13-012 Kurama pre-edit FROM author
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AutoCAD files must be compatible with autoCAD 2007 o Graphs should be the original graphing program, such as Excel. o Please provide a PDF or word document showing the final layout for all figures (do not include the captions in the figure layout). In preparing your final copy, please leave the tracked changes tool on so we can easily identify the changes you have made to the final version of your paper. Thank you. <note to production: this is a peer reviewed article> <title> Seismic design guidelines for solid and perforated hybrid precast concrete shear walls <byline> Brian J. Smith and Yahya C. Kurama <begin summary box> <body> . . . <end summary box> <body> The hybrid precast concrete wall system investigated in this paper (Fig. 1) is constructed by placing rectangular precast concrete panels across horizontal joints at the foundation and floor levels.1-4 The term hybrid reflects that a combination of Grade 400 (Grade 60) mild steel bars and high-strength unbonded posttensioning strands is used for the lateral resistance of the structure across the critical joint between the base panel and the foundation (that is, the base joint). Under the application of lateral loads into the nonlinear range, the primary mode of displacement in a well-designed hybrid precast concrete shear wall occurs through a gap at the base joint, allowing the wall to undergo large nonlinear lateral displacements with little damage. Upon unloading, the posttensioning steel provides a vertical restoring force (in addition to the gravity loads acting on the wall) to close this gap, thus significantly reducing the residual (that is, permanent) lateral displacements of the structure after a large earthquake. The use of unbonded tendons delays the yielding of the posttensioning strands and reduces the tensile stresses transferred to the concrete (thus reducing cracking) as the tendons elongate under lateral loading. The mild steel bars across the base joint are designed to yield in tension and compression, and provide energy dissipation through the gap opening and closing behavior of the wall under reversed-cyclic lateral loading. A predetermined length of these energy-dissipating bars is unbonded at the base joint (by wrapping the bars with plastic sleeves) to limit the steel strains and prevent low-cycle fatigue fracture. Hybrid precast concrete shear walls are efficient structures that offer high-quality production, relatively simple erection, and excellent seismic characteristics by providing self-centering to restore the building toward its original undisplaced position as well as energy dissipation to limit the lateral displacements during an earthquake. Despite these desirable characteristics, hybrid precast walls are classified as nonemulative structures because their behavior is different from that of conventional monolithic cast-in-place reinforced concrete shear walls. Thus, experimental validation is required by the American Concrete Institute’s (ACI’s) Building Code Requirements for Structural Concrete and Commentary (ACI 318-11)5 and Acceptance Criteria for Special Unbonded Post-Tensioned Precast Structural Walls Based on Validation Testing and Commentary (ACI ITG-5.1)6 prior to the use of these structures in seismic regions of the United States. <subhead 1> Research objectives and significance <body> The primary objective of the research described in this paper is to advance precast concrete building construction by conducting the required experimental validation and associated design and analytical studies for the code approval of hybrid wall structures as special reinforced concrete shear walls per ACI 318-11. The project provides new information in accordance with and directly addressing the ACI ITG-5.1 validation requirements, as well as information regarding the behavior of hybrid precast concrete walls featuring multiple wall panels (that is, multiple horizontal joints over the wall height) and panel perforations, both common features in practical building construction. The experimental results demonstrate that hybrid precast walls can satisfy all requirements for special reinforced concrete shear walls in high seismic regions with improved performance, while also revealing important design, detailing, and analysis considerations to prevent undesirable failure mechanisms. It is not the objective of this paper to provide full details on the experiments, which can be found elsewhere. 1-4 Instead, only the relevant experimental results that support the proposed design recommendations are provided. <subhead 1> Overview of proposed design guidelines <body> The design, detailing, and analysis guidelines and recommendations provided in this paper are intended for use by practicing engineers and precast concrete producers involved in the design of hybrid shear walls in moderate and high seismic regions. Where appropriate, ACI 318-11 requirements for special monolithic cast-in-place reinforced concrete shear walls are used to help in the adoption of the recommended guidelines. Furthermore, applicable references and suggested modifications and additions to the design recommendations in Requirements for Design of a Special Unbonded Post-Tensioned Precast Shear Wall Satisfying ACI ITG-5.1 and Commentary (ACI ITG-5.2)7 are given. The recommendations developed by this project include a performance-based design procedure and a prescriptive design procedure. This paper summarizes key guidelines from the performance-based design procedure. A detailed example demonstrating step-by-step application of the design procedure can be found online in the appendix at . Smith and Kurama recommend capacity reduction factors to use with the design. 1 The proposed guidelines can be used to design hybrid walls with height-to-length Hw/Lw aspect ratios equal to or greater than 0.5 in low- to mid-rise structures with a practical height limitation of 36.5 m (120 ft), or approximately eight to ten stories tall. The procedure is applicable to both single-panel wall systems (featuring only the base joint) as well as multi-panel systems (featuring base panel-to-foundation and upper panel-to-panel joints) with or without panel perforations. The design is conducted at two wall drift levels as follows: the design-level wall drift Δwd corresponding to the design-basis earthquake (DBE), and the maximum-level wall drift Δwm corresponding to the maximum-considered earthquake (MCE). The wall drift Δw is defined as the relative lateral displacement at the top of the wall divided by the wall height from the top of the foundation Hw. While all lateral deformations and rotations of the wall due to flexure, shear, and horizontal shear slip are included in the calculation for Δw,6 shear slip across the horizontal joints and gap opening across the upper panel-to-panel joints can be ignored because the design of a wall includes provisions to prevent these undesirable deformation behaviors. Figure 2 shows the idealized base shear force versus roof drift behavior of a properly designed hybrid precast concrete wall subjected to these drift demands. The corresponding expected wall performance at Δwd is as follows: gap opening at the base joint but no gap opening or nonlinear material behavior at the upper panel-to-panel joints no residual vertical uplift of the wall upon removal of lateral loads (that is, the gap at the base joint fully closes upon unloading) no shear slip at the horizontal joints yielding of the energy-dissipating bars posttensioning steel remaining in the linear-elastic range minor hairline cracking in the base panel (for perforated walls, cracking may extend into the upper panels) no observable concrete damage in compression, but cover concrete at the wall toes on the verge of spalling The corresponding expected wall performance at Δwm is as follows: increased gap opening at the base joint but no significant gap opening or nonlinear material behavior at the upper panel-to-panel joints no significant residual vertical uplift of the wall upon removal of lateral loads no significant shear slip at the horizontal joints significant yielding but no fracture of the energy-dissipating bars posttensioning steel in the nonlinear range but with strains not exceeding 0.01 mm/mm (0.01 in./in.) well distributed, still hairline, cracking of the concrete cover concrete spalling at the wall toes, with the confined core concrete on the verge of crushing The proposed design guidelines to achieve these performance objectives were validated using the measured and predicted behaviors of six 0.4-scale wall test specimens (four solid and two perforated walls) subjected to servicelevel gravity loads combined with quasi-static reversed-cyclic lateral loading.2-4 Tables 1 and 2 provide some of the important features of these wall specimens. Specimens HW1 through HW5 were hybrid walls, and specimen EW was an emulative precast concrete wall with no posttensioning steel reinforcement (that is, the entire lateral resistance was provided by mild steel bars).3 While outside the scope of this paper, three analytical models were developed by this project1 as experimentally-validated tools for engineers to design hybrid walls with predictable and reliable behavior under lateral loads. The recommendations for a linear-elastic effective stiffness model (which can be used to estimate the linear-elastic lateral displacement demands) and a nonlinear finite element model (used to conduct nonlinear monotonic lateral load analyses and aid in the design of hybrid walls with perforations) intentionally incorporate several simplifying assumptions appropriate for the design office. Recommendations for a detailed fiber element model (used to conduct reversed-cyclic lateral load analyses and dynamic analyses) were also developed.3 <subhead 1> Determination of seismic forces and drift demands <body> The design of a hybrid wall should be conducted under all applicable load combinations prescribed by Minimum Design Loads for Buildings and Other Structures (ASCE/SEI 7-10),8 including the use of a redundancy factor and torsional effects from accidental and applied eccentricities. The design base shear force can be obtained using any of the procedures allowed in ASCE 7-10, such as the equivalent lateral force procedure or the modal analysis procedure. The most basic approach is the equivalent lateral force procedure (appendix design example), in which the wall design base shear force Vwd is determined by dividing the first mode linear-elastic force demand under the DBE with the prescribed response modification factor R. When selecting R using Table 12.2-1 in ASCE 7-10, the seismic force-resisting system for hybrid walls can be classified as special reinforced concrete shear walls. Therefore, the response modification factors R should be taken as 5.0 and 6.0 for bearing wall systems and building frame systems, respectively. Once the design base shear force Vwd is established, the design base moment Mwd and the other design forces (for example, story shear forces and bending moments) can be found from a linear-elastic analysis of the structure under the vertical distribution of the design base shear force from ASCE 7-10. Appropriate analytical techniques, such as nonlinear dynamic response history analyses under properly-selected DBE and MCE ground motion sets, can be used to determine the design-level drift Δwd and the maximum-level drift Δwm. Alternatively, the ASCE 7-10 guidelines in section 12.8.6 can be used to determine Δwd by multiplying the linear-elastic wall drift Δwe under the design base shear force Vwd with the prescribed deflection amplification factor Cd. When selecting Cd from Table 12.2-1 in ASCE 7-10, the seismic force-resisting system for hybrid walls should be classified as special reinforced concrete shear walls, resulting in Cd equal to 5.0. The linear-elastic wall drift Δwe (flexural plus shear displacements corresponding to Vwd) can be calculated using an effective linear-elastic stiffness model. As described in Smith and Kurama,1 for design purposes, the linear-elastic effective flexural and shear stiffnesses can be determined using an effective moment of inertia Ie and effective shear area Ash, respectively, which are calculated using Eq. (1) and (2). Ie = 0.50Igross ................................................................................................................................................. (1) where Igross = moment of inertia of gross wall cross-section Ash = 0.80Agross .............................................................................................................................................. (2) where Agross = area of gross wall cross section For perforated walls, Igross should be taken at the cross section of the base panel including the perforations. The shear deformations are considerably increased due to the presence of panel perforations; and thus, the effective shear area Ash should be taken as the gross cross-sectional area of only the exterior vertical chord on the compression side of the base panel (that is, the compression vertical chord located outside of the perforations) without the 0.8 factor in Eq. (2). The other vertical chords located on the tension side of the wall and in between perforations do not contribute significantly to the shear stiffness; thus they should not be included in the effective shear area. Unless nonlinear dynamic response history analyses are conducted under properly-selected MCE ground motion sets, the maximum-level wall drift Δwm can be estimated from approximate methods that consider the unique hysteretic characteristics of hybrid precast concrete walls (in particular, the reduced energy dissipation and increased self-centering).9,10 For the test specimens in this project, Δwm was taken equal to the prescribed validation-level wall drift Δwc in ACI ITG-5.1 given as Eq. (3). Δwc = 0.9% Hw 0.8% + 0.5% 3.0% ....................................................................................................... (3) Lw For the tested wall dimensions of Hw equal to 5.49 m (18.0 ft) and Lw equal to 2.44 m (8.00 ft), the validation-level wall drift Δwc was 2.30%, and this value was used as Δwm in the design of the structures. The validity of using Δwm equal to 2.30% for design was supported by nonlinear dynamic response history analyses of the test specimens under selected sets of MCE ground motion records. 11,12 <subhead 1> Design of base joint <body> The design for the base joint of a hybrid precast concrete wall includes the determination of the energy-dissipating and posttensioning steel areas, probable (maximum) base moment strength of the wall, contact length and confinement reinforcement at the wall toes, energy-dissipating steel strains and stresses (including the determination of the unbonded length for the energy-dissipating bars), and posttensioning steel strains and stresses (including the determination of the posttensioning stress losses). <subhead 2> Reinforcement crossing base joint <body> The posttensioning and energy-dissipating steel areas crossing the base joint can be determined1 using fundamental concepts of reinforced and prestressed concrete mechanics (equilibrium, compatibility and kinematics, and design constitutive relationships). As demonstrated in the design example, a key parameter selected by the designer during this process is the energy-dissipating steel moment ratio κd, which is defined as Eq. (4). d M ws .......................................................................................................................................... (4) M wp M wn where Mwn = contribution of wall gravity axial force Nw to satisfy Mwd Mwp = contribution of posttensioning steel to satisfy Mwd Mws = contribution of energy-dissipating steel to satisfy Mwd The κd ratio is a relative measure of the resisting moments from the energy-dissipating force provided by the energydissipating steel reinforcement and the restoring (that is, self-centering) force provided by the posttensioning steel reinforcement plus the gravity axial load in the wall.13 If κd is too small, the energy dissipation of the structure may be small. Conversely, if κd is too large, the self-centering capability of the wall may not be sufficient to yield the tensile energy-dissipating bars back in compression and close the gap at the base joint upon the removal of the lateral loads. The κd values used in the design of the test specimens in this project ranged from 0.50 to 0.85, with the actual κd values for the as-tested structures (using the provided steel areas) ranging from 0.50 to 0.90 (Table 1). Based on the performance of the test specimens,2-4 the κd value used in design should not exceed 0.80 to ensure sufficient self-centering and should not be less than 0.50 to ensure sufficient energy dissipation. Specimens with κd values close to both of these limits were tested. Figure 3 shows the relative energy dissipation ratio β (as defined in ACI ITG-5.1) of the six specimens as a function of wall drift. Specimen HW4, with design κd equal to 0.50 and actual κd equal to 0.54, satisfied the minimum relative energy dissipation ratio βmin of 0.125 prescribed by ACI ITG5.1 with a small margin, leading to the recommended lower limit of κd equal to 0.50 for design. Specimen HW4 was a perforated wall, which increased the shear deformations in the wall panels, resulting in reduced energy dissipation. Although it may be possible to use a reduced value for the lower κd limit for solid hybrid walls, this was not investigated by this research project. The recommended upper limit of κd equal to 0.80 was selected based on the premature failure of specimen HW5 (with design κd equal to 0.85 and actual κd equal to 0.90) prior to sustaining three loading cycles at the ACI ITG-5.1 validation drift Δwc of 2.30%. This specimen suffered from a loss of restoring force and failed due to the permanent uplift of the wall from the foundation (that is, a gap formed along the entire base joint when the wall was unloaded to Δw of 0%), which resulted in the subsequent out-of-plane displacements of the wall base and buckling of the energy-dissipating bars in compression. Wall uplift and excessive out-of-plane displacements (or in-plane slip as was observed in the emulative specimen EW1) resulting from loss or lack of restoring force across the base joint can develop quickly and lead to failure with little warning. Figure 4 shows the measured base shear force Vb versus wall drift Δw behavior of specimen HW5, where the loss of restoring force can be seen by the unloading curves that do not return through the origin. The observed out-of-plane displacements and buckling of the energy-dissipating bars at the base joint are also shown. The goal of the recommended upper limit on κd is to prevent this type of behavior. <subhead 2> Confinement reinforcement at wall toes <body> Confinement reinforcement, comprised of closely-spaced closed hoops, is required at the bottom corners (called toes) of the base panel to prevent premature crushing and failure of the core (inner) concrete prior to the maximumlevel wall drift Δwm. The confinement steel ratio, hoop layout, and hoop spacing can be designed according to sections 5.6.3.5 through 5.6.3.9 in ACI ITG-5.2 (appendix design example). Based on the extent of the cover concrete spalling in the test specimens and accurate measurement of the concrete compression strains using digital image correlation,2-4 the confined concrete region at the wall toes should extend vertically over a height of the base panel not less than the plastic hinge height hp, defined as 0.06Hw per ACI ITG-5.2. Figure 5 shows the north toe of the base panel in specimens HW3 and HW4 (where hp equals 330 mm [13 in.] for each wall) at the conclusion of each test. The observed vertical extent of cover spalling in specimens HW3 and HW4 was approximately 356 (14.0) and 305 mm (12.0 in.), respectively, supporting the design recommendation. Note that since the specimen length and height were not varied in this experimental program, additional research may be needed to validate the plastic hinge height recommendation in ACI ITG-5.2 for different wall dimensions. As required by ACI ITG-5.2, the confined region should extend horizontally from each end of the base panel over a distance not less than 0.95cm and not less than 305 mm (12.0 in.), where cm is the neutral axis length1 (contact length) at the wall base when Δwm is reached. The design and detailing of the confinement reinforcement should satisfy all applicable requirements for special boundary regions as well as the requirements for bar spacing and concrete cover in ACI 318-11. The distance of the first confinement hoop from the base of the wall is critical to the performance of the confined concrete. The first hoop should be placed at a distance from the bottom of the base panel no greater than the minimum required concrete cover per ACI 318. This recommendation was not satisfied in specimen HW1 (Table 2) due to the accidental slanted placement of the hoops during fabrication, resulting in premature confined concrete crushing (Fig. 6). The length-to-width aspect ratio (measured center-to-center of bar) for rectangular hoops should not exceed 2.50. This requirement is slightly more conservative than but similar to the requirements in past seismic design code specifications for concrete confinement (for example, section 1921.6.6.6 of the 1997 Uniform Building Code, Volume 2: Structural Engineering Design Provisions14), which have since been removed from current codes in the United States. As observed from the performance of specimen HW3 (Fig. 6), a large length-to-width ratio for the hoops can cause the bowing of the longer hoop legs in the out-of-plane direction, reducing the confinement effectiveness. Further, intermediate crossties were ineffective in preventing the bowing of the longer hoop legs because in typical construction the crossties do not directly engage the hoop steel. (Rather, the crossties engage the vertical reinforcement within the hoops.) For comparison, specimens HW4 and HW5 followed all of the listed recommendations, resulting in excellent behavior of the confined concrete at the wall toes.3 Figure 6 shows the performance of specimen HW4 and the confinement detailing differences between the hybrid wall specimens. Figure 6 further demonstrates the confined concrete region performance with plots of the measured neutral axis length across the base joint at the south end of the walls. The results are shown for the peak of the first cycle in each drift series, except for the last series where the peaks from all three cycles are shown. During the small displacements of each wall, the neutral axis length went through a rapid decrease associated with gap opening at the base. As each wall was displaced further, the neutral axis length continued to decrease but at a much slower pace. Once deterioration of the concrete at the wall toes initiated, the neutral axis length began to elongate to satisfy equilibrium with the reduced concrete stresses. This effect was more evident during the final drift series for specimens HW1 and HW3, and was especially exacerbated due to the premature crushing of the confined concrete in specimen HW1. In specimens HW4 and HW5, the neutral axis remained stable over a much larger drift range as the elongation of the neutral axis length was not observed in specimen HW5 and was evident in specimen HW4 only after achieving the validation-level drift Δwc of +2.30%. This was primarily due to the confinement steel detailing improvements incorporated into these specimens (Fig. 6). <subhead 2> Energy-dissipating bar strains and unbonded length <body> The performance-based seismic design of a hybrid precast concrete shear wall requires an iterative process where the energy-dissipating bar strains are estimated based on a predicted neutral axis length. As demonstrated in the appendix design example, the elongations of the energy-dissipating bars can be found by assuming that the wall displaces like a rigid body rotating through gap opening at the base joint.1 The strain in each bar can then be calculated by dividing the bar elongation with the total unbonded length, which is taken as the sum of the plasticwrapped length and an additional length of debonding that is expected to develop during the reversed-cyclic lateral displacements of the structure (estimated as the coefficient for additional energy-dissipating bar debonding αs times the bar diameter per ACI ITG-5.2). The coefficient αs can be assumed equal to 0 and 2.0 at Δwd and Δwm, respectively. Concrete cores were taken through the thickness of the base panel around the end of the wrapped length of the energy-dissipating bars in two of the test specimens, supporting the use of αs equal to 2.0 at Δwm. The wrapped unbonded length of the energy-dissipating bars (Fig. 7) should be designed such that the strains of the bars at Δwm are greater than 0.5εsu to ensure sufficient energy dissipation but do not exceed the maximum allowable strain of 0.85εsu (per ACI ITG-5.2) to prevent low-cycle fatigue fracture, where εsu is the monotonic strain capacity of the steel at peak strength. Energy-dissipating bar strains up to 0.85εsu were used in the design of the test specimens (Table 1), with no bar fracture observed during the experiments. The wrapped length can be located in either the bottom of the base panel or the top of the foundation; in either configuration, the energy-dissipating bars should also be isolated from the grout through the thickness of the grout pad at the base joint (that is, the wrapped length should include the thickness of the grout pad). Sufficient development length should be provided at both ends of the wrapped region of the energy-dissipating bars. Due to the large reversed-cyclic steel strains expected through Δwm, Type II grouted mechanical splices specified in section 21.1.6 of ACI 318-11 and permitted by section 5.4.2 of ACI ITG-5.2 should not be used for the energydissipating bars in hybrid precast concrete walls in seismic regions unless the splices have been tested and validated under cyclic loading up to a steel strain of at least 0.85εsu. Pullout of the energy-dissipating bars from the foundation occurred in specimen HW2 due to the failure of the grout within Type II splice connectors prior to Δwm (Fig. 8). The pullout caused the energy-dissipating bar elongations and strains to be smaller than designed, resulting in smaller lateral strength and energy dissipation of the wall (Fig. 3). Figure 8 shows the measured Vb - Δw behavior of specimen HW2, with the gray shaded band indicating the calculated strength of the wall (using the measured posttensioning steel forces, gravity load, and neutral axis length, as well as the ACI 318-11 and ACI ITG-5.2 concrete stress blocks) ignoring the contribution of the energy-dissipating bars. Comparing this range with the measured strength, it was confirmed that specimen HW2 was essentially behaving as a fully-posttensioned wall with no effective energy-dissipating steel reinforcement by the end of the test. The grout used in the energy-dissipating bar splice connectors in specimen HW2 satisfied the splice manufacturer’s specifications, and the splice itself satisfied the performance requirements in ACI 318-11 and AC133: Acceptance Criteria for Mechanical Connector Systems for Steel Reinforcing Bars15 for Type II mechanical connectors. However, the energy-dissipating bars in specimen HW2 were subjected to much greater strains and over a significantly larger number of cycles than required to classify a Type II connection per ACI 318-11 and AC133, causing the pullout of the bars. As a result, it is recommended that in validating Type II connectors for use in energy-dissipating bar splices of hybrid precast concrete shear walls, the bars should first be subjected to 20 loading cycles through +0.95εsy and -0.5εsy, as required by AC133 (where εsy is the yield strain of the energy-dissipating steel). Beyond this point, six cycles should be applied at each load increment, with the compression strain amplitude kept constant at -0.5εsy and the tension strain amplitude increased to a value not less than 1.25 times and not more than 1.5 times the strain amplitude from the previous load increment. Testing should continue until the tension strain amplitude reaches or exceeds +0.85εsu (that is, the maximum allowable energy-dissipating bar strain per ACI ITG5.2) over six cycles. These requirements would result in similar cyclic loading conditions during the validation testing of the splices as the loading conditions that can develop in the energy-dissipating bars during the validation testing of hybrid shear walls based on ACI ITG-5.1. In lieu of Type II mechanical splices, the full development length of the energy-dissipating bars can be cast or grouted (during the erection process) into the base panel and the foundation. Both of these fully-developed connection techniques were successfully used in this project with no observed pullout of the bars from the concrete or grout.2,3 <subhead 2> Posttensioning steel strain limitation <body> The elongations of the posttensioning strands can also be found by assuming that the wall displaces like a rigid body rotating through the gap opening at the base joint (appendix).1 The strain increment in each tendon due to the lateral displacements of the wall can then be calculated by dividing the strand elongations with the unbonded length. Significant nonlinear straining of the posttensioning steel should be prevented to limit the prestress losses at Δwm. Furthermore, the anchorage system for the posttensioning tendons should be capable of allowing the strands to reach the predicted stresses and total strains (that is, prestrain plus the strain increment due to gap opening) at Δwm without strand wire fracture or wire slip. Unless the posttensioning anchors have been qualified for greater strand strains under cyclic loading, the total strand strains (including prestrain) should be limited to a maximum allowable strain of 0.01 mm/mm (0.01 in./in.). Strand wire fractures can occur if tendon strains exceed 0.01 mm/mm (0.01 in./in.).1618 With this strain limit, which was used in the design of the hybrid specimens tested as part of this project, no strand wire fracture or slip was observed and the tendons remained mostly in the linear-elastic range through Δwm. <subhead 1> Wall restoring force <body> Hybrid precast concrete walls must maintain an adequate amount of restoring force (self-centering capability) to ensure that the gap at the base joint is fully closed upon removal of the lateral load after tensile yielding of the energy-dissipating bars. This restoring force, which comprises the gravity axial force and the total posttensioning force (including losses) at Δwm, should be sufficient to yield the tensile energy-dissipating bars back in compression and return them to essentially zero strain. Figure 9 demonstrates this requirement using an idealized stress-strain relationship for the energy-dissipating steel. As the wall is displaced from the initial unloaded position (close to point A), the energy-dissipating bars yield in tension (point B) and reach the maximum strain εsm (point C) at Δwm. Upon unloading of the wall, the restoring force must be able to yield the tensile energy-dissipating bars back in compression (point D) and return the bars to zero strain (point E), resulting in a total stress reversal of fsm plus fsy (where fsm is the energy-dissipating steel stress at Δwm) such that no significant plastic tensile strains accumulate in the steel. The use of the full energy-dissipating bar stress reversal of fsm plus fsy in design means that the elastic restoring force provided by the energy-dissipating steel is conservatively ignored when determining the required wall restoring force. For walls where, for design purposes, the individual energy-dissipating bars can be lumped (that is, combined) into a single steel area (equal to the sum of the individual bar areas) at the wall centerline (which requires that the bars be placed near the centerline, similar to Fig. 1) and the posttensioning tendons on either side of the wall centerline can be lumped into a single tension-side tendon area and a single compression-side tendon area, the design requirement for the restoring force can be written as Eq. (5). Ap f pm 0.5 f p ,loss N w As f sm f sy ......................................................................................... (5) where Ap = total posttensioning steel area (sum of the lumped tension-side and compression-side tendon areas) fpm = predicted stress1 in the lumped tension-side posttensioning tendon under monotonic loading to Δwm fp,loss = predicted stress loss1 in the lumped compression-side posttensioning tendon at Δwm due to nonlinear material behavior associated with reversed-cyclic loading of the wall to ±Δwm As = total lumped energy-dissipating steel area The application of Eq. (5) is demonstrated in the appendix design example. Where the posttensioning tendons or the energy-dissipating bars cannot be lumped, Eq. (5) should be revised to separately consider the steel stress in each tendon or bar. The design requirement given by Eq. (5) is more demanding than that given in section 5.3.1 of ACI ITG-5.2. The restoring force in specimens HW1, HW2, HW3, and HW4 satisfied Eq. (5), while specimens HW5 and EW did not satisfy Eq. (5). To investigate the effect of the wall restoring force, Fig. 9 shows the residual axial elongation (that is, heightening of the wall upon unloading from a lateral displacement) at the centerline of each specimen (measured at the same elevation as the applied lateral load) upon returning of the wall to Δw of 0% from the third cycle in each drift series (where upward displacement is defined as positive). The accumulation of this residual axial elongation represents a reduction (that is, loss) or lack of sufficient axial restoring force in the system. In specimens HW1, HW2, HW3, and HW4, the axial elongation did not start to accumulate until the drift series with Δwm of ±1.55%, which coincided with the initiation of posttensioning stress losses. Among these four walls, the largest residual elongation occurred in specimen HW3, where the maximum elongation upon unloading from Δw of ±2.30% was 1.7 mm (0.067 in.). This small uplift did not adversely affect the performance of the wall. In comparison, the residual axial elongation in specimen HW5 started to accumulate earlier (during the drift cycles with Δw of ±1.15%), and the maximum residual elongation at Δw of 2.30% was almost twice as large (3.0 mm [0.12 in.]). This increased uplift is related to the reduced self-centering in the wall (from the posttensioning force plus the gravity load) and the increased contribution of the energy-dissipating steel to the total base moment strength (due to the use of a large κd in design [Table 1]). Specimen HW5 marginally satisfied the restoring force requirement in ACI ITG-5.2 but did not satisfy Eq. (5). Once the energy-dissipating bars yielded in tension, the restoring force was not sufficient to yield the bars back in compression and bring to essentially zero strain upon returning of the wall to Δw of 0%. Over successive loading/unloading cycles with increasing wall drift, the residual (plastic) tensile strains in the energy-dissipating bars resulted in the complete uplift of the structure at the base joint, overcoming the downward restoring force. This undesirable behavior ultimately caused out-of-plane displacements of the wall base during unloading and buckling of the energy-dissipating bars in compression (Fig. 4), leading to the rapid deterioration and failure of the wall. Specimen EW accumulated significantly greater residual axial elongations compared with the hybrid walls, beginning from the drift cycles with Δw of ±0.27%, with a maximum uplift of 6.1 mm (0.24 in.) after the last cycle to Δw of ±1.15%. The large uplift of the emulative system is related to the lack of posttensioning steel, which resulted in an even smaller restoring force (provided only by the gravity load) than in specimen HW5. The axial elongations of specimen EW accumulated rapidly, leading to the failure of the wall due to excessive horizontal in-plane shear slip at the base joint and localized splitting of the base panel concrete (Fig. 10), with larger strength and stiffness degradation than in specimen HW5.3 <subhead 1> Shear design across horizontal joints <body> To prevent significant horizontal slip of the wall during loading up to Δwm, the shear friction capacity at the horizontal joints should be greater than the joint shear force demand. The joint shear forces should be calculated from the maximum wall base shear force Vwm corresponding to the probable base moment strength Mwm of the wall at Δwm (appendix). The nominal shear friction strength at the base joint can be determined as the assumed shear friction coefficient µss of 0.5 multiplied by the calculated compression force in the contact region at the wall toe. For the upper panel-to-panel joints, µss can be taken as 0.6, and the shear friction design method in section 11.6 of ACI 318-11 can be used by combining the shear friction strength provided by the axial force (from the posttensioning steel and the gravity load) with the shear friction strength from the yielding of the mild steel bars crossing the upper joint at both ends (Fig. 1). As recommended in section 5.5.3 of ACI ITG-5.2, a smaller value of µss is assumed at the base joint due to the deterioration of the concrete and grout pad at Δwm, leading to a smaller slip strength compared with the upper joints. The specimens tested as part of this project were designed using this approach. The horizontal shear slip at the upper panel-to-panel joint of the walls was negligible. Figure 9 shows the measured slip at the centerline of the base joint from the six specimens. For each specimen, only the peak of the third cycle in each drift series is plotted, except for the final drift series where the peaks from all cycles are plotted. For walls that satisfied the axial restoring force requirement described earlier in this paper (specimens HW1, HW2, HW3, and HW4), only a small amount of shear slip occurred along the base joint. In some instances, the measured base slip of these specimens exceeded the maximum allowable slip of 1.5 mm (0.06 in.) per section 7.1.4(3) of ACI ITG-5.1. However, this amount of base slip did not affect the performance of the walls in any undesirable way, indicating that the current allowable slip limit of 1.5 mm (0.06 in.) in ACI ITG-5.1 may be too conservative. A more reasonable allowable slip limit may be 3.8 mm (0.15 in.), which was the largest measured base slip among the specimens whose performance was not affected by slip (that is, specimens HW1, HW2, HW3, and HW4). Specimens HW5 and EW did not satisfy the axial restoring force requirement described previously in this paper, resulting in excessive in-plane slip at the base joint. Under load reversals with increasing in-plane slip, significant out-of-plane displacements also developed at the base of specimen HW5 (Fig. 4), resulting in the buckling of the energy-dissipating bars in compression (Fig. 4). Unlike specimen HW5, no significant out-of-plane displacements occurred in specimen EW. Instead, the concrete around the energy-dissipating bars deteriorated due to the shear force transfer from the bars to the surrounding concrete, ultimately causing failure through the splitting of the base panel concrete around the bars (Fig. 9). <subhead 1> Other wall panel reinforcement <body> The other wall panel reinforcement, not continuous across the horizontal joints, includes shear (that is, diagonaltension) reinforcement, edge reinforcement, reinforcement to control temperature and shrinkage cracks, and reinforcement to support panel lifting inserts. The base panel is expected to develop diagonal cracking; thus, distributed vertical and horizontal steel reinforcement should be designed according to the shear requirements in ACI 318-11, specifically the applicable requirements in sections 21.9.2 and 21.9.4. The specimens tested as part of this project satisfied these requirements and developed well-distributed hairline cracking in the base panel (Fig. 5). The cracks visible in the photographs were highlighted with markers during each test for enhanced viewing. The upper panels of the solid walls developed no cracking; thus, the distributed reinforcement in these panels can be reduced to satisfy the requirements in section 16.4.2 of ACI 318-11. In perforated walls, the reinforcement in both the base and upper panels includes additional mild steel bars for the panel chords around the perforations. The design of this panel chord reinforcement using a simplified finite element analysis of the wall through Δwm is described in Smith and Kurama.1 Section 21.9.6.4(e) of ACI 318-11 should be satisfied for the termination and development of the base panel horizontal distributed reinforcement within the confined boundary regions at the wall toes. The horizontal bars in specimens HW1, HW2, and HW3 were not developed inside the boundary regions (Fig. 6), reducing the effectiveness of these bars. During the larger drift cycles after concrete cover spalling at the wall toes, the ends of the horizontal distributed bars near the base were exposed, causing the bars to delaminate and, subsequently, the cover spalling to extend further along the length of the wall (Fig. 6). When the horizontal bars were terminated inside the confinement hoop cages (which was done in specimens HW4 and HW5), no bar delamination was observed and the concrete cover spalling did not extend beyond the wall toes (Fig. 6). In addition to the distributed steel, reinforcement should be placed around the entire exterior perimeter of each wall panel using mild steel bars placed parallel to each panel edge. As required by section 4.4.10 of ACI ITG-5.2, the mild steel reinforcement along the bottom edge of the base panel (appendix) should provide a nominal tensile strength of not less than 87.6 kN/m (6 kip/ft) along the length of the panel. The objective of this reinforcement is to control vertical concrete cracking initiating from the bottom of the base panel near the tip of the gap (that is, near the neutral axis).19 Thus, the bars should be placed as close to the bottom of the panel as practically possible while also satisfying the ACI 318-11 concrete cover and spacing requirements. The bottom edge bars in the base panel should be anchored using a standard 90° hook at the wall toes with sufficient development length from the critical location at the neutral axis (that is, at a distance cm from each end of the wall). The bottom edge reinforcement in the wall specimens tested as part of this project was designed using this approach. Strain gauges placed on the bars near the critical location indicated strains reaching approximately 0.85εsy at Δwm, supporting the design requirement. <subhead 1> Flexural design of upper joints <body> The energy-dissipating bars crossing the base joint do not continue into the upper panel-to-panel joints (Fig. 1), resulting in a significant reduction in the lateral strength of the wall at these locations. The philosophy behind the flexural design of the upper panel-to-panel joints is to prevent significant gap opening and nonlinear material behavior during lateral displacements up to the maximum-level wall drift Δwm. Except for the base joint, where the wall is designed to rotate about the foundation, the structure should behave essentially as a rigid body through Δwm. Thus, the design of the upper panel-to-panel joints is conducted for the maximum joint moment demands corresponding to the probable base moment strength Mwm of the wall.1 To prevent significant gap opening at the upper panel-to-panel joints, mild steel reinforcement crossing these joints should be designed at the panel ends and placed in a symmetrical layout (Fig. 1). As shown in the appendix, the design of this reinforcement is based on the principles of equilibrium, linear material models, and a linear strain distribution (that is, plane sections remain plane assumption). The design requires that the tension steel strain be limited to εsy to limit gap opening and the maximum concrete compressive stress be limited to 0.5 f c' to keep the concrete linear elastic (where f c' is the compression strength of the unconfined panel concrete). These material limits were used in the design of the wall specimens tested as part of this project with no undesirable behavior developing in the upper joints.2-4 To prevent strain concentrations in the upper panel-to-panel joint steel, a short prescribed length of the bars (approximately 102 to 152 mm [4 to 6 in.] long) should be unbonded (wrapped in a plastic sleeve) at each joint. <subhead 1> Summary and Conclusion <body> This paper describes the key aspects of a performance-based design approach, including a design example, for special unbonded posttensioned hybrid precast concrete shear walls in seismic regions. The proposed design guidelines were developed and validated using the measured behaviors from six 0.4-scale multi-panel wall test specimens (four solid and two perforated walls) subjected to service-level gravity loads combined with reversedcyclic lateral loading. It is concluded that hybrid precast walls can satisfy all requirements for special reinforced concrete shear walls in high seismic regions with improved performance; however, critical design, detailing, and analysis considerations need to be undertaken to prevent undesirable failure mechanisms such as anchorage failure of the energy-dissipating bars, permanent uplift and excessive slip of the wall at the base joint, and premature crushing of the confined concrete at the wall toes. Ultimately, the recommendations in this paper are aimed to allow practicing engineers and precast concrete producers to design ACI-compliant special hybrid shear walls with predictable, reliable, and improved seismic performance. <subhead 1> Acknowledgments <body> This research was funded by the Charles Pankow Foundation and PCI. Additional technical and financial support was provided by High Concrete Group LLC, the Consulting Engineers Group Inc., and the University of Notre Dame. The authors acknowledge the support of the PCI Research and Development Committee, the PCI Central Region, and the members of the project advisory panel, who include Walt Korkosz (chair) of the Consulting Engineers Group Inc., Ken Baur of the High Concrete Group LLC, Neil Hawkins of the University of Illinois at Urbana-Champaign, S. K. Ghosh of S. K. Ghosh Associates Inc., and Dave Dieter of Mid-State Precast LP. The authors also thank Professor Michael McGinnis and undergraduate student Michael Lisk from the University of Texas at Tyler for monitoring the response of the test specimens using three-dimensional digital image correlation. Additional assistance and material donations were provided by Brian Balentine of Builders Iron Works Inc.; Jenny Bass of Essve Tech Inc., Randy Draginis and Norris Hayes of Hayes Industries Ltd.; Randy Ernest of Prestress Supply Inc.; Eric Fries of Contractors Materials Co.; Rod Fuss of Ambassador Steel Corp., Tony Johnson of the Concrete Reinforcing Steel Institute; Martin Koch of Southwestern Supplies Inc.; Chris Lagaden of Ecco Manufacturing; Stan Landry of Enerpac Precision SURE-LOCK; Richard Lutz of Summit Engineered Products; Kristen Murphy of Acument Global Technologies; Shane Whitacre of Dayton Superior Corp.; Phil Wiedemann of PCI Central Region; and Steve Yoshida of Sumiden Wire Products Corp. The authors thank these individuals, institutions, and companies for supporting the project. The opinions, findings, conclusions, and recommendations expressed in this paper are those of the authors and do not necessarily represent the views of the individuals and organizations acknowledged. <subhead 1> References <body> 1. Smith, B., and Y. Kurama, Y. 2012. Seismic Design Guidelines for Special Hybrid Precast Concrete Shear Walls. Research report NDSE-2012-02. Notre Dame, IN: University of Notre Dame. hybridwalls.nd.edu/Project/Downloads/Design_Guidelines.pdf. 2. Smith, B., Y. Kurama, and M. McGinnis. 2013. “Behavior of Precast Concrete Shear Walls for Seismic Regions: Comparison of Hybrid and Emulative Specimens.” Journal of Structural Engineering 139 (11): 1917– 1927. ascelibrary.org/doi/abs/10.1061/%28ASCE%29ST.1943-541X.0000755. 3. Smith, B., Y. Kurama, and M. McGinnis. 2012. Hybrid Precast Wall Systems for Seismic Regions. Research report NDSE-2012-01. Notre Dame, IN: University of Notre Dame. hybridwalls.nd.edu/Project/Downloads/Final_Report.pdf. 4. Smith, B., Y. Kurama, and M. McGinnis. 2011. “Design and Measured Behavior of a Hybrid Precast Concrete Wall Specimen for Seismic Regions.” Journal of Structural Engineering 137 (10): 1052–1062. 5. ACI (American Concrete Institute) Committee 318. 2011. Building Code Requirements for Structural Concrete (ACI 318-11) and Commentary (ACI 318R-11). Farmington Hills, MI: ACI. 6. ACI Innovation Task Group 5. 2007. Acceptance Criteria for Special Unbonded Post-Tensioned Precast Structural Walls Based on Validation Testing and Commentary. ACI ITG-5.1. Farmington Hills, MI: ACI. 7. ACI Innovation Task Group 5. 2009. Requirements for Design of a Special Unbonded Post-Tensioned Precast Shear Wall Satisfying ACI ITG-5.1 (ACI ITG-5.2-9) and Commentary. Farmington Hills, MI: ACI. 8. ASCE (American Society of Civil Engineers) and SEI (Structural Engineering InstituteSEI). 2010. Minimum Design Loads for Buildings and Other Structures. ASCE/SEI 7-10. Reston, VA: ASCE and SEI. 9. Farrow, K., and Y. Kurama. 2003. “SDOF Demand Index Relationships for Performance-Based Seismic Design.” Earthquake Spectra 19 (4): 799–838. 10. Seo, C., and R. Sause. 2005. “Ductility Demands on Self-Centering Systems under Earthquake Loading.” ACI Structural Journal 102 (2): 275–285. 11. Smith, B. 2013. “Design, Analysis, and Experimental Evaluation of Hybrid Precast Concrete Shear Walls for Seismic Regions.” Doctoral diss. Notre Dame, IN: University of Notre Dame. etd.nd.edu/ETDdb/theses/available/etd-12132012-112729/unrestricted/SmithBJ122012D.pdf. 12. Smith, B., and Y. Kurama. 2013. “Seismic Displacement Demands for Hybrid Precast Concrete Shear Walls.” In ASCE Structures Congress: Bridging your Passion with your Profession. Pittsburgh, PA, May 2-4. [Reston, Virginia]: [American Society of Civil Engineers]. 13. Kurama, Y. 2005. “Seismic Design of Partially Post-Tensioned Precast Concrete Walls.” PCI Journal 50 (4): 100–125. 14. ICBO (International Conference of Building Officials). 1997. Uniform Building Code, Volume 2: Structural Engineering Design Provisions. Whittier, CA: ICBO 15. ICC (International Code Council). 2010. AC133: Acceptance Criteria for Mechanical Connector Systems for Steel Reinforcing Bars. Whittier, CA: ICC Evaluation Services. 16. Walsh, K., and Y. Kurama. 2010. “Behavior of Unbonded Post-tensioning Monostrand Anchorage Systems under Monotonic Tensile Loading.” PCI Journal 55 (1): 97–117. 17. Walsh, K., and Y. Kurama. 2012. “Effects of Loading Conditions on the Behavior of Unbonded Post-tensioning Strand-Anchorage Systems.” PCI Journal 57 (1): 76–96. 18. Kurama, Y., B. Weldon, and Q. Shen. 2006. “Experimental Evaluation of Posttensioned Hybrid Coupled Wall Subassemblages.” Journal of Structural Engineering 132 (7): 1017–1029. 19. Allen, M., and Y. Kurama. 2002. “Design of Rectangular Openings in Precast Walls under Combined Vertical and Lateral Loads.” PCI Journal 47 (2): 58–83. <subhead 1> Notation <body> Agross = gross area of wall cross section (taken at section with perforations in case of perforated walls) Ap = total posttensioning steel area As = total energy-dissipating steel area Ash = effective shear area of wall cross section cm = neutral axis length at base joint at Δwm Cd = ASCE 7-10 deflection amplification factor ep = distance of tension-side and compression-side posttensioning tendons from wall centerline es = distance of tension-side and compression-side energy-dissipating bars from wall centerline f ' c = compressive strength of unconfined panel concrete fpd = posttensioning steel stress at Δwd fpi = initial stress of posttensioning steel after all short-term and long-term losses but before any lateral displacement of wall fp,loss = stress loss in compression-side posttensioning tendon at Δwm due to nonlinear material behavior associated with reversed-cyclic loading of wall to ± Δwm fpm = posttensioning steel stress at Δwm fpu = ultimate strength of posttensioning steel fsd = energy-dissipating steel stress at Δwd fsm = energy-dissipating steel stress at Δwm fsy = yield strength of energy-dissipating steel or other mild steel hp = plastic hinge height over which plastic curvature is assumed to be uniformly distributed at wall base Hw = wall height from top of foundation Ie = reduced linear-elastic effective moment of inertia of wall cross section Igross = moment of inertia of gross wall cross section (taken at section with perforations in case of perforated walls) lh = confined region length at wall toes (center-to-center of bars) lhoop = length of individual confinement hoop (center-to-center of bars) Lw = wall length Mwd = wall design base moment at Δwd Mwm = probable base moment strength of wall at Δwm Mwn = contribution of wall gravity axial force Nw to satisfy Mwd Mwp = contribution of posttensioning steel to satisfy Mwd Mws = contribution of energy-dissipating steel to satisfy Mwd Nw = factored design gravity axial force at wall base for design load combination being considered R = ASCE 7-10 response modification factor sbot = first confinement hoop distance from bottom of base panel to center of bar shoop = confinement hoop spacing (center-to-center of bars) Vb = wall base shear force Vwd = design wall base shear force corresponding to Mwd at Δwd Vwm = maximum wall base shear force corresponding to Mwm at Δwm whoop = width of individual confinement hoop (center-to-center of bars) αs = ACI ITG-5.2 coefficient to estimate additional energy-dissipating bar debonding that is expected to occur during reversed-cyclic lateral displacements of wall β = relative energy dissipation ratio per ACI ITG-5.16 βmin = minimum required relative energy dissipation ratio per ACI ITG-5.1 Δw = wall drift, defined as relative lateral displacement at wall top divided by height from top of foundation Δwc = validation-level wall drift based on ACI ITG-5.1 Δwd = design-level wall drift corresponding to design-basis earthquake Δwe = linear-elastic wall drift under Vwd Δwm = maximum-level wall drift corresponding to maximum-considered earthquake εsm = energy-dissipating steel strain at Δwm εsu = strain of energy-dissipating steel or other mild steel at fsu εsy = yield strain of energy-dissipating steel or other mild steel at fsy κd = energy-dissipating steel moment ratio, defining relative amounts of energy-dissipating resistance from Asfsd and restoring resistance from Apfpd and Nw at wall base µss = coefficient of shear friction at horizontal joints <subhead 1> About the authors <body> Brian J. Smith, PhD, PE, is an assistant teaching professor for the Department of Civil and Environmental Engineering and Earth Sciences at the University of Notre Dame in Notre Dame, Ind. Yahya C. Kurama, PhD, PE, is a professor for the Department of Civil and Environmental Engineering and Earth Sciences at the University of Notre Dame. <subhead 1> Abstract <body> This paper presents recommended seismic design and detailing guidelines for special unbonded posttensioned hybrid precast concrete shear walls, including a full-scale worked design example and supporting experimental evidence from six 0.4-scale test specimens. Hybrid precast walls use a combination of mild steel (Grade 400 [Grade 60]) and high-strength unbonded posttensioning steel for lateral resistance across horizontal joints. The mild steel bars are designed to yield in tension and compression, providing energy dissipation. The unbonded posttensioning steel provides self-centering capability to reduce the residual lateral displacements of the structure after a large earthquake. The proposed design guidelines are aimed to allow practicing engineers and precast concrete producers to design American Concrete Institute compliant hybrid shear walls with predictable and reliable seismic behavior. Ultimately, the results from this project support the U.S. code approval of the hybrid precast wall system as a special reinforced concrete shear wall for moderate and high seismic regions. <subhead 1> Keywords <body> Hybrid, joint, posttensioned, seismic, shear wall, testing. <policy note> Review policy This paper was reviewed in accordance with the Precast/Prestressed Concrete Institute’s peer-review process. <request for comments> Please address any reader comments to journal@pci.org or Precast/Prestressed Concrete Institute, c/o PCI Journal, 200 W. Adams St., Suite 2100, Chicago, IL 60606. <end of end of article info> <insert dingbat here> <FIGURE CAPTIONS> Figure 1. Elevation, exaggerated displaced position, and cross section of a hybrid wall. Figure 2. Idealized expected base shear force versus roof drift behavior. Note: Vwd = design wall base shear force at Δwd; Vwm = maximum wall base shear force at Δwm; Δwd = design-level wall drift corresponding to design-basis earthquake; Δwe = linear-elastic wall drift under design base shear force Vwd; Δwm = maximum-level wall drift corresponding to maximum considered earthquake. Figure 3. Energy dissipation ratio. Note: βmin = minimum required relative energy dissipation ratio per ACI ITG-5.1. Figure 4. Performance of specimen HW5. Note: 1 kN = 0.225 kip. <Note to production: put label on left part of figure> Vb-Δw behavior <Note to production: put label on middle part of figure> Out-of-plane displacement <Note to production: put label on right part of figure> Buckling of energy-dissipating bar Figure 5. Performance of base panel north toe. Note: 1 mm = 0.0394 in. <Note to production: put label on left part of figure> Specimen HW3 <Note to production: put label on right part of figure> Specimen HW4 Figure 6. Performance and detailing of confined concrete region. Note: lh = confined region length at wall toes (center-to-center of bars); lhoop = length of individual confinement hoop (center-to-center of bars); whoop = width of individual confinement hoop (center-to-center of bars). 10M = no. 3. 1 mm = 0.0394 in. <Note to production: put label on top left part of figure> Specimen HW1 <Note to production: put label on top middle part of figure> Specimen HW3 <Note to production: put label on top right part of figure> Specimen HW4 <Note to production: put label on bottom left part of figure> Hoop detailing <Note to production: put label on bottom right part of figure> Measured normalized neutral axis length Figure 7. Energy-dissipating bar wrapped length. <Note to production: put label on left part of figure> Specimen HW3 <Note to production: put label on right part of figure> Specimen EW Figure 8. Performance of specimen HW2. Note: 1 kN = 0.225 kip. <Note to production: put label on left part of figure> Energy-dissipating bar splice pullout <Note to production: put label on right part of figure> Vb-Δw behavior Figure 9. Restoring force. Note: fsm = energy-dissipating steel stress at Δwm; fsy = yield strength of energy-dissipating steel. εsm = energy-dissipating steel strain at Δwm; εsy = yield strain of energy-dissipating steel at fsy. 1 mm = 0.0394 in.; 1 MPa = 0.145 ksi. <Note to production: put label on left part of figure> Idealized energy-dissipating steel stress-strain relationship <Note to production: put label on right part of figure> Measured residual wall uplift Figure 10. Horizontal slip along base joint. Note: 1 mm = 0.0394 in. <Note to production: put label on left part of figure> Measured slip <Note to production: put label on right part of figure> Base panel damage in specimen EW <TABLES> Table 1. Selected specimen properties: Part 1 κd Posttensioning tendons Number of Speci men Design Actual 12.7 mm diameter strands HW1 0.50 0.53 3 HW2 0.50 0.53 3 HW3 0.50 0.50 3 HW4 0.50 0.54 3 HW5 0.85 0.90 2 EW † n/a n/a n/a Energy-dissipating bars fpi/fpu Wrapped Eccentricity Eccentricity Detail at Size length, εsm/εsu ep, mm es, mm base mm Total gravity load, kN† 0.54 0.54 0.54 0.54 0.54 ±229 ±229 ±279 ±279 ±140 n/a n/a 19M 19M 19M 19M 22M ±76, 152 ±76, 152 ±89, 191 ±89, 191 ±229, 864 ±788, 914, 22M 1041 254 254 381 381 254, 406 0.64 0.61 0.48 0.49 0.85 Spliced Spliced Continuous Continuous Continuous 361 361 361 361 534 559 0.73 Spliced 361 Total gravity load includes wall self-weight and externally applied gravity load. Note: ep = distance of tension-side and compression-side posttensioning tendons from wall centerline; es = distance of tension-side and compression-side energy-dissipating bars from wall centerline; fpi = average initial strand stress; fpu = design ultimate strength of strand = 1862 MPa; εsm = expected (design) energy-dissipating bar strain at maximum-level wall drift Δwm = 2.30%; εsu = strain at maximum (peak) strength of energy-dissipating bar from monotonic material testing; κd = energy-dissipating steel moment ratio, defining relative amounts of energydissipating resistance (from energy-dissipating steel) and restoring resistance (from posttensioning steel and gravity axial force) at wall base. 1 mm = 0.0394 in.; 1 kN = 0.225 kip; 1 MPa = 0.145 ksi. Table 2. Selected specimen properties: Part 2 Panel Specimen perforations, mm HW1 n/a HW2 n/a HW3 n/a HW4 356 × 508 HW5 456 × 508 EW n/a Confined region Distributed panel details reinforcement lh, lhoop, shoop, sbot, Type End Detail mm mm mm mm Welded Developed outside 406 406 83 114 Wire Fabric confinement region Developed outside 406 406 83 19 10M confinement region Developed outside 406 406 76 19 10M confinement region Developed inside 470 254 64 19 10M confinement region Developed inside 470 254 64 19 10M confinement region Developed outside 203 203 83 19 10M confinement region Note: lh = confined region length at wall toes (center-to-center of hoop bars); lhoop = length of individual confinement hoop (center-to-center of hoop bars); sbot = first hoop distance from bottom of base panel to center of hoop bar; shoop = confinement hoop spacing (center-to-center of hoop bars). 1 mm = 0.0394 in. <subhead 1> Appendix: Design example <body> The following example demonstrates the use of the proposed performance-based procedure to design the reinforcement for a solid hybrid precast concrete shear wall for a four-story parking structure in Los Angeles, Calif. Figure A1 shows the plan view of the building and the elevation view of the wall being designed. The geometric properties of the wall are described in more detail in Smith et al.4 For brevity, only the base panel reinforcement, the reinforcement crossing the base panel–to–foundation joint, and the design of one upper panel–to–panel joint located at the first floor (Fig. A1) are discussed in this design example. <subhead 2> Prototype wall properties <body> Wall dimensions: Wall height from top of foundation Hw = 13.7 m (45 ft) Wall length Lw = 6100 mm = 6.1 m (20 ft) Wall thickness tw = 380 mm (15 in.) = 0.38 m (1.2 ft) Seismic floor and roof weights at ith floor level Wi assigned to the prototype wall: Seismic weight at first floor W1 = 22,512 kN (5061 kip) Seismic weight at second floor W2 = 22,392 kN (5034 kip) Seismic weight at third floor W3 = 22,392 kN (5034 kip) Seismic weight at the roof (fourth floor) W4 = 21,356 kN (4807 kip) Fundamental period determined from modal analysis T1 = 0.39 sec Design spectral acceleration parameters for Los Angeles: 5% damped design spectral response acceleration parameter at short periods SDS = 1.00g 5% damped, spectral response acceleration parameter at a fundamental period T equal to one second SD1 = 0.64g where g = acceleration due to gravity <subhead 2> Material properties <body> Concrete: Compressive strength of unconfined panel concrete f c' = 41 MPa (5.9 ksi) Modulus of elasticity of unconfined concrete Ec = 4730 Shear modulus of concrete Gc = 12,900 MPa (1870 ksi) f c' = 4730 41 = 30,440 MPa (4415 ksi) ACI 318-11 factor relating depth of equivalent rectangular compressive stress block to neutral axis depth β1 = 0.75 ACI ITG-5.2 factor that relates stress of rectangular confined concrete compression block at toe of base panel to confined concrete strength γm = 0.92 ACI ITG-5.2 factor that relates cm to length am of equivalent rectangular compression stress block βm = 0.96 Posttensioning strands (Fig. A2): Ultimate strength of posttensioning steel fpu = 1862 MPa (270 ksi) Yield strength of posttensioning steel fpy = 1620 MPa (235 ksi) Initial stress of posttensioning steel fpi = 0.55fpu = 0.55(1862) = 1029 MPa (149 ksi) Modulus of elasticity of posttensioning steel Epi = 196,500 MPa (28,500 ksi) Energy-dissipating bars (Fig. A2): Ultimate strength of energy-dissipating steel fsu = 655 MPa (95.0 ksi) Yield strength of energy-dissipating or other mild steel fsy = 468 MPa (67.9 ksi) Strain of energy-dissipating or other mild steel at ultimate strength εsu = 0.12 mm/mm (0.12 in./in.) Modulus of elasticity of energy-dissipating or other mild steel Es = 200,000 MPa (29,000 ksi) ACI ITG-5.2 coefficient to estimate additional energy-dissipating bar debonding that is expected to occur during reversed-cyclic lateral displacements of wall αs = 2.0 Other mild steel bars: fsy = 414 MPa (60.0 ksi) εsu = 0.08 mm/mm (0.08 in./in.) Es = 200,000 MPa (29,000 ksi) <subhead 2> ASCE 7-10 design force demands <body> The design of the wall should be conducted under all applicable load combinations prescribed by ASCE 7-10. The response modification factor R is taken as 6.0 for considering the structure as a building frame system. For brevity, this design example is demonstrated under a single load combination, with the following factored design axial force, shear force, and bending moment at the wall base determined using the equivalent lateral force procedure in ASCE 7-10: Wall design base moment Mwd = 24,422 kN-m (18,013 kip-ft) Factored design gravity axial force at wall base Nw = 1076 kN (242 kip) Design wall base shear force Vwd = 2384 kN (536 kip) <subhead 2> Seismic drift demands <body> The linear-elastic drift of the wall can be determined from the linear-elastic flexural and shear deflections. The effective moment of inertia Ie is determined using Eq. (1). 3 1 1 I e 0.5I gross 0.5 tw Lw 0.5 0.38 6.1 = 3.6 m4 (417 ft4) 12 12 The resulting linear-elastic flexural deflection w,flex at the top of the wall assuming a cantilever (with stiffness EcIe) loaded under the equivalent lateral forces corresponding to Vwd can be found. w,flex = 0.012 m = 12 mm (0.47 in.) Similarly, the effective shear area Ash can be found using Eq. (2). Ash = (0.8)(0.38)(6.1) = 1.86 m2 = 1,860,000 mm2 (2880 in2) Then, the shear deflection at the top of the wall w,sh can be determined assuming a shear cantilever with EcAsh. w,sh = 0.001 m =1.3 mm (0.05 in.) The total linear-elastic drift of the wall Δwe can be determined from the sum of the flexural and shear deflections. w, flex w, sh 0.012 0.001 100 = 0.098% Hw 13.7 we The design-level wall drift Δwd can be determined from the linear-elastic drift using section 12.8.6 of ASCE 7-10 (with a recommended deflection amplification factor Cd of 5.0), and the maximum-level drift can be taken as 95% of the drift from Eq. (3).1 Δwd = CdΔwe = (5)(0.098%) = 0.49% 13.7 wm 0.95 0.8% 0.5% = 2.19 6.1 <subhead 2> Posttensioning and energy-dissipating steel areas <body> The posttensioning and energy-dissipating steel areas crossing the base joint can be determined using equilibrium, compatibility and kinematics, and design constitutive relationships. The reinforcement is placed in a symmetrical layout and outside of the confined concrete boundary (toe) regions of the base panel (Fig. A1). To simplify the presentation of the design process, the posttensioning and energy-dissipating steel areas on the compression side and tension side of the wall centerline are assumed to be lumped (Fig. A3). As a further simplification, the moment resistance due to the unequal steel stresses on the “compression-side” and “tension-side” of the wall are also ignored by using the average steel stresses. Then, the following equations can be used to determine the concrete compressive stress resultant C d and the neutral axis length cd at Δwd. Cd 0.85 f c'tw 1cd = (0.85)(41)(0.38)(0.75)cd M wd f L c Cd w 1 d 2 2 where f = axial-flexural capacity reduction factor from ACI 318-11 24, 422 6.1 0.75cd Cd 2 0.9 2 Solving this system of equations provides the following results: C d = 10,159 kN (2284 kip) cd = 1.01 m = 1010 mm (39.8 in.) = 0.17Lw The different stresses in the individual posttensioning tendons and energy-dissipating bars should be incorporated in the final design of the wall. Furthermore, the previously stated simplifications should not be made if the distance from the wall centerline to the extreme posttensioning tendon or energy-dissipating bar is greater than 0.125Lw (instead, estimated values for the steel stresses should be incorporated into the following iteration). If we assume that the design-level wall drift is achieved by the gap opening at the base joint through rigid body kinematics (Fig. A3), the posttensioning and energy-dissipating steel strains (and stresses) can be estimated using the neutral axis length cd. At this stage, the different strains and stresses in the lumped compression-side and tension-side posttensioning and energy-dissipating steel are calculated. The design process requires initial estimations of the lumped posttensioning and energy-dissipating steel eccentricities ep and es (that is, distance from the centerline of the wall). Posttensioning steel: ep = 200 mm (8 in.) (initial estimate) Unbonded length of posttensioning steel strand lpu = 15.2 m = 15,240 mm (600 in.) Elongation of posttensioning strand pd = Δwd(0.5Lw – cd ± ep) = (0.0049)[(0.5)(6100) – 1010 ± 200] = 9.0 or 11.0 mm (0.35 or 0.43 in.) for the lumped compression-side and tension-side tendons, respectively f pd f pi E pi pd l pu where fpd = posttensioning steel stress at Δwd f pd 1029 196,500 9.0 = 1145 MPa (166 ksi) or 15, 240 f pd 1029 196,500 11.0 = 1171 MPa (169 ksi) 15, 240 Both results are in the linear-elastic range (Fig. A2). Average posttensioning steel stress for compression-side and tension-side tendons fpd,avg = 1158 MPa (167.9 ksi) Energy-dissipating steel: es = 560 mm (22 in.) (initial estimate) Wrapped length of energy-dissipating steel lsw = 810 mm (32 in.) Elongation of energy-dissipating bar sd is calcuted: sd = Δwd(0.5Lw – cd ± es) = (0.0049)[(0.5)(6100) – 1010 ± 560] = 7.2 or 12.7 mm (0.28 or 0.50 in.) for the lumped compression-side and tension-side bars, respectively Strain in energy-dissipating steel εsd is calculated. sd sd lsw 7.2 12.7 = 0.0088 or 0.0156 mm/mm (0.0088 or 0.0156 in./in.) or 810 810 Both results are within range of the yield plateau (Fig. A2). fsd,avg = fsy = 468 MPa (68 ksi) for the compression-side and tension-side bars Assuming that the steel stresses on the compression side and tension side of the wall are equal to the above-average design-level stresses (that is, ignoring the differences in the compression-side and tension-side steel stresses), equilibrium along the base joint combined with Eq. (4) can be used to determine the required posttensioning and energy-dissipating steel areas. The designer must select a value for the energy-dissipating steel moment ratio κd within the recommended range (0.50 ≤ κd ≤ 0.80). For this design example, 0.50 is used for κd. C d = Asfsd,avg + Apfpd,avg + Nw where Ap = total posttensioning steel area (includes the tension-side and compression-side tendons) As = total energy-dissipating steel area 10,159 = As(468) + Ap(1158) + 1076 c L As f sd , avg w 1 d 2 2 Lw 1cd Ap f pd , avg N w 2 2 d M ws M wp M wn 0.5 As 468 Ap 1158 1076 Solving this system of equations gives the following results: Ap = 4970 mm2 (7.7 in.2) As = 7100 mm2 (11.0 in.2) For this example, the provided reinforcement crossing the base joint consists of thirty-six 15.2 mm (0.6 in.) diameter posttensioning strands with Ap of 5030 mm2 (7.8 in.2) and fourteen 25M (no. 8) energy-dissipating bars with As of 7140 mm2 (11.1 in.2). All of the subsequent design calculations should be made using these values of Ap and As. Seven energy-dissipating bars are placed at a center-to-center spacing of 75 mm (3.0 in.) on each side of the wall centerline, and the posttensioning strands on each side are placed in a single bundle with eighteen strands in the bundle (Fig. A4). The placement of the energy-dissipating bars and the posttensioning tendons satisfy the previous estimations of the steel eccentricities ep and es. This design step may need to be revised if the eccentricities or unbonded lengths of the posttensioning and energy-dissipating steel are modified later in the design. <subhead 2> Probable base moment strength initial calculation <body> Based on the initial estimations for the eccentricity and unbonded length of the compression-side and tension-side posttensioning and energy-dissipating steel, the material stresses at the maximum-level wall drift Δwm can be determined and used to calculate the probable base moment strength of wall Mwm. This process is iterative and requires an assumed neutral axis length cm at Δwm. As a starting point for the iteration cm is determined: cm = 0.9cd = (0.9)(1010) = 909 mm (35.9 in.) = 0.91 m (3.0 ft) Then, the posttensioning and energy-dissipating steel strains and stresses can be calculated. Posttensioning steel: Calculate elongation of posttensioning strand pm: pm = Δwm(0.5Lw – cm ± ep) = (0.0219)[(0.5)(6100) – 909 ± 200] = 42.4 or 51.3 mm (1.67 or 2.02 in.) for the compression-side and tension-side tendons, respectively Calculate posttensioning steel strain εpm: pm f pi Ep pm l pu 1029 42.4 or 51.3 = 0.0080 or 0.0086 mm/mm (0.0080 or 0.0086 in./in.) + 196,500 15, 240 Posttensioning steel stress fpm = 1503 or 1586 MPa (218 or 230 ksi) This value is based on the assumed posttensioning strand stress-strain relationship in Fig. A2 Average posttensioning steel stress for compression-side and tension-side tendons fpm,avg = 1544 MPa (224 ksi) Energy-dissipating steel: Calculate elongation of energy-dissipating bar sm: sm = Δwm(0.5Lw – cm ± es) = (0.0219)[(0.5)(6100) – 909 ± 560] = 34.5 or 58.9 mm (1.36 or 2.32 in.) for the lumped compression-side and tension-side energy-dissipating bars, respectively Calculate maximum strain in energy-dissipating steel εsm: sm sm lsw s d s 34.5 or 58.9 = 0.040 or 0.068 mm/mm (0.040 or 0.068 in./in.) 810 2.0 25.4 where ds = diameter of energy-dissipating bar Energy-dissipating steel stress fsm = 586 or 634 MPa (85 or 92 ksi) This value is based on the assumed energy-dissipating steel stress-strain relationship in Fig. A2 Average energy-dissipating steel stress for compression-side and tension-side bars fsm,avg = 607 MPa (88 ksi) Using equilibrium along the base joint at Δwm (Fig. A5 has final design values at the end of the iteration), the concrete compressive stress resultant Cm and the probable base moment strength of the wall Mwm can be determined: Cm = Asfsm,avg + Apfpm,avg + Nw 7140 5030 = 6071000 15441000 1076 = 13,215 kN (2971 kip) 6 6 10 10 6.1 0.96 0.91 c L M wm Cm w m m 13, 215 = 34,436 kN-m (25,439 kip-ft) 2 2 2 2 In the previous moment calculation, the differences in the compression-side and tension-side steel stresses are again ignored by using the estimated average steel stresses. At Δwm, the confined concrete in compression is represented using a modified rectangular stress block with the concrete strength parameter γm of 0.92 and the stress block length parameter βm of 0.96, as given in section 5.6.3.8(a) of ACI ITG-5.2. <subhead 2> Confinement reinforcement at wall toes <body> Confinement reinforcement is required at the ends of the base panel and can be designed using sections 5.6.3.5 through 5.6.3.9 of ACI ITG-5.2.9 The confined concrete is designed for a maximum concrete strain εcm at Δwm, determined as follows: Plastic hinge height hp = 0.06Hw = (0.06)(13.7) = 0.822 m (2.7 ft) = 822 mm (32.4 in.) Plastic curvature at wall base wm wm 0.0219 = 0.0000266/mm hp 822 εcm = wmcm = (0.0000266)(909) = 0.0242 mm/mm (0.0242 in./in.) The required confined concrete strength can then be calculated using Table 5.2 of ACI ITG-5.2, which relates the lateral confining stress provided by the transverse reinforcement f l ' , the confined concrete compressive strength f cc' , and the unconfined concrete compressive strength f c' . Assuming that the strain at peak stress of the confinement hoop steel εsu is 0.08 mm/mm (0.08 in./in.): fl ' f cc' cm 0.004 4.6 su f' 0.0242 0.004 4.6 0.08 l' f cc fl ' f ' 0.055 cc Then, using Table 5.2 of ACI ITG-5.2: f cc' f ' 1.48 c f cc' = 1.48 f c' = (1.48)(41) = 61 MPa (8.9 ksi) Using axial force equilibrium at Δwm, the initial assumption of cm equal to 0.9cd should be checked and, if necessary, the previous steps to determine the probable base moment strength and required confined concrete strength be iterated until the calculated cm from equilibrium at the wall base is sufficiently close to the assumed cm. For this design example, a clear cover of 25 mm (1.0 in.) is assumed at the wall toes, resulting in a confined concrete width wh of 330 mm (13.0 in.). Therefore: 7140 5030 Cm = Asfsm,avg + Apfpm,avg + Nw = 6071000 15441000 1076 = 13,215 kN (2971 kip) 6 6 10 10 Cm = γm f cc' whβmcm = 0.92 61 3310 0.96 cm 10 1000 = 13,215 kN (2971 kip) cm = 742 mm (29.2 in.) < 0.9cd = 909 mm (35.9 in.) Based on the difference between the initial assumption of cm and the calculated cm from equilibrium, cm and f cc' should be recalculated. Through this iteration, which is not shown here, the following final design values can be determined: f l ' = 2.85 MPa (0.414 ksi) f cc' = 58.6 MPa (8.5 ksi) cm = 818 mm (32.2 in.) = 0.13Lw These parameters are then used to determine the confinement hoop layout and spacing according to sections 5.6.3.5 through 5.6.3.9 in ACI ITG-5.2. For rectangular confinement hoops with fsy of 414 MPa (60 ksi), the confinement reinforcement ratio h can be calculated using section 5.6.3.8(b). h fl ' 0.414 = 0.02 0.35 f sy 0.35 60 The length of the confined concrete region at each wall end lh is assumed to be equal to 840 mm (33 in.), thus satisfying the following design requirement: lh > 0.95cm 840 mm > (0.95)(818) = 777 mm (33 in. > 30.6 in.) → OK Then, two overlapping 13M (no. 4) hoops are selected for each wall end (Fig. A3) Hoop bar area ahoop = 130 mm2 (0.20 in.2) Hoop width whoop = 330 mm (13 in.) Individual hoop length lhoop = 560 mm (22 in.) This satisfies the following requirement: lhoop > 2.50 whoop 560 = 1.70 > 2.50 → OK 330 The confinement hoop spacing shoop can be determined using the following equation based on the specific geometry of the selected hoop layout. h ahoop 4lhoop 4whoop 0.02 lh wh shoop 130 4 560 4 330 840 330 shoop Solving this equation for the confinement hoop spacing yields: shoop = 83 mm (3.3 in.) For this example, the provided confinement hoop spacing is 82.5 mm (3.25 in.). The confinement hoops should extend vertically over a height of the base panel not less than the plastic hinge height hp of 823 mm. (32.4 in.). Thus, placing 11 layers of hoops at each wall end while providing a distance from the first confinement hoop to the bottom of the base panel sbot of 38 mm (1.5 in.), the total height of the confinement region would be 863 mm (34.0 in.). For ease of handling and placement of the hoops at each wall end, a confinement cage should be constructed by tying the corners of the hoops to 13M (no. 4) longitudinal bars. <subhead 2> Probable base moment strength (final calculation) <body> The probable base moment strength is recalculated using the finalized neutral axis length cm, and confined concrete strength f cc' , resulting in the following: pm = 44.5 to 53.3 mm (1.75 or 2.10 in.) for the compression-side and tension-side tendons, respectively εpm = 0.0082 to 0.0087 mm/mm (0.0082 to 0.0087 in./in.) fpm,avg = 1570 MPa (228 ksi) (average for the compression-side and tension-side tendons) sm = 36.6 to 61.0 mm (1.44 to 2.40 in.) for the lumped compression-side and tension-side energy-dissipating bars, respectively εsm = 0.042 to 0.071 mm/mm (0.042 to 0.071 in./in.) fsm,avg = 621 MPa (90 ksi) Cm = 13,403 kN (3013 kip) Mwm = 35,600 kN-m (26,252 kip-ft) The corresponding wall base shear demand can be calculated as: V 2.384 = 3483 kN (783 kip) Vwm M wm wd 35, 600 24, 422 M wd <subhead 2> Maximum energy-dissipating bar strain and wrapped length The calculated neutral axis length cm, can be used to check the initial assumption for the energy-dissipating bar wrapped length (used to estimate the energy-dissipating steel strains) based on the maximum energy-dissipating bar strain. The maximum energy-dissipating bar strain will occur in the extreme bar at the tension side of the wall (located a distance from the wall centerline ese of 790 mm [31 in.]); therefore, the other bar strains do not need to be checked. The designer must select an allowable energy-dissipating bar strain εsa based on the recommended limits of 0.5εsu ≤ εsa ≤ 0.85εsu. For this design example, it is assumed that: εsa = 0.65εsu = 0.078 mm/mm (0.078 in./in.) Then, based on Fig. A5: sm = Δwm(0.5Lw – cm + ese) = (0.0219)[(0.5)(610) – 81.8 ± 79] = 66.2 mm (2.60 in.) sm sm lsw s d s 6.62 ≤ εsa = 0.078 mm/mm (0.078 in./in.) lsw 2.0 2.54 Therefore, lsw ≥ 798 mm (31.4 in.) The initial assumption for the energy-dissipating bar wrapped length lsw of 810 mm (32.0 in.) is close to and satisfies the listed limit. No further iteration is required. <subhead 2> Maximum posttensioning steel strain <body> Similarly, the calculated neutral axis length 𝑐𝑚 can be used to determine the maximum posttensioning steel strain and ensure that it does not exceed the recommended limit of 0.01 mm/mm (0.01 in./in.). Using the previously defined equations, the strain in the tension-side posttensioning tendon can be determined as: pm = Δwm(0.5Lw – cm + ep) = (0.0219)[(0.5)(610) – 81.8 + 20] = 53.3 mm (2.10 in.) Therefore, pm f pi E pi pm l pu 1029 5.33 = 0.0087 mm/mm (0.0087 in./in.) < 0.01 mm/mm (0.01 in./in.) → OK 196,500 1524 <subhead 2> Posttensioning stress losses <body> A small posttensioning stress loss will occur as the wall is displaced to ±Δwm under fully reversed and repeated cyclic loading and the strand strains exceed the limit of proportionality (that is, as the stress-strain behavior of the strand becomes nonlinear). Figure A6 displays the stress-strain relationship of the tension-side and compressionside posttensioning tendons as the wall is cyclically displaced to ±Δwm. Let the stresses in the tension-side and compression-side tendons upon first loading to +Δwm be fpm,1 and fpm,2, respectively. Upon reversed loading of the wall to -Δwm and reloading back to +Δwm, the tension-side tendon will return to the same point on the stress-strain relationship (that is, the stress and strain of the tendon will equal fpm,1 and εpm,1, respectively) and the tendon will not experience any stress loss due to the nonlinear material behavior. The compression-side tendon will also return to the same strain (that is, εpm,2), but it will incur stress losses because of the greater strain (equal to εpm,1) that it experiences in the reverse loading direction (that is, when the wall is subjected to -Δwm). The stress loss fp,loss when the wall is displaced back to +Δwm can be determined as follows: pm = Δwm(0.5Lw – cm + ep) = (0.0219)[(0.5)(610) – 81.8 ± 20] = 44.5 or 53.3 mm (1.75 or 2.10 in.) for the compression-side and tension-side tendons, respectively pm f pi Ep Therefore, pm l pu 1029 4.45 or 5.33 = 0.0082 or 0.0087 mm/mm (0.0082 or 0.0087 in./in.) + 196,500 1524 εpm,1 = 0.0087 mm/mm (0.0087 in./in.) εpm,2 = 0.0082 mm/mm (0.0082 in./in.) fpm,1 = 1606 MPa (233 ksi) (based on the posttensioning strand stress-strain relationship in Fig. A2) fpm,2 = 1534 MPa (222 ksi) (based on Fig. A2) fpm,avg = 1570 MPa (228 ksi) (average for the compression-side and tension-side tendons) Stress in compression-side posttensioning tendon after repeated loading fpm,2r: fpm,2r = fpm,1 – Epi(εpm,1 – εpm,2) = 1606 – 196,500(0.0087 – 0.0082) = 1509 MPa (219 ksi) fp,loss = fpm,2 – fpm,2r = 1534 – 1509 = 25 MPa (3.6 ksi) <subhead 2> Wall restoring force <body> Eq. (5) must be satisfied in order to provide a sufficient restoring force for the wall. For design purposes, a capacity reduction factor against loss of restoring r of 0.90 is used to prevent the loss of self-centering. Therefore, the design equation is r Ap f pm,avg 0.5 f p ,loss N w As f sm,avg f sy 50.3100 1570 0.5 25 71.4100 621 448 0.9 1076 1000 1000 8019 kN (1803 kip) > 7615 kN (1712 kip) → OK <subhead 2> Shear design of base joint <body> To prevent significant horizontal slip of the wall during loading up to the maximum-level drift Δwm, the shear friction capacity at the horizontal joints Vss should be greater than the maximum joint shear force Vjm: sVss > Vjm where s = capacity reduction factor for shear design At the base joint: Vjm = Vwm = 3483 kN (783 kip) where Vwm = maximum wall base shear force corresponding to Mwm at Δwm Then, using the compressive force at the wall base to determine the shear friction capacity Vss, along with a capacity reduction factor against shear slip failure s of 0.75 and a shear friction coefficient of µss of 0.5 as recommended by section 5.5.3 of ACI ITG-5.2, the design equation can be written as: sµss ( Cm – 0.5Apfp,loss) > Vwm 50.3100 25 1000 0.75 0.5 13, 403 0.5 > 3483 kN (783 kip) 5003 kN (1125 kip) > 3483 kN (783 kip) → OK <subhead 2> Base panel shear reinforcement <body> The base panel is expected to develop diagonal cracking; thus, distributed vertical and horizontal steel reinforcement should be designed based on the shear requirements in sections 21.9.2 and 21.9.4 of ACI 318-11. This distributed reinforcement should be designed to ensure that the shear capacity of the base panel Vn is greater than the maximum wall base shear force Vwm: sVn > Vwm As specified in section 21.9.4 of ACI 318-11, the shear capacity is a function of the gross area of concrete in the direction of the shear force Lwtw, the shear steel reinforcement ratio t, and the material properties of the concrete and steel. For design purposes, a capacity reduction factor s of 0.75 is used. Then, the limiting design equation can be written as: s Lwtw c fc' t f sy Vwm αc = coefficient defining relative contribution of concrete strength to nominal wall shear strength = factor reflecting reduced mechanical properties of lightweight-aggregate concrete relative to normalweight aggregate concrete t = shear steel reinforcement ratio within panel 0.171.0 41 t 414 = 3483 kN (783 kip) 1000 0.75 61010 3810 Solving the design equation for the reinforcement ratio yields t = 0.0023 As required by section 21.9.2.1 of ACI 318-11, the minimum reinforcement ratiot shall not be less than 0.0025. Therefore, using this minimum requirement, the required shear reinforcement steel area At can be determined: At = ttw = (0.0025)(38)(100) = 950 mm2/m (0.45 in.2/ft) For this example, the provided base panel shear reinforcement consists of 13M (no. 4) mild steel bars spaced 250 mm (10 in.) on center and located at each panel face in both the horizontal and vertical directions provided At equals 1020 mm2/m (0.48 in.2/ft). Section 21.9.6.4(e) of ACI 318-11 should be satisfied for the termination and development of the base panel horizontal distributed reinforcement within the confined boundary regions at the wall toes. <subhead 2> Base panel bottom edge reinforcement <body> The bottom edge reinforcement in the base panel controls the initiation of vertical concrete cracking near the tip of the gap (that is, at the neutral axis) at the base panel–to–foundation joint. As required by section 4.4.10 of ACI ITG5.2, the total mild steel area along bottom edge of base panel Aedge should provide a nominal tensile strength Tedge of not less than 87.6 kN/m (6 kip/ft) along the length of the panel. Therefore, the required area of reinforcement along the bottom edge of the base panel can be calculated: Tedge = 87.6Lw = (87.6)(6.1) = 534 kN (120 kip) Aedge Tedge f sy 5341000 414100 = 1290 mm2 (2.0 in.2) For this example, the provided reinforcement at the bottom of the base panel consists of two 29M (no. 9) mild steel bars with an Aedge equal to 1290 mm2 [2.0 in.2]). <subhead 2> Flexural design of upper panel–to–panel joints <body> The philosophy behind the flexural design of the upper panel–to–panel joints is to prevent significant gap opening and nonlinear behavior of the material during lateral displacements up to the maximum-level drift Δwm. For this example, the design gravity axial force at upper panel–to–panel joint Nw,u and upper panel–to–panel joint bending moment Mwm,u at the first-floor panel-to-panel joint (Fig. A1) can be calculated: Nw,u = 934 kN (210 kip) Mwm,u = 22,925 kN-m (16,905 kip-ft) (calculated from the probable base moment strength Mwm at Δwm) To prevent significant gap opening at the upper panel–to–panel joints, mild steel reinforcement should be designed ' at the panel ends (where As,u and As ,u represent the tension and compression mild steel areas crossing the upper ' panel–to–panel joint, respectively) and placed in a symmetrical layout (where d and d represent the centroid location of the tension and compression mild steel bars crossing the upper panel–to–panel joint, respectively). For design purposes, a capacity reduction factor for axial-flexural design of upper panel–to–panel joints f,u of 0.90 is used. Using equilibrium, compatibility/kinematics, and design constitutive relationships at the upper panel–to–panel joints when Δwm is reached (Fig. A7 for the first floor joint), the following equations can be used to solve for the concrete compressive stress resultant Cm ,u , neutral axis length cm,u, stresses in the tension and compression mild steel bars fs,u ' and f s ,u , respectively, and maximum concrete compression stress fc,u: Cm ,u = As,ufs,u – As' ,u f s',u + Ap(fpm,avg – 0.5fp,loss) + Nw,u Cm ,u = 0.5fc,utwcm,u cm,u L L L Cm,u w As ,u f s ,u w d As' ,u f s',u w d ' 2 2 3 2 M wm,u f ,u Lw cm,u d E f s ,u s f c ,u cm,u Ec cm,u d ' E f s',u s fc ,u Ec cm,u It is required that the tension steel yield strain be limited to εsy to limit gap opening and the maximum concrete compressive stress be limited to 0.5 f c' to keep the concrete linear elastic. The designer would select an initial steel area and location. For this design example, the following steel area and location are used: As,u = As' ,u = 2040 mm2 (3.16 in.2) (four 25M [no. 8] bars at a center-to-center spacing of 50 mm (2.0 in.) located at each end) d = d ' = 150 mm (6 in.) Then Cm,u 20.4100 f s,u 20.4100 f s',u 50.3 100 1570 0.5 25 1000 Cm,u 1000 1000 934 0.5 fc ,u 3810 cm,u 10 1000 22, 9251000 C 0.9 20.4100 f s',u 610 20.4100 f s ,u 610 610 cm,u 15 10 15 10 10 m ,u 2 3 1000 1000 2 2 610 cm,u 15 200, 000 f c ,u 30, 440 cm , u f s ,u cm ,u 15 200, 000 f c ,u 30, 440 cm , u f s ,u ' By solving this system of equations, the steel and concrete strain limitations are satisfied: Cm ,u = 9221 kN (2073 kip) cm,u = 1870 mm (73.6 in.) = 0.31Lw ' fc,u = 19.3 MPa (2.8 ksi) < 0.5 f c → OK fs,u = 372 MPa (54 ksi) tension < fsy → OK ' f s ,u = 159 MPa (23 ksi) compression < fsy → OK Although not shown in this design example, this design step would be repeated for each upper panel–to–panel joint in the wall. To prevent strain concentrations in the upper panel–to–panel joint steel, a short prescribed 150 mm (6 in.) length of the bars should be unbonded at each joint. <subhead 2> Shear design of upper panel–to–panel joints <body> For this design example, the design shear force at the first-floor joint can be calculated from the probable base moment strength of wall Mwm. Vjm = Vwm,u = 3109 kN (699.0 kip) where Vwm,u = upper panel–to–panel joint shear force corresponding to Vwm At the upper panel–to–panel joints, a larger shear friction coefficient of µss equal to 0.6 can be used, and the design equation can be written as sµss[(As,u + As ,u )fsy + Ap(fpm,avg - 0.5fp,loss) + Nw,u] > Vwm,u ' Then, for the first floor joint: 20.4100 20.4100 414 50.3100 1570 0.5 25 934 > 3109 kN (699.0 kip) 0.75 0.6 1000 1000 4706 kN (1058 kip) > 3109 kN (699.0 kip) → OK While not shown in this example, this design step would be repeated for each upper panel–to–panel joint in the wall. <subhead 2> Final wall reinforcement details <body> Figure A4 shows the reinforcement details of the wall based on the above design calculations. Final design checks should be conducted (not shown here) considering the individual strains and stresses in the posttensioning tendons and energy-dissipating steel bars. <subhead 1> Notation <body> ad = length of ACI 318-11 equivalent rectangular concrete compression stress block at Δwd ahoop = confinement hoop steel bar area am = length of ACI ITG-5.2 equivalent rectangular concrete compression stress block at Δwm Aedge = total mild steel area along bottom edge of base panel Ap = total posttensioning steel area As = total energy-dissipating steel area Ash = effective shear area of wall cross section As,u = total tension mild steel area crossing upper panel–to–panel joint at panel end As' ,u = total compression mild steel area crossing upper panel–to–panel joint at panel end At = shear reinforcement area within panel cd = neutral axis length at base joint at Δwd cm = neutral axis length at base joint at Δwm cm,u = neutral axis length at upper panel–to–panel joint at Δwm Cd = ASCE 7-10 deflection amplification factor Cd = concrete compressive stress resultant at base joint at Δwd Cm = concrete compressive stress resultant at base joint at Δwm Cm ,u = concrete compressive stress resultant at upper panel–to–panel joint at Δwm d = centroid location (from panel end) of tension mild steel bars crossing upper panel–to–panel joint d' = centroid location (from panel end) of compression mild steel bars crossing upper panel–to–panel joint ds = diameter of energy-dissipating bar ep = distance of tension-side and compression-side posttensioning tendons from wall centerline es = distance of tension-side and compression-side energy-dissipating bars from wall centerline ese = distance of extreme energy-dissipating bar from wall centerline Ec = modulus of elasticity of unconfined concrete Epi = modulus of elasticity of posttensioning steel Es = modulus of elasticity of energy-dissipating (or other mild) steel f c' = compressive strength of unconfined panel concrete f cc' = compressive strength of confined concrete at toes of base panel fc,u fl ' = maximum concrete compression stress at upper panel–to–panel joint at Δwm = effective lateral confining stress provided by confinement reinforcement at toes of base panel fpd = posttensioning steel stress at Δwd fpd,avg = average posttensioning steel stress for compression-side and tension-side tendons at Δwd fpi = initial stress of posttensioning steel after all short-term and long-term losses but before any lateral displacement of wall fp,loss = stress loss in compression-side posttensioning tendon (at Δw = Δwm) due to nonlinear material = behavior associated with reversed-cyclic loading of wall to ± Δwm fpm = posttensioning steel stress at Δwm fpm,1 = stress in tension-side posttensioning tendon at initial and repeated loading to +Δwm fpm,2 = stress in compression-side posttensioning tendon at initial loading to +Δwm fpm,2r = stress in compression-side posttensioning tendon after repeated loading to +Δwm fpm,avg = average posttensioning steel stress for compression-side and tension-side tendons at Δwm fpu = ultimate strength of posttensioning steel fpy = yield strength of posttensioning steel determined at limit of proportionality point on strand stress-strain relationship fsd,avg = average energy-dissipating steel stress for compression-side and tension-side bars at Δwd fsm = energy-dissipating steel stress at Δwm fsm,avg = average energy-dissipating steel stress for compression-side and tension-side bars at Δwm fsu = ultimate strength of energy-dissipating or other mild steel fs,u = stress in tension mild steel crossing upper panel–to–panel joint at Δwm f s',u = stress in compression mild steel crossing upper panel–to–panel joint at Δwm fsy = yield strength of energy-dissipating or other mild steel g = acceleration due to gravity Gc = shear modulus of concrete hp = plastic hinge height over which plastic curvature is assumed to be uniformly distributed at wall base Hw = wall height from top of foundation Ie = reduced linear-elastic effective moment of inertia of wall cross section Igross = moment of inertia of gross wall cross section taken at section with perforations in case of perforated walls lh = confined region length at wall toes (center-to-center of bars) lhoop = length of individual confinement hoop (center-to-center of bars) lpu = unbonded length of posttensioning steel strand lsw = wrapped length of energy-dissipating steel Lw = wall length Mwd = wall design base moment at Δwd Mwm = probable base moment strength of wall at Δwm Mwm,u = upper panel–to–panel joint moment corresponding to Mwm at Δwm Nw = factored design gravity axial force at wall base for design load combination being considered Nw,u = design gravity axial force at upper panel–to–panel joint R = ASCE 7-10 response modification factor sbot = first confinement hoop distance from bottom of base panel (to center of bar) shoop = confinement hoop spacing (center-to-center of bars) SD1 = ASCE 7-10 5% damped, spectral response acceleration parameter at a fundamental period T equal to one second SDS = ASCE 7-10 5% damped design spectral response acceleration parameter at short periods tw = wall thickness T = fundamental period T1 = fundamental period determined from modal analysis Tedge = total tensile force for design of base panel bottom edge reinforcement Vjm = maximum shear force at horizontal joint at Δwm Vn = nominal shear strength of base panel Vss = shear friction capacity at horizontal joint Vwd = design wall base shear force corresponding to Mwd at Δwd Vwm = maximum wall base shear force corresponding to Mwm at Δwm Vwm,u = upper panel–to–panel joint shear force corresponding to Vwm wh = width of confined concrete region at wall toes whoop = width of individual confinement hoop (center-to-center of bars) W1 = seismic weight at first floor W2 = seismic weight at second floor W3 = seismic weight at third floor W4 = seismic weight at roof (fourth floor) Wi = seismic weight at ith floor αc = ACI 318-11 coefficient defining relative contribution of concrete strength to nominal wall shear strength αs = ACI ITG-5.2 coefficient to estimate additional energy-dissipating bar debonding that is expected to occur during reversed-cyclic lateral displacements of wall β1 = ACI 318-11 factor that relates cd to length ad of equivalent rectangular concrete compression stress block at Δwd βm = ACI ITG-5.2 factor that relates cm to length am of equivalent rectangular concrete compression stress block at Δwm βmin = minimum required relative energy dissipation ratio per ACI ITG-5.1 γm = ACI ITG-5.2 factor that relates stress of rectangular confined concrete compression block at toe of base panel to confined concrete strength f cc' pd = elongation of posttensioning strand at Δwd pm = elongation of posttensioning strand at Δwm sd = elongation of energy-dissipating bar at Δwd sm = elongation of energy-dissipating bar at Δwm w,flex = wall displacement due to flexural deformations in linear-elastic effective stiffness model w,sh = wall displacement due to shear deformations in linear-elastic effective stiffness model Δwd = design-level wall drift corresponding to design-basis earthquake Δwe = linear-elastic wall drift calculated using linear-elastic effective stiffness model Δwm = maximum-level wall drift corresponding to maximum-considered earthquake εcm = maximum concrete compression strain at base joint at Δwm εpm = strain in posttensioning steel at Δwm εpm,1 = strain in tension-side posttensioning tendon at initial and repeated loading to +Δwm εpm,2 = strain in compression-side posttensioning tendon at initial and repeated loading to +Δwm εpy, = yield strain of posttensioning steel, defined at limit of proportionality point on strand stress-strain relationship εsa = allowable strain in energy-dissipating steel at Δwm εsd = strain in energy-dissipating steel at Δwd εsm = maximum strain in energy-dissipating steel at Δwm εsu = strain of energy-dissipating or other mild steel at fsu εsy = yield strain of energy-dissipating or other mild steel at fsy κd = energy-dissipating steel moment ratio, defining relative amounts of energy-dissipating resistance from Asfsd and restoring resistance from Apfpd and Nw at wall base = ACI 318-11 factor reflecting reduced mechanical properties of lightweight-aggregate concrete relative to normalweight aggregate concrete µss = coefficient of shear friction at horizontal joints h = confinement steel reinforcement ratio at wall toes t = shear steel reinforcement ratio within panel f = capacity reduction factor for axial-flexural design of base joint f,u = capacity reduction factor for axial-flexural design of upper panel–to–panel joints r = capacity reduction factor against loss of restoring s = capacity reduction factor for shear design wm = plastic curvature at wall base at Δwm <FIGURE CAPTIONS> Figure A1. Plan view of full-scale structure and elevation view of full-scale wall. Note: 1 mm = 0.0394 in.; 1 m = 3.28 ft. Figure A2. Stress-strain behavior of posttensioning steel strand and energy-dissipating bars. Note: fpy = yield strength of posttensioning steel determined at limit of proportionality point on strand stress-strain relationship; fsu = ultimate (maximum) strength of energy-dissipating or other mild steel; fsy = yield strength of energy-dissipating or other mild steel; εpy = yield strain of posttensioning steel, defined at limit of proportionality point on strand stressstrain relationship; εsu = strain of energy-dissipating or other mild steel at fsu; εsy = yield strain of energy-dissipating or other mild steel at fsy. 1 mm = 0.0394 in.; 1 MPa = 0.145 ksi. Figure A3. Free body diagram and gap opening of base joint at Δwm. Note: ad = length of ACI 318-11 equivalent rectangular concrete compression stress block at Δwd; Ap = total posttensioning steel area; As = total energydissipating steel area; cd = neutral axis length at base joint at Δwd; Cd = concrete compressive stress resultant at base ' joint at Δwd; es = distance of tension-side and compression-side energy-dissipating bars from wall centerline; f c = compression strength of unconfined panel concrete; fpd,avg = average posttensioning steel stress for compression-side and tension-side tendons at Δwd; fsd,avg = average energy-dissipating steel stress for compression-side and tension-side bars at Δwd; lh = confined region length at wall toes (center-to-center of bars); lhoop = length of individual confinement hoop (center-to-center of bars); Lw = wall length; Mwd = wall design base moment at Δwd; Nw = factored design gravity axial force at wall base for design load combination being considered; wh = width of confined concrete region at wall toes; whoop = width of individual confinement hoop (center tocenter of bars); sd = elongation of energy-dissipating bar at Δwd; Δwd = design-level wall drift corresponding to design-basis earthquake; f = capacity reduction factor for axial-flexural design of base joint.1 mm = 0.0394 in.; 1 m = 3.28 ft; 1 kN = 0.225 kip; 1 MPa = 0.145 ksi. Figure A4. Cross section and elevation of wall reinforcement details. Note: d = centroid location (from panel end) of tension mild steel bars crossing upper panel–to–panel joint; d ' = centroid location (from panel end) of compression mild steel bars crossing upper panel–to–panel joint; ep = distance of tension-side and compression-side posttensioning tendons from wall centerline; es = distance of tension-side and compression-side energy-dissipating bars from wall centerline; ese = distance of extreme energy-dissipating bar from wall centerline; lh = confined region length at wall toes (center-to-center of bars); lhoop = length of individual confinement hoop (center-to-center of bars); sbot = first confinement hoop distance from bottom of base panel (to center of bar); wh = width of confined concrete region at wall toes; whoop = width of individual confinement hoop (center-to-center of bars). 13M = no. 4; 25M = no. 8; 29M = no. 9; 1 mm = 0.0394 in.; 1 m = 3.28 ft. Figure A5. Free body diagram and gap opening of base joint at Δwm. Note: ad = length of ACI 318-11 equivalent rectangular concrete compression stress block at Δwd; Ap = total posttensioning steel area; As = total energydissipating steel area; cm = neutral axis length at base joint at Δwm; Cm = concrete compressive stress resultant at base joint at Δwm; es = distance of tension-side and compression-side energy-dissipating bars from wall centerline; ' f cc = compression strength of confined concrete at toes of base panel; fpm,avg = average posttensioning steel stress for compression-side and tension-side tendons at Δwm; fsm,avg = average energy-dissipating steel stress for compression-side and tension-side bars at Δwm; lh = confined region length at wall toes (center-to-center of bars); lhoop = length of individual confinement hoop (center-to-center of bars); Lw = wall length; Mwm = probable base moment strength of wall at Δwm; Nw = factored design gravity axial force at wall base for design load combination being considered; wh = width of confined concrete region at wall toes; whoop = width of individual confinement hoop (center-to-center of bars); β1 = ACI 318-11 factor that relates cd to length ad of equivalent rectangular concrete compression stress block at Δwd; γm = ACI ITG-5.2 factor that relates stress of rectangular confined concrete compression block at toe of base panel to confined concrete strength f cc ; sm = elongation of energy-dissipating bar ' at Δwm; Δwm = maximum-level wall drift corresponding to maximum-considered earthquake; f = capacity reduction factor for axial-flexural design of base joint. 1 mm = 0.0394 in.; 1 m = 3.28 ft; 1 kN = 0.225 kip; 1 MPa = 0.145 ksi. Figure A6. Posttensioning steel stress-strain relationship under repeated cycles to ±Δwm. Note: Epi = modulus of elasticity of posttensioning steel; fp,loss = stress loss in compression-side posttensioning tendon (at Δw = Δwm) due to nonlinear material = behavior associated with reversed-cyclic loading of wall to ± Δwm; fpi = initial stress of posttensioning steel after all short-term and long-term losses (but before any lateral displacement of wall); fpm,1 = stress in tension-side posttensioning tendon at initial and repeated loading to +Δwm; fpm,2 = stress in compression-side posttensioning tendon at initial loading to +Δwm; fpm,2r = stress in compression-side posttensioning tendon after repeated loading to +Δwm; Δwm = maximum-level wall drift corresponding to maximum-considered earthquake; εpm,1 = strain in tension-side posttensioning tendon at initial and repeated loading to +Δwm; εpm,2 = strain in compressionside posttensioning tendon at initial and repeated loading to +Δwm. 1 mm = 0.0394 in.; 1 MPa = 0.145 ksi. Figure A7. Free body diagram of first-floor upper panel–to–panel joint at Δwm. Note: Ap = total posttensioning steel ' area; As,u = total tension mild steel area crossing upper panel–to–panel joint at panel end; As ,u = total compression mild steel area crossing upper panel–to–panel joint at panel end; cm,u = neutral axis length at upper panel-to-panel joint at Δwm; Cm ,u = concrete compressive stress resultant at upper panel–to–panel joint at Δwm; d = centroid location (from panel end) of tension mild steel bars crossing upper panel–to–panel joint; d ' = centroid location (from panel end) of compression mild steel bars crossing upper panel–to–panel joint; fc,u = maximum concrete compression stress at upper panel–to–panel joint at Δwm; fp,loss = stress loss in compression-side posttensioning tendon at Δw = Δwm due to nonlinear material = behavior associated with reversed-cyclic loading of wall to ± Δwm; fpm,avg = average posttensioning steel stress for compression-side and tension-side tendons at Δwm; fs,u = stress in tension mild steel crossing upper panel–to–panel joint at Δwm; f s',u = stress in compression mild steel crossing upper panel–to–panel joint at Δwm; Lw = wall length; Mwm = probable base moment strength of wall at Δwm; Mwm,u = upper panel–to–panel joint moment corresponding to Mwm at Δwm; Nw,u = design gravity axial force at upper panel–to–panel joint; Δwm = maximum-level wall drift corresponding to maximum-considered earthquake. 1 mm = 0.0394 in.; 1 m = 3.28 ft; 1 kN = 0.225 kip; 1 MPa = 0.145 ksi.
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