CSCE 2009 Annual General Conference Congrès annuel générale annuelle SCGC 2009 St. John’s, Newfoundland and Labrador / St. John’s, Terre-Neuve et Labrador May 27-30, 2009 / 27-30 mai 2009
MODELLING OF SEISMIC REPAIR AND RETROFIT OF CONCRETE SHEAR WALLS 1
2
W. L. Cortés and D. Palermo 1 University of Ottawa, Assistant professor, (
[email protected]) 2 University of Ottawa, M.A.Sc Candidate, (
[email protected]) Abstract: This paper is focused on modelling and assessment of seismically repaired and/or retrofitted reinforced concrete (RC) shear walls through nonlinear finite element analysis. The modelling process involved reproduction of the geometry of the original, undamaged wall, simulation of the loading protocol, nonlinear analysis of the model up to failure, modelling of the repair/retrofitting intervention, and finally, nonlinear analysis of the repaired/retrofitted wall. In general, there was excellent agreement with the experimental results, and the finite element (FE) method was able to capture behavioural aspects such as maximum strength, displacement, ductility and failure mechanisms. Keywords: Nonlinear finite element modelling; seismic repair, seismic retrofitting; reinforced concrete; shear walls.
1
INTRODUCTION
Repair and/or retrofitting of existing buildings have emerged as a viable solution for seismic upgrading to meet new requirements in modern seismic design provisions, such as ductility and energy dissipation. Failure of a structural element, such as a shear wall, due to an inappropriate repair/retrofit scheme can be catastrophic; therefore, detailed assessment of repair/retrofitting techniques is necessary. In this respect, analysis tools based on the FE method have become useful and economic for academics and practicing engineers. These tools, however, should be investigated and corroborated against experimental data in order to demonstrate their applicability and reliability. Several programs are capable of performing nonlinear analysis of RC structures: however, few have the capabilities to properly model repair and/or retrofitting techniques. To date, significant research has focused on developing repair/retrofitting techniques experimentally, and lagging are complementary analytical tools. An accurate analysis of a repaired/retrofitted RC structure requires nonlinear algorithms that simulate the construction sequence of the repair/retrofitting intervention. One such program is VecTor2, as described by Vecchio and Bucci (1999), which allows changes in the structural model by adding (engaging) unstressed elements and eliminating (disengaging) previously stressed elements. Additionally, the program permits modelling of the interface between the new and old material. The conceptual bases of VecTor2 are the Modified Compression Field Theory (MCFT) and the Disturbed Stress Field Model (DSFM). The program has an extensive library of models for concrete, steel and fibre reinforced polymers (FRP), which can be extended to other emerging repair/retrofitting materials. The focus of this study was to analyze and assess, by means of nonlinear FE analysis, the following repair and retrofitting techniques of shear walls previously tested and available in the literature: concrete replacement, reinforcing steel replacement, externally bonded steel plates, and externally bonded FRP sheets. The FE modelling and analysis included simulation of the load history, assessment of damage, modelling of the repair/retrofitting intervention, and analysis of the repaired/retrofitted walls. The analyses included predictions of salient features of response: strength capacity, displacement capacity, and failure mode.
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2 2.1
FINITE ELEMENT BASES Conceptual Bases
To predict the seismic behaviour of RC structures requires significant attention to the material models. In general, commercial FE programs implement sophisticated finite elements; however, the conceptual models may not necessarily capture the behaviour of RC structures satisfactorily. The nonlinear FE program VecTor2 uses simple finite elements, while employing rational models of the MCFT and the DSFM. The MCFT, developed by Vecchio and Collins (1986), is an analytical model for representing the nonlinear behaviour of RC structures subjected to in-plane normal and shear stresses. The RC element is modelled as a solid continuum orthotropic material using a smeared rotating crack approach. The formulation of the MCFT comprises compatibility relationships, equilibrium relationships, and realistic constitutive models for cracked concrete and reinforcement based on experimentally observed phenomena. On the other hand, the DSFM (Vecchio, 2000) addresses systematic deficiencies of the MCFT and extends it in several respects. Most importantly, the DSFM augments the compatibility relationships of the MCFT to include crack shear slip deformations. Additionally, by explicitly calculating crack slip deformation, the DSFM eliminates the crack shear check as required by the MCFT. The analyses of RC shear walls presented herein are based on the DSFM. A preliminary parametric study of the original (undamaged) walls demonstrated that differences in the MCFT and the DSFM were negligible in shear walls subjected to reverse cyclic loading and containing orthogonal reinforcement. In addition to the conceptual models, VecTor2 has an extensive library of constitutive models for the materials: reinforced concrete, steel reinforcement, FRP reinforcement, and bonding adhesive or epoxy that work similarly for both the MCFT and the DSFM. 2.2
Material Constitutive Models and Finite Elements
Constitutive models are linear and nonlinear functions of stress and strain that describe the response of the materials: concrete, reinforcement (steel and FRP), and bonding (adhesive and epoxy). Since adequate selection of the material models is important in predicting the response of RC structures, it is helpful to the user to have a criterion facilitate this selection process. In the analysis of the shear walls presented herein, default models were selected, except for the compression curve (pre-peak and postpeak response) and hysteretic models for concrete. These exemptions, used herein and elsewhere (Palermo and Vecchio, 2007), have demonstrated satisfactory results for shear walls. Reinforced concrete was modelled with plane stress rectangular elements; however, constant strain triangular elements were employed to meet geometric constraints where necessary. Three models for the concrete pre-peak response were selected according to the cylinder compressive strength f’c of the concrete. The Smith-Young model was used for concrete with f’c lower than 22 MPa, Popovics normalstrength for concrete with f’c ranging from 22 MPa to 45 MPa, and Popovics high-strength concrete for f’c greater than 45 MPa. For the concrete post-peak response, the base curve was selected, except for walls B9R, B11R and W11R, which were analyzed with the modified Park-Kent model due to the increasing compressive concrete strength attributed to the confinement reinforcement. Additionally, the hysteretic response model of concrete proposed by Palermo and Vecchio (2004) was selected. Typical constitutive models for concrete are shown in Figure 1a. Other relevant concrete models were selected as default: Vecchio’s 1992-A model for compression softening, modified Bentz tension stiffening model for tension stiffening effect, linear model for tension softening, Kupfer/Richard model for confined strength effect, modified Kupfer for lateral expansion or dilation of concrete, Mohr-Coulomb stress model for cracking criterion, 20% of the aggregate size limit for crack width check, and Vecchio-Lai for slip distortion. Reinforcement in the shear walls was generally modelled as smeared within the concrete element. The repair/retrofitting methods required other element types to model the reinforcement: 1) ductile steel reinforcement in cases where the internal reinforcement was modelled as discrete truss elements in the concrete, and in cases where external steel plates were externally bonded to the wall; 2) tension-only
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reinforcement for modelling externally bolted steel plates; and 3) externally bonded FRP reinforcement for modelling FRP sheets or strips. The default constitutive models were selected regardless of the element used in the modelling. A typical response of ductile steel is shown in Figure 1b. The Seckin model with Bauschinger effect was selected to capture the hysteretic response of the reinforcement. The Tassios model was selected for dowel action of the reinforcing bars. Finally, buckling of reinforcement was not considered. Buckling requires modelling the reinforcement as truss elements with bond elements. -f
c
fp = 5 0 M P a
f
s
f
u
f
y
fp = 4 0 M P a fp = 3 0 M P a
E
sh
1
fp = 2 0 M P a E u
sh
s
y s y
-
- f
y
-f
u
sh
u
c
a) b) c) Figure 1. Material Responses models: a) Concrete, b) Steel, and c) Bonding. (Wong and Vecchio 2002) Bonding materials were modelled with two-nodded non-dimensional link elements. Link elements can be conceptualized as two orthogonal springs that simulate the transfer of shear and normal stresses as well as the slippage in the interface between concrete and reinforcement. Before slippage, the two nodes must share the same coordinate, one of them connecting the concrete element and the other connecting the reinforcement, which is modelled as a truss element. In addition to the link elements, linear elastic with linear post-peak descending branch bond-slip relationship based on the fracture energy method (Sato and Vecchio, 2003) was selected for modelling bond between the external steel plates and FRP sheets to the concrete (See Figure 1c). The parameters in this relationship include the maximum shear stress, the slip at the occurrence of the maximum bond stress, and the slip in which the shear stress is diminished to zero after reaching the maximum shear stress. In addition to the selection of the relationship for bond-slip, the Eligehausen concrete bond model was selected to capture the hysteretic response of the element. Further details of the constitutive models are available elsewhere (Wong and Vecchio 2002).
3
FINITE ELEMENT MODELLING AND ANALYSIS METHODOLOGY
The modelling and analysis process consists of four basic steps and a post processing step as shown in Figure 2. In the post-processing, salient features such as maximum strength, maximum lateral displacement and failure mechanism are assessed and compared with the experimental results. To illustrate the aforementioned methodology, four shear walls, repaired and/or retrofitted with different techniques are presented. The first wall, B9R (Fiorato et. al., 1983), was repaired by means of concrete replacement after damage of the original wall; the second wall, B11R (Fiorato et. al., 1983), was repaired similarly to wall B9R, but in addition to the concrete replacement, two diagonal steel bars were incorporated; the third wall, W11R (Tadghi, 2000), was retrofitted with two diagonal plates and two vertical plates externally bolted to both sides of the shear wall; and the fourth wall, FRPLSW1 (Antoniades et al., 2003), was repaired after damage of the original wall with vertical and horizontal FRP sheets in addition to concrete replacement of the lower zone of the wall. 3.1
Modelling of Original Walls
Modelling the original walls begins with selecting the constitutive models and materials. Constitutive models are linear and nonlinear functions of stress and strain that describe the response of the materials: concrete, reinforcement (steel and FRP), and bonding (adhesive, epoxy).
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Original Wall Retrofitting Definition of Intervention
Repair 1
Modelling of original wall
2
Analysis of original wall until damage
3
Modelling of repaired/retrofitted wall
4
Analysis of repaired/retrofitted wall until damage
End of non-linear FE modelling and analysis
5
Post processing: Analysis of results
Figure 2. Finite element modelling and analysis procedure The concrete base curve was selected for wall FRPLSW1 for both the pre-peak and post-peak response, while the modified Park-Kent model was chosen for the post-peak regime for walls B9R, B11R and W11R. In walls B9R and B11R, the modified Park-Kent was selected due to the confinement effect of the hoop reinforcement in the boundary zones of the walls. Even though the confinement-effect criterion did not apply for wall W11R, the modified Park-Kent post-peak model provided a better prediction in the postpeak loading stages. For the hysteretic response of the concrete, the Palermo and Vecchio (2004) model was used in all analyses. The default material models were employed for all other concrete phenomena. The internal reinforcement of the walls was modelled as smeared in the concrete elements by defining reinforcement ratios in the horizontal, vertical and out-of-plane directions. In addition to the smeared reinforcement, truss elements were used as follows: 1) ductile steel reinforcement for the new diagonal bars in wall B11R, 2) tension-only reinforcement for the vertical carbon FRP used in wall FRPLWS1, and bonded steel plates in wall W11R, and 3) externally bonded FRP reinforcement for modelling FRP sheets or strips. The externally bonded FRP element is similar to the tension only reinforcement element, in that the reinforcement does not exhibit compressive stress. Additional modifications allow the local crack stresses in externally bonded FRP reinforcement to be more accurately calculated. Even though the bolted steel plates used in the retrofitting of wall W11R were ductile, it was decided to use tension only elements due to the early buckling of the plates between bolts. Tension-only elements were used in wall FRPLSW1 since the carbon FRP was modelled as perfectly bonded without using link elements. In all the walls, reinforcement constitutive models were set as default. Bonding materials were modelled with two-nodded non-dimensional link elements and a linear elastic with linear post-peak descending branch bond-slip relationship based on the fracture energy method (Sato and Vecchio, 2003) Following the selection of the constitutive models and materials, the geometry and the finite element mesh are created. Three of the four shear walls presented in this paper were model before damage
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(original walls): B9, B11 and LSW1. Wall W11R was not damaged before retrofitting; therefore no original wall was considered in the modelling and analysis process. Original wall B9 (called B9R after repair) was modelled with five homogeneous concrete zones with smeared steel, according to the geometry and material properties of the specimen: the first corresponding to the web; the second and third corresponding to the upper and lower portion of the boundary columns (the latter with more transverse and confinement reinforcement); the fourth corresponding to the foundation (fully restrained horizontally and vertically at the base) and top beams; and the fifth corresponding to the concrete to be replaced (Figure 3a). The mesh, excluding the foundation and top beams, consisted of 14 rectangular elements horizontally and 30 rectangular elements vertically. The elements used to define the zone of the new concrete in the repair process were initially disengaged during the analysis of the original wall; thus they did not contribute to the stiffness or strength. Original wall B11 (called B11R after repair) was modelled with five homogeneous concrete zones with smeared steel similar to wall B9 and according to the geometry and material properties of the specimen. Furthermore, four truss bar elements were used to model the four diagonal bars to be added in the repair process. Elements used to define the new concrete and diagonal steel bars in the repair process were initially disengaged during the analysis of the original wall (Figure 3b).
a) b) c) Figure 3. Finite element mesh of original walls: a) B9, b) B11, and c) LSW1 The model corresponding to original wall LSW1 (called FRPLSW1 after repair) was divided into six homogeneous concrete zones: web, boundary elements, foundation beam, top loading beam, and new concrete in the lower 250 mm of the web and boundary elements. The total number of rectangular concrete elements was 728, in which 442 corresponded to a 17 x 23 mesh of the web and boundary elements, including 51 double meshed elements used to model the concrete replacement. For the vertical CFRP strips, 30 perfectly bonded truss elements (15 in each edge) were used based on the assumption of adequate anchorage. Wrapping of the wall with GFRP was simulated with 16 rows of 17 horizontal truss elements along the wall. Elements, representing the GFRP, were connected to the concrete by means of bond-link elements, except for the extreme nodes at the wall edges which were modelled as perfectly bonded. Finite elements corresponding to the new elements incorporated during the repair were initially disengaged (Figure 3c). After construction of the geometric model, the materials are assigned to the finite elements, and the loading protocol to be used in the analysis is defined. The loading of the walls included lateral displacements and vertical axial load following the same patterns used in the experiments.
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3.2
Analysis of Original Walls
Analysis of a FE model using program VecTor2 is an iterative process for each load step, in which internal strains and stresses are calculated until a convergence criteria is achieved, which is based on secant moduli, reactions, or displacements. The analyses conducted were terminated at the onset of failure of the shear walls, which was assumed to correspond to a drop of more than 20% in the lateral load carrying capacity of the wall. Two types of failures were observed in the original walls presented herein: concrete crushing with significant shear distortion for walls B9 and B11 and shear sliding in wall LSW1. Wall B9 was loaded with five alternating small and long single (one repetition) displacements to 13 mm, 133 mm, 38 mm, 121 mm and 108 mm. Wall B11 was loaded with initial small amplitude cycles followed by loading up to 102 mm of displacement. Thereafter, three repetitions of loading were imposed from 51 mm of displacement up to failure at 152 mm of displacement. The loading history of wall LSW1 consisted of incremental cyclic lateral displacements of 2 mm until failure, applied along the top beam. Three repetitions of loading were imposed at each displacement level. 3.3
Modelling of Repaired/Retrofitted Walls
Modelling of repaired and/or retrofitted walls is a process in which distressed elements of the original damaged wall are disengaged and elements representing the new materials are engaged. Engaged elements represent portions of the structure that are active and contribute to the strength and stiffness of the structure. Conversely, disengaged elements represent portions the structure that are not active and do not contribute to the strength and stiffness of the structure. In cases where regions of a structure will be replaced by repair materials, engaged and disengaged elements occupy the same space in the mesh, resulting in a double meshed region (Wong and Vecchio, 2002). For disengaged elements, the total strain of the element is compatible with the surrounding elements; the total strain is retained as a plastic offset strain. Once engaged, the element begins to contribute to the strength and stiffness of the structure from a zero elastic strain. The effectiveness of the repair procedure depends upon the load sharing between the original and newly added portions of the structure. In turn, this depends not only upon the final configuration of the structure and loads, but also the extent of damage prior to repair and the strain differentials between the original material and repair materials at the time of repair (Wong and Vecchio, 2002). Similarly to the original walls, the modelling begins with the definition of the constitutive models and materials. The criterion used for the selection of the constitutive models is the same as discussed before for the original walls, and according to the properties of the new materials. Modelling of repaired wall B9R was based on original wall B9 where five homogeneous concrete zones were defined. Wall B9R was repaired with new concrete in the lower portion of the web up to 2600 mm. Elements in zone one (web) located in the same position of elements in zone five (new concrete in lower portion of the web) were disengaged after damage, and the originally disengaged elements in zone five were engaged (See Figure 4a). Similar to wall B9R, wall B11R was based on original wall B11. In addition to the new concrete in the lower portion of the web up to 2600 mm, four diagonals ties (two in each direction) as shown in Figure 4b were added. The engage-disengage process of concrete elements in zone one and five was the same as in wall B9R. Additionally, truss elements used to model the diagonal reinforcement were engaged in the analysis of the repaired wall. Wall W11R was modelled with five homogeneous concrete zones and additional truss bars representing internal reinforcement and externally bonded plates. The first zone was defined as plain concrete for the web in which three vertical truss bars were included to simulate the internal vertical steel reinforcement. Heavily reinforced concrete areas were defined as zones two and three for the I-shape foundation (zone three at the edges was wider than zone two). Zone four was defined for the top beam. Finally, the fifth
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zone with concentrated horizontal smeared reinforcement was defined to model the horizontal reinforcement. This zone was located in specific rows in the meshing to avoid premature shear sliding specially in the vicinity with the foundation (Figure 4c). To model the externally bonded diagonal steel plates, three ductile truss bars, perfectly bonded along the wall, were attached to the model from opposite corners in both directions. Finally, the externally bonded vertical steel plates were modelled with perfectly bonded tension-only truss bars attached along the edges of the wall. Although the steel used in the vertical steel plates was ductile, tension-only elements were specified as it was observed during testing that buckling occurred at early stages of loading. An additional model with link elements between the truss bars and the concrete was investigated in an attempt to capture buckling of the plates; however, this phenomenon was not well captured, owing in most part to the link properties not reflecting the bolting technique used to attach the steel plates to the concrete. Finally, concrete elements with high steel ratios were defined to model anchorage of the steel plates to the foundation and to avoid undesired stress concentrations not observed during testing.
c)
a)
b)
d) Figure 4. Finite element mesh of repaired/retrofitted walls: a) B9R, b) B11R, and c) W11R, d) FRPLSW1 Meshing of the web of wall W11R was a combination of rectangular and triangular elements in which 672 elements were used, equivalent to a grid of 22 by 20 elements. Triangular elements were necessary due to geometric constrains (Figure 4d). Modelling of wall FRPLSW1 involved disengaging of homogeneous concrete zones one and two, and engaging homogeneous concrete zones five and six. Zones five and six represented the new concrete in the lower 250 mm of the web and boundary elements of the wall. The 30 perfectly-bonded truss elements (15 in each edge), representing the vertical CFRP strips, and the 16 rows of 17 horizontal truss elements, representing the horizontal GFRP wrapping, were engaged, as well as the bond-link elements connecting the horizontal GFRP to the concrete. 3.4
Analysis of Repaired/Retrofitted Walls
The same iterative loading process described for the original walls was used for the analysis of the repaired/retrofitted walls. Additionally, the seed file parameter was included in the analysis of the
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repaired/retrofitted wall. This parameter ensures that the new analysis starts from the previous damage state of the original wall allowing proper simulation of the loading history and repair/retrofitting process. The criterion for stopping the analysis was a drop of more than 20 % of the lateral load carrying capacity, which was the same as the initial analyses. Loads defined for the repaired/retrofitted walls B9R, B11R and FRPLSW1 followed the same pattern used in the original walls; except for wall B9R in which the axial load was not included. Wall W11R was subjected to three repetitions of lateral reverse cyclic displacements and constant axial compression. The first three lateral reverse cycles were applied at 0.1%, 0.2% and 0.5% drift, and thereafter the lateral displacement was incremented by 0.5% until failure (2.5% in the experiment). Three types of failures were observed in the repaired/retrofitted walls presented herein: concrete crushing with shear distortion in walls B9R and B11R, concrete crushing and steel rupture in wall W11R, and steel rupture in wall FRPLSW1. 3.5
Post Processing: Analysis of Results
During the post processing stage, nodal displacements and element stresses are evaluated and further analyses are performed. Salient features of seismic behaviour such as strength, ductility (displacement), energy dissipation and failure modes are assessed. Post processing of repaired/retrofitted shear walls is performed first after damage of the original wall, and then after damage of the repaired/retrofitted wall. Post processing of the numerical results in this study focused on three seismic parameters: strength, ductility and failure modes; therefore, maximum load, maximum displacement and damage at ultimate were investigated for both the original and the repaired/retrofitted walls. In general, the analytical responses of the repaired/retrofitted shear walls compared very well to the experimental responses as demonstrated in Figures 5, 6, 7 and 8. Maximum strength capacity of 954 KN at approximately 169 mm was predicted for wall B9R. This maximum load was sustained until failure, which occurred at 188 mm. The predicted failure of wall B9R was concrete crushing with significant shear distortion of web concrete near the interface of the web and the boundary elements with significant cracking up to 1200 mm from the foundation. Additionally, heavy cracks were predicted in the upper 1000 mm of the web. The analytical response of wall B11R predicted a maximum strength of 672 KN at a corresponding displacement of 151 mm. The predicted damage in wall B11R was similar to wall B9R; crushing and shear distortion in the web, specifically near the boundary elements, and flexural cracks in the upper portion of the wall. The analytical results for Wall W11R included yield strength of about 500 KN at 0.5% lateral drift, which was sustained up to 1.5% drift. At this displacement, the strength was reduced to approximately 460 KN due to crushing of concrete between the diagonal plates and vertical plates around the quarter-height of the wall, and shear sliding in the plate anchorage zone. From 1.5% to 2.5% drift the strength was approximately 460 KN. At 3.0% drift, rupture of the vertical steel plates was predicted and the strength of the wall was reduced to 307 KN. A strength capacity of 387 KN at a corresponding displacement of 7 mm was predicted for FRPLSW1. At the second excursion to 7 mm, rupture of the internal vertical reinforcement at the base of the wall near the interface with the foundation was predicted, as well as cracking above the ruptured reinforcement. Discrepancies between the predicted damage and the observed damage were evident. The predicted failure mechanism did not capture failure of the anchor of the vertical CFRP observed during testing; however, steel rupture of the vertical steel after rupture of the anchor was properly simulated.
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(a) (b) Figure 5. Load-Deformation Responses of Wall B9R: a) Analytical; b) Experimental
(a) (b) Figure 6. Load-Deformation Responses of Wall B11R: a) Analytical; b) Experimental
(b) (b) Figure 7. Load-Deformation Responses of Wall W11R: (a) Analytical; (b) Experimental
(c) (b) Figure 8 Load-Deformation Responses of Wall FRPLSW1: (a) Analytical; (b) Experimental
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CONCLUSIONS
Nonlinear finite element programs such as VecTor2 are useful tools for modelling and assessment of emerging repair and/or retrofitting techniques. The FE methodology discussed in this paper allows assessment of the repair/retrofitting intervention through the following four steps: 1) modelling of the original wall, 2) analysis of the original wall, 3) modelling of the repaired/retrofitted wall and 4) analysis of the repaired/retrofitted wall. Additionally, a post-processing stage allows analysis of salient seismic parameters of reinforced concrete such as strength, ductility, energy dissipation and failure modes. Modelling of the original shear walls involves definition of the constitutive models of the materials used, definition of the geometry and finite element (FE) meshing, and definition of the load patterns. After modelling, the shear wall is analyzed by means of an iterative process in which the shear wall is loaded according to the load pattern defined in the modelling. Modelling of repaired/retrofitted shear walls is based on the model of the original wall and includes engagement and disengagement of unstressed new materials and damaged elements, respectively. The engagement-disengagement process ensures proper calculation of strains and stresses in the new materials. Analysis of the repaired shear wall is similar to the analysis of the original wall; however, it starts at a stage in which the shear wall is already damaged and strains and stresses are carried forward in the elements that remain engaged. Three seismic parameters were studied in the nonlinear FE modelling and analysis of four walls repaired/retrofitted with three common techniques: concrete replacement, steel replacement and FRP bonding. Even though predictions using this simple nonlinear FE modelling and analysis methodology have demonstrated reliability and have compared very well to the experimental observations, further work is necessary to improve the accuracy of the predictions. Additionally, further studies are required to address other novel repair/retrofitting techniques applied to different seismic resisting RC elements.
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REFERENCES
Antoniades, K.K., Salonikos, T.N., and Kappos, A.J. 2003. Cyclic Tests on Seismically Damaged Reinforced Concrete Walls Strengthened Using Fiber-Reinforced Polymer Reinforcement, ACI Structural Journal, 100(4): 510-518. Fiorato, A.E., Oesterle, R.G., and Corley W.G. 1983. Behavior of Earthquake Resistant Structural Walls Before and After Repair, ACI journal, September-October 1983: 403-413. Palermo, D., and Vecchio, F.J. 2004. Compression Field Modeling of Reinforced Concrete Subjected to Reverse Loading: Verification, ACI Structural Journal, 101(2): 155-164. Palermo, D., and Vecchio, F.J. 2007. Simulation of Cyclically Loaded Concrete Structures Based on the Finite-Element Method, Journal of Structural Engineering, ASCE, 133(5): 728-738. Sato, Y., and Vecchio, F.J. 2003. Tension Stiffening and Crack Formation in Reinforced Concrete Members with Fiber-Reinforced Polymer Sheets, Journal of Structural Engineering, ASCE, 129(6): 717724. Taghdi, M., Bruneau, M., Saatcioglu, M. 2000. Seismic Retrofitting of Low-Rise Masonry and Concrete Walls Using Steel Strips, Journal of Structural Engineering, ASCE, 126(9): 1017-1025. Vecchio, F.J. and Collins, M.P. 1986. The Modified Compression Field Theory for Reinforced Concrete Elements Subject to Shear, ACI Journal, 83(2): 219-231. Vecchio, F.J. and Collins, M.P. 1993. Compression Response of Cracked Reinforced Concrete, Journal of Structural Engineering, ASCE, 119(12): 3590-3610. Vecchio, F.J., and Bucci, F. 1999. Analysis of Repaired Concrete Structures. Journal of Structural Engineering. ASCE, 125(6): 644-652. Vecchio, F.J. 2000. Disturbed Stress Field Model for Reinforced Concrete: Formulation, Journal of Structural Engineering, ASCE, 126(9): 1070-1077. Wong, P.S., and Vecchio, F.J. 2002. VecTor2 and formworks user’s manual, Rep., Civil Engineering, University of Toronto, Toronto. ON, Canada.
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