragm with welds and no washers as deck-to-frame connections
have the lowest ductility. On the other hand, diaphragms using B-deck profiles with screwed
side lap fasteners and nails or welds with washer have the highest strength during cyclic
loading.
2.4.1 Olund (2009)
Olund (2009) proposed a concrete tilt-up model using the finite element software PERFORM
3D Version 4.0.3 (Computers and Structures, 2007). An analysis was performed following the
main steps of the ATC-63 methodology (ATC, 2008). This methodology is a systematic way to
assess the performance of a seismic force-resisting system. It is based on the Incremental
Dynamics Analysis method proposed by Vamvatsikos and Cornell (2002).
The first part of this study was to design archetypical tilt-up buildings. A study amongst tilt-up
designers in North America was performed in order to determine the most common
characteristics of the roof diaphragm (material, kind of fastener, deck profile, deck thickness,
ratio of the roof mass to the mass of the out-of-plane panels, girder and joist span, dead load),
29
wall panels (dimensions , number, arrangement), and connections (type, arrangement). Results
were employed to design two typical buildings: one using solid wall panels and one using
panels with large openings.
Panels were modeled using concrete and steel nonlinear fibre. Roof diaphragm and its
connections to the panels were assumed to be linear elastic. Contact and friction at the panel-
slab and at the panel-panel interfaces were modeled using nonlinear materials (a compression
only elastic spring material for contact and an elastic perfectly plastic material for friction).
The monotonic behaviour of the panel-slab and panel-panel connectors was modeled using the
results from the first test conducted by Devine (2009). Coupling between horizontal and
vertical motions was not modeled. Since the connection model was monotonic, the seismic
behaviour of the building could not be investigated. As a result, most of the work on solid
panels was performed without panel-slab connectors, a situation that would occur when all the
connectors have failed.
Three models were derived to study possible energy dissipating mechanisms:
• A panel sliding model, where panel rocking was restrained and the panel-slab
connections of the in-plane panels were removed.
• A panel rocking model, where panel sliding was restrained and all the connectors of the
in-plane panels were removed.
• Frame action, for panels with wide openings.
A set of 8 ground motion records was applied to the building. Records were scaled such that
the spectral acceleration at the fundamental period ranges from zero to 3g. Figure 2.14 shows
the median maximum drift of panel top and the drift at the middle of the roof. The
displacements are presented for both the sliding model and the rocking model. The sliding drift
was found to be smaller than the rocking drift. However, it also leads to a high residual
displacement that would be nearly impossible to repair.
30
Figure 2.14 IDA drift results from 8 ground motion records (Olund 2009):
The model was also used to propose a value of the RdRo factor for the rocking and the frame
mechanism. The 2005 NBCC method assesses the RdRo value based only on the connectors
behaviour without taking the building mechanisms into account. This study showed that an
RdRo value of 2.1 for the rocking mechanism resulted in a collapse probability lower than 10%
for a hazard of 2% in 50 years. It was concluded that a rocking mechanism would be more
practical because of the residual displacement of the sliding mechanism. For the frame
mechanism, the value of Ro =1.3 and Rd=1.5 that are commonly used in design were found to
be unconservative because they resulted in high shear demand in the roof diaphragm.
The natural period of the sliding model was 0.58s, which is within 10% of the period predicted
by ASCE41-06 when half of the mass of the out-of-plane panels is added to the mass of the
roof. This period is much higher than the one predicted by 2010 NBCC in Eq (2.6), which is
0.26s.
31
2.4.2 Devine (2009)
In a study by Devine (2009), strength, ductility, and failure mode of tilt-up panel systems using
various connection configurations were studied. The results of the experimental tests described
in 2.2.2 were used to develop a backbone model of tilt-up connectors. This model was then
incorporated in a panel row model. A monotonic pushover study was performed using this
model. The model was two dimensional with the assumption that the panels are rigid, and the
effect of the steel angle and of the connections to out-of-plane panels were neglected. Studied
configurations include the combination of 1 to 4 panels, 2 to 4 panel-slab connectors, and 2 to
4 panel-panel connectors.
The strength of the different systems was compared. It was found that adding panel-panel
connectors increases significantly the strength of a system. However, this strength was found to
be very brittle.
A total of four failure mechanisms were predicted using this static monotonic pushover model,
each of them having resulting in a specific behaviour:
• Overturning without uplift: this failure mode occurs when there are not sufficient panel-
panel connectors to uplift any panel before they fail.
• Overturning with panel uplift: this failure mode occurs when there are enough panel-
panel connectors top uplift one or two panel during the rocking motion. Contrary to the
previous failure mechanism, all the panel-panel connectors do not fail simultaneously.
As a result, they increasing the system resistance in a larger drift range.
• Sliding: this failure mode typically occurs when there is a lot of panel-panel connectors
and few between panels and slab. This failure mode occurs in 13 configurations out of
25: this is the most common one.
• Extremely brittle failure: this failure mode occurs when there are so many panel-panel
connectors that panel-slab connectors fail before the panel-panel ones. This results in
failure at a very low panel drift.
32
A study was then led on the performance of ideal connectors. It was suggested that two
characteristics would be desirable: increasing the uplift displacement capacity of the
connectors, and having a minimum coupling between the shear and the axial behaviour of the
connectors.
2.4.3 Need for a new model
In the past, Devine (2009) and Olund (2009) have each developed a numerical model to
simulate the seismic behaviour of tilt-up buildings.
Devine’s model was developed using simple empirical equations to estimate the force-
deformation backbone of the EM5 connector. This model was used to simulate the peak force-
deformation response of the tilt-up panels with EM5 connections under monotonic loading
condition. The model, however, was unable to simulate the force-deformation response of the
EM5 connection under cyclic response.
Olund (2009) used the Perform3D software to developed a full three dimensional nonlinear
model for a tilt-up building, however the EM5 connectors were not implemented.
To overcome deficiencies of the past models, a numerical model of the EM5 connector was
developed in Chapter 3 using the experimental tests conducted by Devine (2009) and Lemieux
et al. (1998). This model was then incorporated in a building model using the software
MATLAB, as explained in Chapter 4.
33
EM5 CONNECTOR
The seismic behaviour of concrete tilt-up buildings is highly influenced by the cyclic behaviour
of the connectors. Amongst all the connectors used in the tilt-up industry to link concrete
elements to each other, the EM5 connector is the only connection recognized by the Concrete
Design Handbook to have a significant ductility. In this chapter, a numerical model of the EM5
connector is developed based on the results obtained from the experimental tests from Devine
(2009) and Lemieux et al. (1998).
3.1 Summary of experimental tests used in this study
All the available tests on the EM5 connectors in the shear and uplift direction were used to
calibrate the nonlinear EM5 model. They are summarized in Table 3.1.
34
Table 3.1 Summary of the experimental tests used to develop the EM5 connection model
Specimen name
5 Monotonic Devine (2009)
6 Cyclic Devine (2009)
12 Monotonic Devine (2009)
15 3 vertical cycles at 25mm, brought back to 0,
then cyclic shear Devine (2009)
16 3 vertical cycles at 25mm, held up, then cyclic shear Devine (2009)
Four tests were available to calibrate the model coefficients for the axial behaviour, and six for
the uncoupled shear behaviour. No test was available from the study of Lemieux et al. (1998)
regarding the axial behaviour of the EM5 connector. Seven tests were available to calibrate the
coupled model.
In the experimental study of Devine (2009), two connection details were tested: one using a
typical rounded filler bar, and one using a square filler bar. Those two connection details were
treated in the same way in the model. In fact, the force-deformation plots were very similar, the
only significant difference being the axial displacement cyclic capacity.
35
3.2 General mechanical behaviour of the EM5 connector
In tilt-up buildings, EM5 connections are mostly utilized in the shear and axial directions. The
responses in these two directions are quite different and hence the response in these two
directions will be calibrated separately. It should be noted that, based on the experimental data
from Devine (2009), the shear response is affected by the axial response. However, the axial
behaviour is assumed to be unaffected by the shear response.
3.2.1 Monotonic and cyclic envelope
Figure 3.1 and 3.2 show the force-deformation response of the EM5 connection under shear
and axial load, respectively.
36
As shown in these figures:
• The force-deformation envelope of cyclic tests is very close to the monotonic tests. For
shear tests under cyclic load, the force envelope tends to cap beyond the yielding
displacement, while the monotonic force-deformation plot tends to increase gradually
with increasing deformations. On the other hand, for axial tests, the cyclic envelope is
almost identical to the monotonic envelope. A difference can only be observed between
40 and 90 mm. This difference is due to spalling of the concrete cover of the connector.
This phenomenon occurs at very random displacements depending on the axial test, so
it was ignored in the modelling.
• For both monotonic and cyclic cases, the force-deformation envelope can be
approximated well using a bilinear model.
37
3.2.2 Cyclic behaviour and damage
The force-deformation response of the EM5 connection under cyclic loading can be
approximated using a tri-linear line as shown in Figure 3.3. The three parts of this trilinear
model can be described as unloading, transition, and reloading. Axial and shear behaviour both
follow similar patterns, hence they can be modeled in a very similar way. The main difference
comes from the fact that negative displacement in the axial behaviour is impossible, hence
restricted.
Figure 3.3 Trilinear model for cyclic behaviour
The strength and the stiffness of the tri-linear lines tend to degrade significantly after repeated
cycles. When multiple cycles are performed at the same displacement, the reloading part starts
at a higher displacement and a lower force. This shows the EM5 connection is not capable of
retaining the strength after it suffers the inelastic damage. Nevertheless, when the connection is
pushed to a new maximum displacement, the connection seems to be able to resist higher force
similar to the case of the monotonic loading condition. This shows that modelling of the
damage is a key part of the numerical model.
38
3.2.3 Coupling
Based on the experimental tests conducted by Devine (2009), when the EM5 connection is
damaged under axial loading, then the stiffness, strength, and degradation rate in the shear
direction are affected. For example, the cyclic shear strength, which is about 230kN in
uncoupled tests, drops to 140kN in tests that have experienced an axial uplift of 100 mm
beforehand.
3.2.4 Symmetry
Most of the tests experience nearly symmetrical response, where the maximum negative
strength is within 10% of the maximum positive strength. However, in some special cases such
tests 15 and 16 from Devine (2009) are shown on Figure 3.4, the behaviour can be
unsymmetrical.
(a) , (b) Test 15 and Test 16 from Devine (2009)
Figure 3.4 Unsymmetric connection behaviour
39
In Test 15 (Figure 3.4 (a)) the maximum positive force is 25% stronger than the maximum
negative force (+239kN vs. -191kN). In Test 16 (Figure 3.4 (b)), the maximum negative force
is 50% higher than the maximum positive force (-249kN vs. +166kN). However, since the
design and loading history for specimen 15 and 16 is relatively symmetrical and without
additional data to back up the difference, it is assumed that the force-deformation response of
the EM5 connector will always be symmetrical.
3.3 Proposed analytical model
A numerical model was developed to model the cyclic response of the EM5 connector. Both
the coupled and uncoupled response were modeled.
3.3.1 Mathematical rules for the uncoupled model
i) General description
The model takes a displacement as the input, and returns the corresponding resisting force. The
resisting force will account for the strength and stiffness degradation from the preceding
cycles. To get the restoring force for any displacement, the force-displacement domain is
separated in three regions, namely Region 1, Region 2 and Region 3. The force-deformation
response in each region is calculated based on the stiffness rule defined in each region. Figure
3.5 shows the detail as where regions 1, 2 and 3 are defined. It can be noted that the regions
presented in Figure 3.5 represent the case where the displacement is increasing (positive
increase). Symmetrical rules apply if the displacement decreases (i.e. goes towards the negative
direction).
40
Figure 3.5 Force-deformation response regions of the EM5 connector model
The regions are defined as follows:
• Region 1 includes the following three sub areas: 1) all the area below the force level FB;
2) the area where displacement is higher than DA and force is lower than FA; 3) The
triangular area below the point A, which is defined using the elastic stiffness.
• Region 2 is the area where the displacement is lower than DA and the force is higher
than FB, except the triangular area described in Region 1. Because of the model
response rules, the force in this region is always lower than FA.
• Region 3 is the area where the displacement is higher than DA and the force is higher
than FA.
In general, if the force-deformation response is in region 1, the response follows the elastic
stiffness kel. If the force-deformation response is in region 2, the response goes toward point A.
If the force-deformation response is in region 3, the response follows the reloading stiffness kr.
41
The force should neither exceed the backbone curve, and it should also not be outside the
coloured areas of Figure 3.5.
In order to deal with decreasing displacements, a second point A is defined on the negative
side. The name of A in the positive and negative direction is defined as A+ and A-,
respectively. The position of the points A+ and A- depends on the displacement history and the
dissipated energy. Eq. 3.1 to 3.4 show the rules to calculate the position of point A+ and A-:
DA+ = rDisp * Dmax+ (3. 1)
DA- = rDisp * Dmax- (3. 2)
FA+ = rForce+ * FBB+ (3. 3)
FA- = rForce- * FBB- (3. 4)
Where:
• rDisp, rForce+ and rForce- are dimensionless coefficients which are calculated using
Eq. (3.7), (3.8) and (3.9), respectively. The values of these two parameters depend on
the dissipated energy and the past displacement history.
• Dmax+ and Dmax- are the maximum positive and negative displacement experienced by
the connector, respectively.
• FBB+ and FBB- are the values of the backbone at Dmax+ and Dmax-, respectively.
The same rules apply to the modeling of the connector in the axial direction, except DA- is set
to zero and FA- is set to at a constant value FAn.
Using the rules described above, the force-displacement point follows the backbone during the
first cycle. Indeed, displacement and forces are always higher than the ones of the point A
during the first cycle. As a result, the stiffness is given by the reloading slope, which is the
same as the elastic slope in the first cycle. When the yielding displacement is reached, the force
becomes limited by the backbone force.
42
As shown on Figure 3.5, the backbone is defined by its yielding point (force F1, displacement
D2) and a second point on the yielding slope (Force F2, displacement D2 , D2 being arbitrary).
ii ) Cyclic damage parameters
Two parameters have been found to be relevant to model damage: the energy dissipated by the
connection and the maximum displacement.
The dissipated energy is stored on an internal variable W. At every displacement step, this
energy is calculated by the area under the force-deformation response curve, which can be
simplified using a trapezoidal rule as shown in Eq. 3.5.
W(t+1) = W(t)+0.5*(F(t+1)+F(t))*(D(t+1)-D(t)) (3. 5)
where F(t), D(t), and W(t) are the force, displacement, and dissipated energy at time step t,
respectively. At t=0, W(0) is set to zero.
The value of the dissipated energy affects the reloading slope (in region 3) as well as the
position of the points A+ and A-.The value of Dmax+ and Dmax- affects only the position of the
points A+ and A-, respectively.
A constant reference energy, Wref, is defined to normalize W every time it appears in a formula.
This parameter enables the change of the rate of all the phenomena due to degradation at the
same time. It will be used to model the coupled behaviour of the connectors, as explained in
section 3.3.3.
iii) Reloading slope
When multiple cycles are performed, the reloading slope in Region 3 decreases. This
phenomenon is computed using Eq. 3.6 to get the reloading slope kr:
Wref*
krkr krkr (3. 6)
In this equation, kr1 and kr2 are constant coefficients which represent the reloading slope in the
initial and final cycles, respectively. In short, as intended in the equation, at the earlier phase of
43
the loading, the value of W is small, kr reduces to kr1. As W increases, kr reduces to kr2. The
values of kr1 and kr2 can be easily measured from the force-deformation curve of the
experimental data. The coefficient α represents the degradation rate of the reloading slope.
iv) Unloading slope
The unloading slope is modeled using the elastic stiffness kel, which is constant.
v) Position of the point A
* Coefficient rDisp
The coefficient rDisp represents the ratio of the displacement of the point A to the maximum
displacement. To simplify the model, the same coefficient rDisp is used for both the positive
and negative direction. Figure 3.6 (a) and 3.6 (b) show how the coefficients rDisp was fitted as
a function of W/Wref for the axial and shear tests, respectively. These plots are obtained from
Test 5 and Test 3 from Devine (2009), respectively.
(a) Axial test #3 ; (b) Shear test #5
Figure 3.6 Optimum coefficient rDisp for each cycle
As shown in this figure, the value of rDisp increases almost linearly as a function of W/Wref.
Linear regression was used to compute the relationship of rDisp as a function of W/Wref , as
shown on Eq 3.7:
rDisp = W/Wref*rDisp1+rDisp2 (3. 7)
where rDisp1 and rDisp2 are the coefficients obtained from the linear regression. Their value
is shown in Table 3.2.
* Coefficient rForce
The force level of the point A was found to depend on more than one parameter, and a linear
regression model (as for the coefficient rDisp) gives a poor approximation. The force position
of the point A gets lower when multiple cycles are performed at the same maximum
displacement, but it gets higher when a new maximum displacement is achieved. As a result,
the parameter rForce needs to account for both the dissipated energy W and the variation of the
maximum displacement.
Equations 3.8 and 3.9 shows the equation developed to calculate rForce+ and rForce-,
respectively:
rForce+ = rForceRef*(1-rForceM+) (3. 8)
rForce- = rForceRef*(1-rForceM-) (3. 9)
where rForceRef is a constant coefficient which represents the ratio of the force A with respect
to the value on the backbone curve.
The value of rForceM+ and rForceM- is calculated using Eq 3.10 and 3.11. The value of
rForceM- and rForceM+ depends on the maximum displacement, Dmax, and the energy
dissipated by the connector, W.
+
+
++ +++
∂
∂ +
∂
−
−
−− −−−
∂
∂ +
∂
Where:
• rForceM+(t) and rForceM-(t) are the values of rForceM+ and rForceM- at time step t,
respectively. The initial value of those two coefficients is zero.
45
• W
rForceM
∂ + )( is the derivative of rForceM+ with respect to W.
• +
+
∂
∂
rForceM is the derivative of rForceM+ with respect to Dmax+.
• +maxDδ is the increment of Dmax+ between time t+1 and time t.
The force level of the point A was found to decrease quickly in the first cycle, and more slowly
in the later cycles. As a result Eq. 3.13 is used to describe the variation of rForceM+ and
rForceM- with W:
Where:
• f(Dmax+) is a term unrelated to W.
Taking derivative of Eq 3.12 and 3.13 with respect to W:
Wref
WrefW
W
rForceM
β
∂ + (3. 15)
The variation of rForceM+ and rForceM- with an increase of the maximum displacement is set
using the following rule: if the maximum displacement is increased, the force increases as well;
the ratio of the force increase to the force on the backbone (at that displacement) is used to
increase rForce+. This is explained on the example sketched in Figure 3.7, where γ is a fixed
coefficient.
46
Figure 3.7 Model for the increase of the coefficient rForce when maximum displacement
is increased
During the increase of the maximum displacement, the force increases by a value dF. The point
A, which was at the position Aold before the increase, goes up to the position Anew. For
example, if the value of dF is 20% of the backbone force, then A goes up by γ*20%. The
mathematical formula that appears on Figure 3.7 is written for rForce+ and rForce- in Eq 3.16
and 3.17:
(3. 17)
+
+
+ +
+
+
∂
∂ −=
∂
47
Eq 3.20 and 3.21 are derived by substituting Eq 3.18 and 3.19 into Eq 3.16 and 3.17:
frForceR
The variable Dmax is then introduced using the reloading slope:
dF=kr*Dmax (3. 22)
This leads to these final equations:
+
+
+ +
+
+ −= ∂
γ δ (3. 24)
In this process, the value of rForceM+ and rForceM- is forced to remain positive. As a result,
the point A is never higher than the backbone force multiplied by rForceRef.
One could argue that this model is quite complicated. However, a basic linear model where
only the dissipated energy is taken into account was also tried. It resulted in a very poor
approximation of the force level of the point A.
* Coefficient FAn
The force level of the point A- was assessed for all the cyclic axial tests. The average value
was used to define the coefficient FAn.
48
3.3.2 Coefficients of the uncoupled model
Model coefficients for the uncoupled shear and axial behaviour were calibrated using the tests
described in section 3.1. They are summarized in Table 3.2.
Table 3.2 Coefficients used for the uncoupled model
Shear
Wref (kN.m) 21.1 16.7
FB (kN) -14 0
3.3.3 Model of coupling
As shown in the experimental results presented in Devine (2009), the shear force-deformation
response is affected by the axial deformation. Hence, a coupling model is developed. Note that
this coupling model does not describe how axial response is affected by the shear response as
no experimental test is available in this aspect.
49
The uncoupled shear behaviour is modified in the following way to take coupling into account:
• The backbone curve is lowered to model the strength and stiffness reduction. This is
done by modifying the value of the envelope parameters F1 and F2.
• The reference energy Wref is lowered to take into account the fact that shear damage
occurs more quickly after the connection has undergone some axial damage.
• The height of the point B is adjusted to fit the average of all the shear tests instead of
fitting only the uncoupled tests.
i) Modification of the backbone
For each coupled test from Devine (2009), the values of F1 and F2 that fit best the envelope of
the cycles were determined. Those optimum parameters F1 and F2 have been plotted versus the
maximum vertical displacement and the vertical dissipated energy. Using either of those
vertical damage parameters, a linear approximation of the coefficients F1 and F2 was found to
work well. In order to keep the model simple, the maximum vertical displacement was used as
the axial damage parameter. Figure 3.8 shows the values of the optimum parameters F1 and F2
plotted versus the maximum axial displacement.
0
50
100
150
200
250
Maximum axial displacement (mm)
Maximum axial displacement (mm)
Linear (Held up)
Figure 3.8 Optimum backbone parameters F1, F2 VS Maximum vertical displacement for
the coupled tests
50
The difference between tests where the connector is kept at the maximum vertical displacement
and tests where the connector is brought back to zero vertical displacement was found to be
very small. This is a good validation for the choice of the maximum vertical displacement as
the only axial damage parameter.
The following equations for F1 and F2 were derived using linear regression:
F1 = -1.08 kN/mm * max(Dv) + 205kN
F2 = -1.15 kN/mm * max(Dv) + 258kN (3. 25)
Where max(Dv) is the maximum axial displacement of the connector.
ii) Modification of the shear damage rate
For each test from Devine (2009), the degradation rate Wref that gave the best fit was
determined. Here are the optimum values of Wref plotted versus the maximum vertical
displacement:
Figure 3.9 Optimum degradation rate parameter Wref Vs Maximum vertical
displacement
These points can be well approximated by a linear model. A least square fitting gives the
following relation:
51
iii) Adjustment of the unloading end
The ideal position of the point B has been estimated for all the shear tests. In tests by Devine
(2009), the height of B does not change in a noticeable way when the damage due to uplift is
increased. Therefore, this parameter is chosen independently of damage.
The average heights on the negative and positive sides are close to each other and therefore the
average height is chosen:
FB = 33.1kN
This average force is higher than the one derived in the uncoupled model. However, this is
unlikely to be due to axial damage. Indeed, FB is nearly the same for all of F.Devine's tests, and
the value of FB in those tests is much higher than in K.Lemieux's tests. As a result, this force
increase is due to the fact that more of F.Devine's tests are taken into account in this new
average.
3.3.4 Model of failure
All of the specimens described in Table 3.1 were tested until failure. For most of them, failure
happens when the displacement is increased to a value higher than the maximum displacement
of the previous cycle (tests 14, 16, 10, 13, 6 and 7). However, failure sometimes occurs in the
middle of a cycle (test 15).
Such failure can be simulated in the numerical model in two ways:
• Based on the amount of dissipated energy;
• When a certain displacement is reached.
Those modelling approaches were studied for both the shear behaviour and the axial behaviour.
i) Coupled shear failure
52
In the coupled tests, the shear failure might be affected by the axial force-displacement history.
Potential trends for the two failure parameters described above (dissipated energy in shear and
maximum shear displacement) were studied using the following axial response variables:
• The maximum vertical displacement,
• The current vertical displacement,
• The energy dissipated vertically.
Using the maximum vertical displacement as the axial response variable gave the best results.
No trend could be observed when the current vertical displacement or the energy dissipated
vertically was used. Hence, the maximum vertical displacement was chosen as the variable
describing the axial response in the coupled shear failure model. Figure 3.10 shows how the
dissipated energy at failure and the maximum horizontal displacement at failure vary with the
maximum axial displacement for all the shear tests (coupled and uncoupled):
0
5
10
15
20
25
30
Max Dv (mm)
Max Dv (mm)
Figure 3.10 Horizontally dissipated energy at failure (left) and shear displacement at
failure (right) for all the coupled tests as a function of the maximum vertical
displacement
Points are very sparse when the dissipated energy in shear is used as the variable describing
failure. However, a trend can be observed if the maximum shear displacement is used instead.
This parameter seems much more reliable to predict the horizontal failure of the connector.
Therefore, shear displacement is chosen as the failure parameter.
53
The maximum displacement does not depend significantly on the vertical displacement since
the trend line is nearly horizontal. As a result, shear failure was modeled to occur at a shear
displacement that does not depend on the vertical displacement history. This displacement was
chosen as 16.5mm, which is the average value of the maximum displacement at failure for all
the shear tests considered in this model.
ii) Axial failure
The study of axial failure is simpler since the axial behaviour is uncoupled. Only four axial
tests were conducted until failure (namely tests 1, 2, 3 and 11 from F.Devine (2009)). Similarly
to the shear behaviour, using the maximum axial displacement as the failure parameter resulted
in a smaller dispersion than when the dissipated energy is used. As a result, the failure
displacement was chosen as the average failure displacement of those four tests, which is
98mm.
iii) Failure implementation
If the connection displacement is higher than the failure displacement at any time step, then the
force returned for any displacement is zero. However, the post-failure stiffness is bounded in
order to avoid having to deal with a very high negative stiffness in the building numerical
model. The value chosen to limit the post-failure stiffness is twice the elastic stiffness.
3.3.5 Model of paired connectors
In buildings, connectors are always assembled two by two. Typically, EM3-EM5 connections
are used to link panels to the slab, and EM5-EM5 connections are used to link panels to each
other. Tests have been performed only on EM3 and EM5 connectors alone, so the behaviour of
paired connectors has to be assumed.
54
It is assumed that, in an EM5-EM5 connection, each connector has the same force-deformation
response as observed in the experimental tests. Most likely, the behaviour will be a little bit
different because of the different restraints between an EM5-EM5 connection and the
experimental setup. However, this difference is likely to be small, and no experimental data is
available in this aspect. As a result, the EM5-EM5 force-deformation response was obtained
using the EM5 model with doubled backbone displacement parameters (D1 and D2).
A detailed model of the EM5-EM3 connection would be complicated since a model of the
EM3 connection would have to be developed first. However, testing results from Lemieux et
al. (1998) show that the EM3 connector has a very small displacement capacity and that its
shear strength is lower than the EM5's shear strength - the EM3 shear strength being however
higher than the EM5 axial strength. As a result, the EM3 connector will much likely fail before
the EM5, thus triggering failure at a much lower displacement than the one predicted by the
EM5 model.
Contrary to the EM3 connector, the EM4 connector is significantly stronger than the EM5
connector. Hence, it can be assumed that the EM4 connector will always remain elastic when
paired in series with an EM5 connector. As result, the EM5-EM4 connection model can be
obtained from the EM5 model by increasing the backbone parameters D1 and D2.
The EM5-EM4 is very rarely used in practice to connect the panels to the slab. However, this
model enables the study of the seismic response of tilt-up buildings where the whole shear
ductility capacity of the EM5 connector is used in the panel-slab connector.
It can be noted that using EM4 connectors to link the panels to the slab would be expensive.
However, in the two following situations, the ductility of the EM5 connection could be better
used without having to resort to EM4 connectors:
• Using connectors with six studs. According to the tests from Lemieux et al. (1998), a
connector having a strength intermediate between the EM3 and the EM4 connectors
would be sufficient to keep the EM5 connector as the ductile fuse.
55
• Having thick enough panels. In the tests from Lemieux et al., panels are 140mm thick.
One of the tested EM3 connectors was found to fail in a cone failure mechanism.
Having a thicker panel would thus increase the EM3 connector strength.
Figure 3.11 shows the backbone of the actual EM5-EM3, EM5-EM4, and EM5-EM5
connections. The axial behaviour of the EM5-EM5 model is not plotted since it will not be
used in the tilt-up model.
Figure 3.11 Backbones of the paired connector model
56
3.4 Validation of the model
The analytical model was validated for each of the test in Table 3.1. The displacement history
of each test was extracted and smoothed. This displacement was used as an input in the model
developed and calibrated in the previous sections. The force given by the model was then
compared with the experimental force.
3.4.1 Shear behaviour
Figure 3.12 shows how the envelope of the shear model compares to results from monotonic
shear tests from Devine (2009) and Lemieux et al. (1998):
-50
0
50
100
150
200
250
300
Displacement (mm)
F o
rc e
( k N
Numerical model
Figure 3.12 Comparison of the shear model with monotonic shear test results
Figure 3.13 shows the comparison of the force given by the model with the cyclic shear
response of three uncoupled tests.
57
Top left: Test 6 from Devine (2009) ; Top right: Tests S2 from Lemieux et al. (1998)
Bottom left: Test S3 from Lemieux et al. (1998)
Figure 3.13 Comparison of uncoupled shear model with experimental curves
The model response follows closely enough the experimental curve. The comparison to the
monotonic test is satisfying, and the strength degradation at each cycle is well approximated. In
test S3 (Figure 3.13 (c)), the degradation occurs too fast on the positive side. However, it
occurs also a bit too slowly in the other tests.
58
3.4.2 Axial behaviour
Results from Test 2 from Devine (2009) are plotted on Figure 3.14 along with the axial force-
deformation envelope of the model:
Figure 3.14 Comparison of axial model with monotonic test results
Figure 3.15 and 3.16 show the comparison of the axial model response with the axial cyclic
tests from Devine (2009). Tests on Figure 3.16 were coupled, so the axial loading was not
applied until failure. The numerical curves are close enough to the experimental ones. In
particular, the strength at each cycle and the initial stiffness are well approximated.
left: test 3 - right: test 8
Figure 3.15 Comparison of model with pure axial cyclic tests
59
2nd line left: test 13 - 2nd line right: test 14
3rd line left: test 7 - 3rd line right: test 9
Bottom left: test 10
Figure 3.16 Comparison of axial model with seven uplift tests
60
3.4.3 Coupled behaviour
Figure 3.17 and 3.18 show how the force output of the model compares to the held-up and
brought down test results from Devine (2009), respectively. The x-axis shows the displacement
history of the connector, in meters.
Top left: test 16 (Uplift 25mm) - Top right: test 4 (Uplift 50mm)
Bottom left: test 7 (Uplift 75mm) - Bottom right: test 15 (Uplift 100mm)
Figure 3.17 Comparison of model with coupled tests where the connection is held-up
61
Top left: test 6 (Uplift 0mm) - Top right: test 15 (Uplift 25mm)
Bottom left: test 14 (Uplift 50mm) - Bottom right: test 10 (Uplift 100mm)
Figure 3.18 Comparison of model with coupled tests where the connection is brought
down to zero axial displacement
62
For each test, the general behaviour is well approximated and the force from the model is
satisfyingly close to the experimental force.
In some tests, the strength is not approximated w

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