INFLUENCE OF LARGE AXIAL LOADS IN ROCKINGWALLS AND REINFORCED CONCRETE WALLS
Rene Francisco Nunez Orellana
MEE17716

Supervisor:

Dr. Susumu Kono

ABSTRACTThis work introduces the concept of rocking structures with the property of self–centering, with a special
focus on Rocking Walls (RWs), as damage–controlling structures, to incorporate them into the newSeismic Design Code of El Salvador. In this regard, analytical models using finite element methods
based on the multiple axial spring macro model and the shear flexure interaction multi–vertical line
element model (SFI–MVLEM) are developed in the software OpenSees, to predict the non–linear
behavior of both RWs and Reinforced Concrete (RC) walls, in order to analytically evaluate theinfluence of large axial loads in terms of energy and provide design recommendations for RWs to control
axial–flexural behavior and to calculate design capacities and demands. The analysis of the influence ofaxial load revealed that base shear increases with axial load ratio, which is more accentuated in RWs
than in RC walls with the 86%. The hysteretic damping, in contrast to base shear, reduces for RC walls,
while for RWs increases, this due to the inelastic energy dissipation, which depends on both strengthand ductility, and is essentially the same in spite of the axial load ratio for RC walls and increases withaxial load ratio in RWs, because of hysteretic behavior.
Keywords: Self–Centering, Rocking, OpenSees, Hysteretic Damping, Residual Drift.
1. INTRODUCTION
A high earthquake disaster risk potential in the world has been demonstrated in the last 20 years, whichis exacerbated by the current high levels of hazard, vulnerability and exposure, as indicated byearthquake statistics of USGS, the emergency events database of CRED and the UN’s world population
prospects. In this regard, in the framework of the UN SDGs, developing resilient and sustainableinfrastructure to make cities safe is one of the major world’s commitments. In this context, on May 18th
2018, the government of El Salvador started the execution of the project “Action Plan for theImplementation of the Governability Index and Public Politics for Risk Disaster Management”, which
is a two–year project whose main objective is to reduce the seismic disaster risk of constructions byupgrading the national seismic design code. In this context, this work introduces the concept of damage
controlling structures with the property of self–centering, with a special focus in RWs in order to takeadvantage of the desirable seismic characteristics related to the self–centering ability and capability ofundergoing nonlinear lateral displacements with little damage, to develop resilient and sustainable cities.
Since RWs are more suited to DDBD than to Force Based Design (FBD), within PBEE, as damage ismore directly correlated to displacement than to forces, the DDBD is proposed as the design method forSRWs. Since in El Salvador RC walls are mostly used, in mid–rise to high–rise buildings, with bareframes as lateral force resisting systems, they are subjected to large axial loads especially in bottomfloors. Given the importance of equivalent stiffness to the maximum displacement and of equivalent
damping to define the equivalent SDOF in the DDBD; and considering the limited research on theinfluence of axial loads in terms of energy dissipation, for the implementation of RWs in the frameworkof PBEE, it is essential to evaluate the influence of axial loads in RWs and RC walls in terms of energy
dissipation, and demonstrate the higher performance of RWs.
Ministry of Public Works, Transportation and Housing and Urban Development (MOPTVDU), El Salvador.
Professor, Department of Architecture and Building Engineering, Tokyo Institute of Technology, Japan.

12. DEFINITION AND BEHAVIOR OF ROCKING WALLS
Unbounded Post–Tensioned Precast Concrete Walls are precast concrete structural walls, which arecomposed by one (uncoupled walls) or more (coupled walls) vertical wall panels. Each vertical wall
panel can be cast as a single element, or as several separate wall elements that are subsequently joinedby rigid connections. Wall panels are connected to the foundation by continuous unbounded post–
tensioned tendons, which along with gravity loads, act as a restoring force to provide the self–centering
property. Supplemental energy
Force
Force
dissipators such as mild steel
Fyreinforcing bars across the wall
FRockto foundation joint in uncoupledwalls, or shear connectors along
vertical joints in coupled walls
Rotationcan be included to increaseb)a)energydissipation.
RWs
F
Reversalreferredinthisstudy
are
Force
Force
Unbounded
Post–Tensioned
Precast Concrete Walls that arecomposed by one vertical panel
cast as a single element without
Rotation
Rotationenergy dissipators. The behaviord)c)of unbounded post–tensioned
precast concrete walls is flag–Figure 1. Behavior of Rocking Walls.
shaped, as typical in prestressed
concrete structures. The flag–shape behavior can be summarized in four steps, as shown in Figure 1,
assuming wall–foundation interface has no flexibility and energy dissipation devices are rigid–perfectlyplastic. In the absence of lateral force, a uniform stress distribution develops wall–foundation interface
due to initial post–tensioning force and self–weight (restoring forces). As the lateral force is applied, thestress distribution shifts to a critical point (FRock), where the resultant force at the outermost fiber of thewall is zero (Figure 1 a)). At this state, if no energy dissipators are provided, wall uplift begins,otherwise, as lateral load continues to increase, energy dissipators activate, and take the load and forcesin them until yield (Figure 1 b)). If energy dissipators are perfectly rigid before yielding, the gap will
not open until yielding of devices occurs. After yielding of energy dissipators the gap opens, and thesystem initiates the non–linear stiffness range, which is essentially controlled by elongation of post–tensioning tendons (Figure 1 c)). When the lateral load is removed, the structural wall returns to theoriginal position by the restoring forces without any structural damage or residual deformation (Figure1 d)). The same behavior occurs when the wall is loaded in the opposite direction. The structural limits
states in the Performance Based Seismic Design (PBSD) are defined from the behavior RWs.
3. ANALYTICAL MODELS FOR ROCKING WALLS AND RC WALLS
3.1. Analytical Modeling of Rocking Walls
The multiple axial spring macro model was developed in OpenSees to simulate the cyclic behavior ofRWs, as shown in Figure 2. Since RWs concentrate the inelastic demand at the wall to foundationinterface, the behavior of wall panel is expected to be essentially elastic; therefore, the wall panel ismodeled as an elastic beam–column element. The critical interface, where inelastic demand isconcentrated through a gap opening and closing, is modeled using an array of uniaxial (compressiononly) nonlinear contact springs (zeroLength elements), distributed in X– direction, whose force–
deformation nonlinear behavior is defined from the modified Kent–Park stress–strain model,
considering the influence area of each spring and the effective length according to the recommendationsof Perez et al. (2007). The post–tensioning (PT) tendons and the corresponding initial prestressing force
are modeled using corotational truss elements and uniaxial initial strain materials respectively, and the2nonlinear stress–strain behavior
defined by the Menegotto–Pinto
model. In order to create a double
rocking interface to reproduce
the test behavior, boundary
conditions
andelements
connectivity are critical. In thisregard, at the base of the wall,
springs are fully constrained at
the bottom end to the floor, while
are rigidly connected to the initialnode of the wall panel at the top
end by rigid links. The springs at
the top interface, to model thedouble rocking, are rigidly
connected to the final node of thewall panel, while at the top,translational and rotational DOFsof the springs are constrained tobe those of the rigid beam
element which simulates the stub.
Figure 2. Multi Axial Spring Macro Model.
Since there is no restoring force
to control and reverse the shear slip displacements that may occur due to lateral loads, shear slip along
the wall to foundation joint is to be avoided, therefore shear springs with large stiffness are provided byusing a zeroLength element modeled in the middle of the rocking interface. The unbounded behavior ofPT tendons is modeled by kinematically constraining lateral and rotational DOFs to the corresponding
element nodes of the wall at each rocking level, also, in order to simulate the anchorage of PT tendons
to the rigid upper stub, translational and rotational DOFs are constrained with that of the top node of therigid beam element. The accuracy of the 2D multiple axial spring macro model is verified against theexperimental results of the specimen NSW6A tested in TIT, as shown in Figure 3. The results show agood accuracy of prediction for initial stiffness and base shear in both, positive and negative sides for
global behavior, which error is below 2%. The accuracy of the model to predict the local behavior isinvestigated with the behavior of PT force, the predictions showed a good agreement with experimental
results, and since the analytical model considers losses due to elongation effects only and does notconsider losses due to friction and anchorage wedge seating, it can be demonstrated that elongation ofPT tendons dominate the PT force variation.

Figure 3. Comparison of Analytical and Experimental Results.

33.2. Analytical Modeling of RC WallsThe Shear Flexure Interaction Multi Vertical Line Element Model (SFI–MVLEM, Kolozvari et al.,2015), which is a finite element modeling approach that captures the experimentally observed shear
flexure interaction, was developed in OpenSees to simulate the cyclic behavior of RC walls. The wall
was modeled as a stack of five SFI–MVLEM elements placed upon one another, containing fivereinforced concrete panels each, as shown in Figure 4. The reinforced concrete panels represent a two–dimensional constitutive model relationshipthat relates the strain field imposed on eachof them (𝜀𝑥 , 𝜀𝑦 and 𝛾𝑥𝑦 ) to the resulting
Plainstress field on concrete (𝜎𝑥 , 𝜎𝑦 and 𝜏𝑥𝑦 ).
Concrete Thereby, the coupling of axial and shear
responses is achieved at a panel level, whichfurther incorporates interaction between
flexural/axial and shear forces anddeformations at the model level (Kolozvariet al., 2015). The flexural response of themodel element is captured through the axial
deformation of the reinforced concrete
Confined panels in the vertical direction (Y-), and theConcrete relative rotation between top and bottom
faces (curvature) of the wall element, as
well as the shear deformations areconcentrated at the center of rotation in thecentral axis of the element. Thus making thecoupling of axial/flexural and shear to occur
Figure 4. Shear Flexure Interaction Multi Verticalat a panel level. The accuracy of the SFI–
Line Element Model.
MVLEM is verified against theexperimental results of the specimen NSW6 tested in TIT, as shown in Figure 5. The results show agood accuracy of prediction for initial stiffness and base shear in both, positive and negative sides for
global behavior, which error is below 1.5%. The accuracy of the model to predict the local behavior isinvestigated with the crack pattern, since it is essentially controlled by strain field at a panel level, andis representative of the failure mode and the interaction between shear and axial/flexure behavior. The
results show that the model reproduces the cracking pattern with a good agreement in the orientation ofcracks, which means that SFI–MVLEM is capable of predicting the failure mode with reasonableaccuracy and can capture the shear flexure interaction experimentally observed.
900

1800

Figure 5. Comparison of Analytical Predictions wit Experimental Results.

44. INFLUENCE OF AXIAL LOADS IN RWs AND RC WALLS𝝃𝑯𝒚𝒔𝒕 =𝟐𝑬𝑫𝝅𝑬𝑺

The hysteretic damping is evaluated according tothe approach introduced by Jacobsen (1930)(Figure 6). The analysis shows that as axial load
ratio increases, inelastic energy dissipation (ED)and hysteretic damping (𝜉𝐻𝑦𝑠𝑡 ) increase in RWs,
while in RC walls ED remains essentially the samein spite of the axial load ratio and consequently𝜉𝐻𝑦𝑠𝑡 reduces. This is due to the cyclic behaviorof RC walls, as axial load increases, the behavior
resembles a flag shape hysteretic behavior, since
residual deformation reduces.

EDES

Figure 6. Equivalent Viscous Damping
According to Jacobsen’s Theory.

Rocking Wallsa) Base Shear (Q)–Lateral Drift (%).

b) PT Force – Lateral Drift (%).

c) Inelastic Energy Dissipation.

d) Hysteretic Damping.

e) Definition of Relative Energy
Dissipation Ratio.

f) Influence of Relative Energy Dissipation
Ratio.

Figure 7. Influence of Axial Load in Rocking Walls.

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Reinforced Concrete Wallsa) Base Shear (Q)–Lateral Drift (%).

b) Inelastic Energy Dissipation.
The results show the dependence ofED on strength, ductility and residual
deformation, which are the parameters defining
the area within the hysteresis loop. In both RWs
and RC walls base shear increases with axial load
ratio, but the increase is more accentuated in RWs
with the 86% than in RC walls, however, ductility
reduces in RWs. RC walls have shown greater
𝜉𝐻𝑦𝑠𝑡 than RWs, but because of the residual
deformation, which are a major indicator ofdamage, RWs as damage controlling structures,
which have negligible residual deformation, andconsequently less damage, are deemed assuperior seismic performance structures.

c) Hysteretic Damping.
Figure 8. Influence of Axial Load in RC Walls.

5. CONCLUSIONSThe base shear for RC walls, such as for RWs, increases with axial load ratios below the balance
point of the interaction curve, however, the influence of axial load ratio in the base shear is more
accentuated in RWs with an increase of 86% than in RC walls with an increase of 37%. Also,regarding post–tensioning force in tendons of RWs, it was found not to be affected by axial loadratio, but instead by the gap opening, therefore failure of the RWs is controlled by concrete
crushing at compression toe.
The ED increases with axial load ratio for the same level of lateral drift in RWs, while for RC
walls reduces for axial load ratios below the balance point of the interaction curve, due to thehysteretic behavior influenced by the reduction of residual drift. On the other hand, the relative
energy dissipation ratio (βh) (Figure 7 f)) according to ACI ITG–5.1–07 acceptance criteria,indicates that to have satisfactory seismic performance in terms of energy dissipation, βh should
be greater than 12.5%, therefore, additional energy dissipation devices should be provided.
ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to Dr. Susumu KONO and Dr. Tatsuya AZUHATA for theircontinuous support, valuable suggestion and instruction during my study.
REFERENCES
Kolozvari, K., Orakal, K., Wallace, J., 2015, PEER, Berkeley, California
Perez, F.J., Sause, R., Pessiki, S., 2007, Journal of Structural Engineering (ASCE).
Zhang, C., 2015, McMaster University, Ontario, Canada.

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