SEISMIC ASSESSMENT OF UNREINFORCED MASONRY BUILDINGS IN BOSTON'S BACK BAY NEIGHBORHOOD ARCHNES by MASSACHUSETTS U' Emily D. Spencer JUL 02 2015 B.S. Civil and Environmental Engineering University of Houston, 2014 LIBRARIES SUBMITTED TO THE DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING IN PARTIAL FULFILLMENT OF THE REQUIREMENTS OF THE DEGREE OF MASTER OF ENGINEERING AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY JUNE 2015 C 2015 Emily D. Spencer. All Rights Reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. / Signature of Author: II Signature redacted Department of Civil a nvir&115'ental Engieepn May 11; Certified by: Signature redacted r John A. Q(fh'sendorf Professor of Civil and Environmental Engineering and Architecture Thesis Supervisor Accepted by: Signature redacted IHeidi Nepf Donald and Martha Harleman Professor of Civil and EnvironmentAl Engineering Chair, Department Committee for Graduate Students TUTE SEISMIC ASSESSMENT OF UNREINFORCED MASONRY BUILDINGS IN BOSTON'S BACK BAY NEIGHBORHOOD by Emily D. Spencer Submitted to the Department of Civil and Environmental Engineering on May 11, 2015 in Partial Fulfillment of the Requirements for the Degree of Master of Engineering in Civil and Environmental Engineering ABSTRACT This thesis presents a seismic evaluation of the unreinforced masonry buildings in Boston Massachusetts's historical Back Bay neighborhood. This Boston district, famous for its rows of Victorian brownstone residences is considered to be one of the best preserved examples of 1 9 'h century urban design. There are a few specific reasons to speculate at the vulnerability of this neighborhood to seismic events. First, in 1755, one hundred years before the Back Bay started to be built, the most massive earthquake of New England's history occurred, damaging unreinforced masonry structures in Boston. Approximately eighty percent of the Back Bay neighborhood is now made up of unreinforced masonry structures. Second, seismic design was not required in Boston until 1975, which means a staggering majority of the buildings in the Back Bay were constructed without any kind of anti-seismic lateral system. The aim of this thesis is to assess the structural response of the unreinforced masonry homes of the Back Bay to seismic activity due to these issues. A case study of an unreinforced masonry building in the neighborhood is assessed through structural analysis of its fagade and party walls. The performance of this building is extrapolated to represent the state of the unreinforced masonry buildings of the Back Bay. Thesis Supervisor: John A. Ochsendorf Title: Professor of Civil and Environmental Engineering and Architecture 3 Acknowledgement I would like to thank a few people in particular that helped in the completion of this thesis. First I would like to thank my advisor Professor John Ochsendorf for not only introducing me to historical preservation engineering but also for his dedicated guidance and expertise through the research and writing of this thesis. I would also like to thank Post-Doctoral Student Ornella Iuorio for her leadership, insight, friendship, and day to day assistance in this project. Thank you to Professors Pierre Ghisbain and Jerome Connor for their incredible engineering classes, knowledge, and advice. Thank you to my friend Rosalie Bianquis, whose moral support during my time at MIT was unprecedented and irreplaceable. Especially, I would like to thank my parents, Bruce and Sandra, my brother Ian, and my sister Samantha, for their love and encouragement during my years of school. I would also like to thank my grandmother, Muriel Paschall, who always believed, more than I, how far I could go. Finally I would like to thank all of my classmates in the Masters of Engineering program. I will always remember their vehement support and friendship. 5 6 Table of Contents Acknow ledgem ent ........................................................................................................................................ 5 Table of Figures ............................................................................................................................................ 9 Table of Tables ........................................................................................................................................... 11 1.0 Introduction.....................................................................................................................................13 1.1 Research M otivations and Objectives..................................................................................... 13 Classification of Unreinforced M asonry Structures................................................................... 15 Characteristics of URM buildings.......................................................................................... 15 2.1.1 M asonry Walls ........................................................................................................................... 15 2.0 2.1 2.1.2 Flooring......................................................................................................................................16 2.1.3 Structural Behavior .................................................................................................................... 2.2 Failure M echanism s of URM Walls ........................................................................................... 17 17 2.2.1 In-Plane Failures ................................................................................................................. 18 2.2.2 Out-of-Plane Failures..................................................................................................... 19 Earthquake Perform ance of URM W alls .................................................................................... 2.3 Earthquake H istory of Boston..................................................................................................... 3.0 20 21 H istorical Boston Earthquakes................................................................................................. 21 3.1.1 The Cape Ann Earthquake .............................................................................................. 21 3.1.2 Earthquake Risk in Boston.............................................................................................. 23 3.1 3.2 Boston Geology .......................................................................................................................... 23 3.3 H istory of Boston Seism ic Building Codes ............................................................................ 24 The Back Bay .................................................................................................................................. 4.0 25 4.1 History.........................................................................................................................................25 4.2 Neighborhood Building Typologies........................................................................................ 26 4.2.1 Unreinforced M asonry ..................................................................................................... 27 4.2.2 Transitional M asonry ..................................................................................................... 27 4.2.3 M odem Construction .......................................................................................................... 28 M ethod of Structural Analysis ..................................................................................................... 29 General M odel Inputs.................................................................................................................. 29 5.0 5.1 5.1.1 Estim ated M aterial Properties.......................................................................................... 29 5.1.2 Expected Ground M otion................................................................................................. 30 5.1.3 Estim ated Base Shear..................................................................................................... 30 Cantilever Beam M ethod ............................................................................................................ 31 5.2 7 5.3 Finite Element M odel ................................................................................................................. 32 5.4 Global Behavior Evaluation Criteria........................................................................................ 35 5.5 Local Behavior Evaluation Criteria ......................................................................................... 36 In Plan e ............................................................................................................................................... 36 O ut o f Plan e ........................................................................................................................................ 37 6.0 Case Study Structural Analysis Results ............................................................................................ 6.1 40 M odelling Considerations..............................................................................................................42 6.2 Cantilever Beam Results...................................................................................................................43 6.3 Finite Element M odel Results...........................................................................................................45 S cen ario 1............................................................................................................................................ 45 S cen ario 2 ............................................................................................................................................ 53 6.4 Local Elem ent Failure Results..........................................................................................................57 In-Plane Failure Results ...................................................................................................................... 57 Out-of-Plane Failure Results...............................................................................................................63 6.5 Analysis M odel Conclusions.............................................................................................................. 7.0 7.1 66 Conclusion......................................................................................................................................68 Areas of Future W ork ................................................................................................................. Documentation References ......................................................................................................................... Image References........................................................................................................................................71 8 69 70 Table of Figures Figure 1 A brick masonry wall is made of multiple wythe layers of stretcher and header brick courses 15 (tp ub .co m )................................................................................................................................................... 16 Figure 2 Pier and spandrel elements define sections of a perforated masonry wall. .............................. 16 Figure 3 Pier and spandrel elements define sections of a perforated masonry wall. .............................. Figure 4 Joist pockets in the masonry allow timber joists to transfer floor loads to load bearing walls.... 16 Figure 5 Joist pockets in the masonry allow timber joists to transfer floor loads to load bearing wallss.. 17 17 Figure 6 Pier and spandrel elements define sections of a perforated masonry wall. .............................. Figure 7 Unreinforced masonry wall failures from varying ground accelerations where liquefaction was an issue. (a) Damage from a quake in Nisqually, Washington with peak ground acceleration of 0.31g (Washington Surveying and Rating Bureau) (b) Long Beach, CA full fagade failure with PGA of 0.22g (Historical Society of Long Beach) (c) The famous Northridge earthquake caused corner failure due to a PGA of 1.Og (Bruneau)(d) The Loma Preita, CA earthquake with a PGA of 0.16g causing out of plane failure of a fagade wall (EERI) (e) Recent damage from the Napa Valley, CA earthquake with PGA of 0.35g causing this corner failure (ZFA Structural Engineers) (f) One of the strongest earthquakes in history, the Christchurch earthquake caused this full facade failure with a PGA of 1.8g (USGS)......................20 Figure 8 Notable earthquakes affecting Boston from the 1 7 th century to the 2 1St century and their 21 ep icen ters. ................................................................................................................................................... Figure 9 Historical drawings accounting the Cape Ann earthquake of 1755 (National Information Service for E arthquake E ngineering).......................................................................................................................22 Figure 10 The Cape Ann earthquake of 1755 aligns with the facades of the Back Bay residences in B oston , M assachu setts................................................................................................................................23 Figure 11 The Back Bay residences have a similar building organization and structure (Bunting)...........26 Figure 12 The Back Bay consists of four general building types: unreinforced masonry (yellow), reinforced masonry (red), modern construction (blue), and churches (green)........................................27 Figure 13 The cantilever beam model lumps masses at the floor levels where portions of the base shear 31 are app lied ................................................................................................................................................... 32 Figure 14 Bad w all to w all connection .................................................................................................. Figure 15 The four wall sections make up four different SAP2000 models to analyze.......................... 33 Figure 16 An example of the data collection process for the SAP2000 models. Proportional base shear values are applied at the joints at floor level and deflection values are read from the joints on the right side 34 of the m odel also at floor level.................................................................................................................... Figure 17 SAP2000 models, from right, considering 10%, 50%, and 100% of the building acting together, 34 w ill be also necessary to analyze. ............................................................................................................... 37 Figure 18 Rigid block free-body diagram (DeJong).............................................................................. 40 Figure 19 Front and rear faqades of 37 Commonwealth........................................................................ Figure 20 Counterclockwise from left, interior brick courses, vaulted brick basement floor, and joist 41 pockets of 39 C omm onw ealth .................................................................................................................... Figure 21 Original structural drawings of 37 Commonwealth (Boston Athenaeum N. Bradlee Collection). . ................................................................................................................................................................... Figure Figure Figure Figure Figure Figure Figure 22 Modeling simplifications for scenario 1................................................................................ 23 Modeling simplification for scenario 2................................................................................... 24 Pushover curve for the cantilever beam hand calculations...................................................... 25 Pushover curve from SAP2000 for scenario 1. ..................................................................... 26 Base shear vs. interstory drift for the basement level............................................................ 27 Base shear vs. interstory drift for the first floor...................................................................... 28 Base shear vs. interstory drift for the second floor.................................................................49 9 41 42 43 43 45 47 48 Figure 29 Base shear vs. interstory drift for the third floor. .................................................................... 50 Figure 30 Base shear vs. interstory drift for the fourth floor. ................................................................. 51 Figure 31 Base shear vs. interstory drift for the roof level. .................................................................... 52 Figure 32 Pushover curve from the cantilever beam hand calculations for scenario 2.......................... 53 Figure 33 Pushover curve from SAP2000 for scenario 2. ...................................................................... 54 Figure 34 Base shear vs. interstory drift for the basement level............................................................ 54 Figure 35 Base shear vs. interstory drift for the first floor..................................................................... 55 Figure 36 Base shear vs. interstory drift for the second floor ................................................................ 55 Figure 37 Base shear vs. interstory drift for the third floor. .................................................................... 56 Figure 38 Base shear vs. interstory drift for the fourth floor. .................................................................. 56 Figure 39 Base shear vs. interstory drift for the roof level. .................................................................... 57 Figure 40 Most of the piers will fail from diagonal tension failure in the SAP2000 models. Bed joint sliding is observed in the full wall and rocking is observed in the curved wall section. ........................ 58 Figure 41 Local shear failure examination for full wall. ........................................................................ 59 Figure 42 Local shear failure examination for flat wall section. ........................................................... 60 Figure 43 Local shear failure examination for the curved wall section ................................................. 61 Figure 44 Local shear failure examination for the long pier section of the fagade.................................62 Figure 45 C ase study chim ney .................................................................................................................... 64 10 Table of Tables Table 1 Masonry properties used throughout all models and calculation methods in this thesis. .......... 29 Table 2 Ground motion and response spectrum values necessary to determine expected building base shear. 31 .................................................................................................................................................................... Table 3 FEMA 356 interstory drift limits for Collapse Prevention, Life Safety, and Immediate Occupancy 35 perform ance levels. ..................................................................................................................................... Table 4 To violate the FEMA 356 performance levels, these are the interstory distances that each floor is allow ed to drift............................................................................................................................................36 Table 5 Expected base shear for wall sections in scenario 1 and corresponding max roof displacements. 44 Table 6 Expected base shear for wall sections in scenario 2 and corresponding max roof displacements. 44 Table 7 Base shear values necessary to exceed FEMA 356 performance levels for the overal pushover 46 cu rv e ............................................................................................................................................................ Table 8 Base shear values necessary to exceed FEMA 356 performance levels for basement level. ........ 47 Table 9 Base shear values necessary to exceed FEMA 356 performance levels for first floor..............48 49 Table 10 Base shear values necessary to exceed FEMA 356 performance levels for second floor. ..... Table 11 Base shear values necessary to exceed FEMA 356 performance levels for third floor...........50 Table 12 Base shear values necessary to exceed FEMA 356 performance levels for fourth floor......51 Table 13 Base shear values necessary to exceed FEMA 356 performance levels for roof level............52 Table 14 Terms needed for rigid block analysis of full fagade...............................................................63 63 Table 15 Stability results for full fagade rigid block calculation. ........................................................... 64 Table 16 Stability results of roof w all..................................................................................................... Table 17 Stability analysis of chimney that is one wythe thick..............................................................65 65 Table 18 Shear capacities at different failure plane heights. ................................................................. 66 ......................................... fagade portions. interstory to crack Table 19 Minimum ground acceleration 11 12 1.0 Introduction Low rise unreinforced masonry buildings are one of the most common structural types in use in the United States. Typically built before the early twentieth century, unreinforced masonry buildings are constructed as four load bearing multi-wythe brick walls with flexible diaphragm timber floors. Inherently, earthquake resistance in these buildings comes from the massive gravity load of the bearing walls, but a lack of lateral anti-seismic design condemns them to be one of the most vulnerable building types to seismic damage. The unreinforced masonry building type describes the majority of the structures in the historical neighborhood of the Back Bay in Boston, Massachusetts. As one of the best examples of nineteenth century urban design in the United States, this neighborhood is lined with beautiful irreplaceable Victorian brownstone rowhouses. The Back Bay is faced with several issues. In a region that is seemingly not susceptible to earthquakes, seismic design was not required for new construction in Massachusetts law until 1975, which means virtually all of the unreinforced masonry homes in the neighborhood were traditionally built without any kind of anti-seismic design. Although rare, Boston has been affected by a handful of damaging earthquakes in the past, namely one of intensity VIII (MMI) off of the coast of Cape Ann, Massachusetts in 1755. Also, the Back Bay was built on timber piles through an infilled area of the Charles River that is susceptible to liquefaction. By considering these structural issues, the building performance can be assessed to make an appropriate seismic assessment of the neighborhood. Research Motivations and Objectives 1.1 The aim of this thesis is to understand the structural performance of the unreinforced masonry buildings in the Back Bay during imminent future earthquakes. The motivations for this thesis are evident in the subsequent points. 1. Back Bay unreinforced masonry homes were built without anti-seismic design on infilled land. Major earthquakes in the 1 71h and 18t centuries damaged the existing brick buildings. It has been more than 250 years since an earthquake that had the intensity of Cape Ann's and it is reasonable to think that Boston is overdue for an earthquake that can happen at any time. 13 2. Politics has prevented a publically available study on the seismic vulnerability of the Back Bay neighborhood as most of the residences are privately owned by the affluent or landlords who choose either not to reveal structural analyses on their homes or are scared to see the results of such a study. 3. Many engineers in the northeast don't believe in a seismic risk to Boston and the Back Bay large enough to substantiate mandatory action. It is necessary to cogitate the long list of risks facing the Back Bay to fully understand the earthquake vulnerability. To honor these motivations, this thesis will focus on the following objectives: * Collect in one piece of literature the primary seismic issues facing the Back Bay " Determine seismic performance of Back Bay structures to facilitate future retrofitting in order to save money and lives This thesis will begin with a classification of non-specific unreinforced masonry structures, their failure modes, and observed earthquake performance. Structural analysis methods and evaluation criteria for this type of building will be presented. A case study building in the Back Bay neighborhood will then be analyzed through hand calculations and a finite element model. Expected ground accelerations for the Back Bay area will be applied to the calculations. The performance of the case study will be determined by global behaviors such as interstory drift and local behaviors such as pier deterioration and chimney dislocation. The conclusion of this thesis will extrapolate a basic seismic assessment for the Back Bay neighborhood. 14 2.0 Classification of Unreinforced Masonry Structures Characteristics of URM buildings 2.1 Typically built before the early twentieth century when the utilization of steel frame design had not become commonplace, an unreinforced masonry building is defined as a building whose walls are constructed of stacked masonry units bonded by mortar without steel reinforcement. The type of masonry used can consist of brick, hollow clay tiles, stone, concrete blocks, or adobe (FEMA 2009). This thesis will focus on unreinforced brick buildings. Defining characteristics of unreinforced masonry buildings include: use of lime and sand mortar, lack of steel reinforcing, vertically tapering wall thicknesses, perforated walls, decorative parapets and chimneys that extend higher than the occupiable structure, timber floors, and an exterior masonry veneer. These characteristics will be explained in detail. 2.1.1 Masonry Walls The walls in unreinforced masonry buildings, regardless of masonry material, act as the main vertical resistance system, supporting their own weight and dead and live loads emanating from the floors and roof. These load bearing walls are considered to be engineered and not designed in that they were designed empirically from tables published in local building laws that defined wall thickness as a function of building height in order to maintain the compressive stress in the masonry walls to be below the maximum allowed stress values, which inherently caused very conservative factors of safety in the walls (Buntrock 2010). In particular, unreinforced brick masonry buildings consist of walls made of a number of brick wythes, which is a vertical layer of bricks. Typically, two or three layers of structural brick layers reside behind a veneer exterior wythe, which is a layer of better looking brick. Every few layers of the veneer brick layer, there are header bricks which STRETCHER NEAM are turned ninety degrees to connect the outer wythe to the inner wythes. Sometimes there is a whole course of header bricks, while at other Figure 1 A brick masonry wall is made of multiple wythe layers of stretcher and header brick courses (tpub.com). 15 times a brick course may have a header brick separated by a regular brick, or a stretcher. Header bricks are one of the main characteristics to find when identifying an unreinforced masonry building from the exterior (FEMA 2009). These terms are manifested in Figure 1. Perforations, or holes for windows, are found in most unreinforced brick structures and their placement and size can dictate the structural integrity of the wall, so their dimensions are nowadays confined by codes. The perforations SPAN separate the walls into new building elements that can be categorized further. A spandrel is a section of a wall between perforations of different stories, Figure 2 Pier and spandrel elements define sections of a perforated masonry wall. and a pier is the section of the wall between perforations on the same floor. Parapets are a portion of the masonry wall that extends above the top floor of the building. Their heights range from a few inches to a few feet. The purpose of the parapet is to protect the roofing materials from wind, to act as a guardrail when the roof is occupied, and to make the building appear taller. Chimneys, like parapets, extend higher than the roof and can be many feet tall. As part of an unreinforced building, these elements are also unreinforced brick, which can make them very vulnerable to failure (FEMA E-74). 2.1.2 Flooring Joists and sheathing made of timber span between the masonry walls. The joists are usually a foot in depth, a few inches thick, and are spaced one foot on center. If the masonry walls are vertically tapered and reduce thickness with every floor, there is a shelf for the joists to sit on. But if the walls are not tapered, which is characteristic for low-rise masonry buildings, instead of a shelf there exists a joist pocket which is a rectangular shaped hole in the masonry that the joist slides Figure 3 Joist pockets of unreinforced masonry building (280 into. An example of a joist pocket is in Figure commonwealth). 16 3. In both of these cases the joist simply sits on a masonry base creating a simply supported condition, but in some situations the joist pockets may be more confining which can create a fixed condition at the joist end when the timber floor deflects. On top of the joists is timber sheathing which are sheets of plywood Figure 4 Arched brick vaults separated by iron beams are sometimes seen in unreinforced masonry buildings (Friedman). one fourth to one half inches thick and can be in one or more layers. Some unreinforced masonry buildings don't have bottom floors made out of timber, but of vaulted brick with iron beams, as seen in Figure 4 (Friedman 1995). 2.1.3 Structural Behavior As aforementioned, the masonry walls are load bearing and the codes that led to their construction deem them to be able to withstand high vertical compressive stresses. The timber floors deflect under live loads and transfer force to the adjoining walls, but depending on the boundary condition it may be a bending stress or just a floor bearing load. Depending on the configuration of the building, timber joists may only be transferring loads to the party walls if there are only joist pockets in one direction. In this case, the force and deflection of the party walls will continue to distribute force through the wall-to-wall corner connection, creating a box effect, assuming the connection is uncracked, in good condition, and can transfer loads. The walls of unreinforced masonry buildings were not designed to resist wind loads and earthquake loads. The compressive stress of the walls generally compensates for lateral wind forces and it is not common to observe building failures due to wind, but that is out of the scope of this thesis. A more controversial aspect about the erection of these buildings is that they are observed to have been built without earthquake design. Historic seismic events have proven that the vertical compressive stress that may resist wind loads well are not enough to prevent the walls from failing from earthquakes (Buntrock 2010). Failure Mechanisms of URM Walls 2.2 Unreinforced masonry walls are highly indeterminate and because of this have been observed to fail in a multitude of in-plane and out-of-plane modes. According to FEMA 306, for a perforated wall with weak piers and strong spandrels, which is the condition assumed for this thesis, there are 17 four modes of in-plane and two out of plane failure mechanisms. For in-plane, these are flexural rocking, bed joint sliding, diagonal tension cracking, and toe crushing. Pictorial examples of these failures are in Figure 5. For out of plane there is one way bending and two way bending, depicted in Figure 6. FEMA 306 provides a shear capacity equation for each failure mode. The failure mode of wall elements is dictated by certain parameters and is typically governed by the lowest shear value. It is observed that through calculation later in this document that the failure mode can be said to be dictated by the pier height to width ratio and compressive stress caused by the masonry above the pier. 2.2.1 1. In-Plane Failures Flexural Rocking The flexural rocking mode is characterized by cracking at the top and bottom of a perforated wall's piers. It is observed to occur when piers are slender, spandrels are strong, and the compressive stress within the masonry is low. In this behavior mode, the piers rock and develop hairline fractures at the top and bottom of the piers. Although cracking deformations can be large, this failure is still considered to be stable since it can still handle vertical loads. If rocking continues, the pier can degrade, overturn in-plane, or slowly 'walk' out-of-plane, causing instability. 2. Bed Joint Sliding The bed-joint sliding mode is characterized by cracking that occurs in the mortar bed-joints between masonry units in the horizontal plane and in a stair-stepping pattern. This failure mode occurs when the shear strength of the masonry is higher than that of the mortar. 3. Toe Crushing Toe-crushing occurs when shear stress is concentrated at the toe of a pier during rocking, causing cracking and material spalling to occur. As the rocking mode degrades the pier, bed-joint sliding occurs in the middle, and diagonal cracks form from the toe of the pier to the upper corners. Failure by toe-crushing occurs when the toe of the wall has weakened to a point when the vertical load carrying capacity of the pier has been compromised. 4. Diagonal Tension Cracking When a pier with strong mortar and weak masonry units is subjected to high compressive stress, the diagonal tension cracking mode can be observed. This behavior is characterized by an 'X' 18 shape that develops through the masonry units, not around, in the pier. Sometimes, with strong masonry units and weak mortar, this mechanism can occur but with cracking in a stair-stepping shape, similar to that of bed-joint sliding, but in an 'X' shape. Diagonal tension cracking is considered to be a failure more than the three aforementioned modes since it cannot handle vertical loads after its initiation. 2.2.2 1. Out-of-Plane Failures One-way bending In this failure mode, a wall overturns in one direction, such as a parapet, chimney, or the top story wall of a structure cracking at the penultimate floor diaphragm, and falling to the ground. 2. Figure 5 From top left, clockwise, examples of rocking failure (Javed), bed joint sliding Two-way bending Two-way bending is very similar to one way bending in (FEMA), diagonal tension cracking (FEMA), and toe crushing (FEMA). terms of cause but occurs when the location of the crack development is at the central area of the story wall, causing the two portions of the wall to burst outward at opposite rotation angles either horizontally or vertically. Figure 6 From the left, a one way bending parapet failure (Welliver), a one way bending fullfloor failure (FEMA), and a two way bending failure of two floors (Bruneau). 19 Perforated walls with weak spandrels and strong piers have a similar method of failure mode determination, but due to height to width ratios of the piers and the height of the spandrels of the case study building analyzed in Chapter 5, this condition is out of the scope of this thesis. 2.3 Earthquake Performance of URM Walls This section will parameterize unreinforced masonry earthquake damage in terms of ground acceleration in order to visually imagine expected damage for the Back Bay. It is estimated that the Cape Ann earthquake in 1755 had a ground acceleration of 0.18-0.21g (Whitman 1975). The examples of damage in Figure 7 are all from sites with loose soil where liquefaction occurred, which is estimated to be the conditions in the Back Bay. (a) (d) Figure 7 Unreinforced masonry wallfailures from varying ground accelerations where liquefaction was an issue. (a) Damage from a quake in Nisqually, Washington with peak ground acceleration of 0.31g (Washington Surveying and Rating Bureau) (b) Long Beach, CA fullfagade failure with PGA of 0.22g (Historical Society of Long (b) Beach) (c) Thefamous Northridge earthquake caused corner failure due to a PGA of 1.Og (Bruneau) (d) The Loma Preita, CA earthquake with a PGA of 0.16g causing out of plane failure of a fagade wall (EERI) M) (e) Recent damage from the Napa Valley, CA earthquake with PGA of 0.35g causing this f) "77 20 corner failure (ZFA Structural Engineers) (f) One of the strongest earthquakes in history, the Christchurch earthquake caused this full fagadefailure with a PGA of 1.8g (USGS) 3.0 Earthquake History of Boston Historical Boston Earthquakes 3.1 Extensive records of seismic activity have been taken in the New England area since it was settled in the early 1600s. Since 1602, nineteen earthquakes, of intensity 5 or greater, have occurred with their epicenter in Massachusetts. Between the years of 1668 and 2007, there have been 355 total earthquakes in Massachusetts. Many other moderate earthquakes, centered in Maine, New York, New Hampshire, and Canada have affected Massachusetts in damaging ways. Figure 8 shows a number of earthquakes with available information to have damaged Boston between the 1 7 th century and twenty first century and their epicenters (USGS). HISTORICAL EARTHQUAKES TO HAVE AFFECTED BOSTON, MA 12 100 zz 00 cc z -bX 160 65 10015 0 0 801 195200 4YEA Figure 8 Notable earthquakes affecting Boston from the 1 7(X century to the 21"~century and their epicenters. 3.1 .1 The Cape Ann Earthquake The most destructive earthquake to have ever occurred in New England had a max intensity (Modified Mercalli Intensity scale) of 8 centered 25 miles east of Cape Ann, Massachusetts on November 18, 1755. Felt over 300,000 square miles, from Halifax, Nova Scotia to the north, Winyah, South Carolina to the south, and even 250 miles out to sea, the earthquake caused a tsunami in the West Indies. The area felt several aftershocks and tremors, notably north of Boston. 21 Seventeen days before, on November 1, Lisbon, Portugal was devastated by a massive earthquake and tsunami. It is believed, however not proven, that it triggered the Cape Ann earthquake (USGS). It is important to note that the majority of the damage was concentrated in the small Boston harbor areas that had been infilled, as the Back Bay neighborhood is now. Having said that, a lot of damage still occurred on the natural land portions of the city, which is concerning since most of Boston now is built on non-engineered fill. It was recorded in Boston that approximately 1500 chimneys were damaged, stone walls and fences collapsed, and roofs were damaged (Ebel 2006). An account from a Bostonian named John Hyde in 1755 stated, "Many chimnies.. .not fewer than 12 or 1500 are shattered, and thrown down in part; so that in some places, especially on the low loose ground, are dislocated, or broken several feet from the top.. .the streets are almost covered with the bricks that have fallen...the roofs of some houses are quite broken in by the fall of the chimnies" (Philisophical Transactions). An account from John Adams, future President of the United States, stated of the quake, "The house[s] seemed to rock and reel and crack as if it would fall in ruins..." (Adams 1856). Figure 9 Historical drawings accounting the Cape Ann earthquake of 1755 (National Information Service for Earthquake Engineering) Modern analysis of the Cape Ann earthquake suggests that based on masonry chimney damage, the ground acceleration might have been as large as 0.18-0.21g (Whitman). It is estimated that these ground motions equate to 2% chance of exceedance every 50 years (USGS). Small earthquakes continue to occur in a cluster in this area and are believed to be very late aftershocks to the 1755 quake (Ebel 2006). The earthquake epicenter was about 30 miles east off the coast of Cape Ann. If it is assumed that ground motion emanates radially from that location, it approaches the Back Bay neighborhood at an angle of sixty degrees, if lines of latitude constitute as zero degrees. The neighborhood is also 22 oriented along the Charles River at an angle very close to sixty degrees. This means that for structural analysis done in Chapter 6, applying the earthquake load in-plane with the fagade is acceptable. Figure 10 displays this finding. CAPE ANN EPICENTER BACK BAY 37 COMMONWEALTH Figure 10 The Cape Ann earthquake of 1755 aligns with the facades of the Back Bay residences in Boston, Massachusetts. 3.1.2 Earthquake Risk in Boston Although Boston and Massachusetts in general do not lie on major active tectonic boundaries like California or Japan, they do lie on ancient faults that formed hundreds of millions of years ago and are now considered to be geological zones of weakness. It is believed that these zones of weakness are areas that stresses from present day faulting action can be released, reactivating the ancient faults and causing intraplate earthquakes. It is near impossible to predict where, when, and how strong the next earthquake in Boston will be. There is a general correlation between the locations of earthquake epicenters in New England from 1602-2014, and it is expected that there is a 2% chance that in any given period of 50 years an earthquake will occur that is potentially damaging (Kafka). 3.2 Boston Geology Boston geology can be simplified into two strata. First, the bedrock is made up of a rock formation called Cambridge Argillite, which is a typically weak rock that is slightly metamorphic (USGS). Second, a great majority of the Boston landmass is made up of non-engineered, cohesionless, 23 saturated infill. These two weak layers can be the cause of ground failure during seismic activity. A geologic phenomenon called liquefaction occurs when the strength and stiffness of a soil deteriorates from seismic shaking and begins to behave like a liquid. Devastating ground failure can results from liquefaction (USGS). Of the historic earthquakes that have had epicenters near Boston, four had ground accelerations that had the potential to cause liquefaction (1638, 1663, 1727, and 1755). Further analysis of the effects of liquefaction are out of the scope of this thesis, but it is important to note its potentiality when considering seismic risk for the Back Bay. History of Boston Seismic Building Codes 3.3 In the United States, the first provisions made for seismic design were first seen in the appendix to the 1927 Uniform Building Code (UBC). The first national seismic zone hazard map was seen in the UBC in 1949. By the 1950s, cities in California had begun to adopt their own seismic resistant design measures. After the San Fernando earthquake in 1971, many revisions were made to the UBC in 1973 (FEMA 1998). In 1975 Massachusetts became the first state in the east to implement seismic design provisions that were particular for the state. A study entitled 'Seismic Design Decision Analysis' done by Robert Whitman of MIT determined that the maximum intensity of earthquake that was probable for the state was similar to the UBC's Zone 3 classification in California (instead of the current Zone 2A classification), but with lower return periods (Nordenson). The zoning classification is vestige of the old UBC building code. Zone 2A corresponds to a potential peak ground acceleration of 0.1 5g and Zone 3 corresponds to 0.30g. A committee of engineers used the MIT study as justification to recommend the 1975 provisions, which included seismic design requirements for new construction. It wasn't until 1980 that rehabilitation requirements were made for existing buildings. 24 4.0 The Back Bay 4.1 History of the Back Bay neighborhood came about not long after Boston, Massachusetts institution The decided to transform itself from a harbor town to a more prosperous, lively city. To do this the city decided that they needed start harnessing energy from the surrounding rivers in Boston in order to become a boom town like certain cities in California and Texas (Bunting 1967). From the year 1818 to 1821, the construction of the Mill Dam took place. As one of Boston's greatest engineering achievements, this dam project's intent was to increase Boston's water power and industrial prowess and to provide a toll road to and from the city. Because the Charles River becomes nothing more than a slow flowing estuary that rose and fell with the tides, the Mill Dam did not end up producing the expected amount of water power that was predicted. Even though a great feat of engineering, the Mill Dam was a huge failure. The course of the Mill Dam confined a bay next to Boston, named the Back Bay. After being dammed up for years, the bay turned into a stagnant lagoon of raw sewage and trash that didn't allow much inflow and outflow to dilute the waste. In 1849 the Health Department demanded that the bay be filled in for public health reasons. Meanwhile, early 1 9 th century Boston saw a substantial rise in population. In fact, the single decade of the 1840s saw a population increase of 33 percent. Due to a need to cover up the environmental hazard that the Back Bay waters were becoming and the need for more housing developments, it was decided that the Back Bay waters would be filled in (Creating Land in Boston's Back Bay). In 1857, the Back Bay began to be infilled with glacial granite from West Needham, Massachusetts. This ice-age gravel was transported on train tracks built especially for the project that are part of the present day Green Line in Boston and Commonwealth Avenue in the Back Bay neighborhood. The Mill Dam was covered completely by the gravel and resides under present day Beacon Street. By the start of the Civil War, the bay had been infilled as far west as Clarendon Street. It was around this time that investors built the first brownstone houses. As more land was filled in, more houses were built. War prosperity caused an acceleration in the infilling process and home construction. By 1870, the infill was complete through Dartmouth Street up to Exeter Street. In 1872, a large fire destroyed 65 acres of Boston, delaying construction and causing required fire escapes in the newly constructed homes. By the late 1880s, the infill project was complete up to the Fens. In the decade following, the remainder of the Back Bay was filled in. At the end of the project 450 acres were added to the original 783 acres of Boston (Bunting 1967). 25 4.2 Neighborhood Building Typologies Back Bay structures have defining characteristics. A Back Bay house is considered to be a Type II structure according to the Boston Building Department. A Type II structure has exterior walls made of masonry and interior floors and partitions made of wood. The high quality seasoned timber floor joists are overdesigned by modem standards and are typically 3"x 12" spaced 1 foot on center. Brick party walls describe the longer walls perpendicular to the road and are engaged by the transverse floor loads. The facade and rear elevations, parallel to the road, are only susceptible to gravity loads, out of plane loading, and roof loads. Each fagade and rear elevation is represented by multiple, tall windows. A typical foundation is comprised of wooden piles ranging from 20-35 feet deep, spaced from 1.5-3 feet on center, that are positioned under the stone foundation walls, which are in turn atop granite leveling blocks (Bunting 1967). Figure 11 shows these typical properties. Figure 11 The Back Bay residences have a similar building organization and structure (Bunting). Three typologies, specified in Figure 12 for each residence, specifically describe the majority of the buildings of the Back Bay: unreinforced masonry, transitional masonry, and modem construction buildings. The materials, construction technique, and structural behavior of the buildings are dictated by these categories. 26 4.2.1 Unreinforced Masonry The unreinforced masonry typology encompasses about eighty percent of the Back Bay neighborhood. Built approximately between the years of 1860 and 1910, these buildings are typically three to five stories in height. During this time there were city ordinances limiting the height of the neighborhood buildings to keep the fagade looking continuous. The masonry walls, typically 16 inches thick at street level and decreasing in size with increasing floor number, are made out of layered brick wythes. Two or three wythes reside behind an ornamental wythe of better looking brick. These thick, heavy walls represent the main vertical structural resistant system. The horizontal floors are timber joists. Joist-pockets in the masonry walls allow the insertion of the transverse joists so that they may transfer loads to the walls. It has been observed that the floor between the basement and first floor are not wooden joists, but vaulted brick flooring with iron beams. The roofs are flat with tar and gravel or slanted wooden mansards roofs. 4.2.2 Transitional Masonry Transitional masonry describes a generation of buildings that present a combination of old and new building materials and were built in the Back Bay beginning in the twentieth century until about the year 1970. Structural steel was introduced to building construction and allowed the buildings in the neighborhood to realize heights of seven to ten stories. Each building comprises of steel frames with masonry infilled multi-wythe walls. Flooring materials have been observed to 0 lo Or'-w r I ~ ~ ~ TYCCG - Lrm3@!tdM Figure 12 The Back Bay consists offour general building types: unreinforced masonry (yellow), reinforced masonry (red), modern construction (blue), and churches (green). 27 be timber joists or concrete. The joist to wall connection in transitional masonry is no longer a joist pocket situation but instead a more modem connection exists where the joists connect to the horizontal steel beams. The roofs are also flat or wooden mansard. 4.2.3 Modem Construction Modem construction in the Back Bay is represented by structures containing reinforced concrete that rise higher than about ten stories. Further, the modem construction typology can be separated into two categories: those built before the implementation of the Boston seismic code in 1970 and those after, to the new code. However, the majority of the modem construction buildings were constructed before the advent of the code. 28 5.0 Method of Structural Analysis There are many assumptions and means of approach for the structural analysis of unreinforced masonry buildings under lateral loading, which causes them to be incredibly complex. Therefore there is not one accepted analysis method. Some codes, such as those from FEMA, suggest ignoring out of plane wall stiffness when analyzing the URM building response and only considering wall mass, while some codes deem it necessary to include out of plane stiffness especially if the out of plane walls carry most of the diaphragm load (Park 2009). This thesis will explore different analysis methods to evaluate the seismic response of the unreinforced masonry buildings of the Back Bay. General Model Inputs 5.1 First, all geometries and measurements used in hand calculations and finite element models were taken straight from the original structural drawings by N. Bradlee from the Boston Athenaeum. Pictures were taken of the drawings and by using a few key dimensions, scaled CAD drawings were able to be made. Every dimension used for this thesis was taken from those scaled drawings. 5.1.1 Estimated Material Properties It is incredibly difficult to choose correctly the properties of existing masonry buildings, especially those that are over 150 years old. Structural properties cannot be inferred solely from age and can be determined by material testing or code default values (FEMA 1997). Since the case study property is a private residence, under no circumstance would it have been possible to carry out invasive or even non-invasive measures in order to obtain material samples to determine material properties from. Therefore it was necessary to use default values suggested by the FEMA 356 and ASCE 41 codes which are essentially one and the same. Table 1 summarizes the estimated material properties and their sources. Table 1 Masonry properties used throughout all models and calculation methods in this thesis. Source Notes Value Units Compressive Strength of Brick (f'm) 780 psi FEMA 356 Table 7-1 & 7-2 Assume Fair Condition Elastic Modulus in Compression (E) 429000 psi FEMA 356 Table 7-1 & 7-2 Assume Fair Condition Property Flexural & Diagonal Tensile Strength (f'dt) 13 psi FEMA 356 Table 7-1& 7-2 Assume Fair Condition Shear Strength of Masonry Bed Joint (Vie) 26 psi FEMA 356 Table 7-1 & 7-2 Assume Fair Condition 120 lb/ft3 - Masonry Specific Gravity (y) Poisson's Ratio for Masonry (v) 0.25 29 - - In accepted range for property Accepted mid-range value 5.1.2 Expected Ground Motion The peak ground acceleration values for the Back Bay area are estimated from maps and equations presented in Chapters 11 and 22 of the ASCE 7-10 code. By reading the typological lines on the maps, the expected ground acceleration is determined to be 14.32% the force of gravity for soil class B. But the Back Bay is built on infilled land and can be assumed to be of soil class D (accepted default value when conditions unknown), which increases the peak ground acceleration to a value of 22.92% the force of gravity. The exact value of 0.2292g is provided by ASCE 7-10 for the Back Bay zip code 02116 and soil class D and is directly used in the hand calculations for the cantilever beam method. A paper written by Robert Whitman, a former professor of MIT, well before ASCE 7-10 determined these values, suggests that it is likely that the Cape Ann earthquake had a ground acceleration of 0.18-0.21g. In this thesis the ground acceleration of 0.2292g will be used due to the support of Whitman's hypothetical ground accelerations. 5.1.3 Estimated Base Shear The ASCE 7-10 code provides a method to estimate the base shear values that will be applied to the base of the four wall sections. First the section's fundamental period must be estimated: T = Ctho(1) Where Ct and x are period parameters taken from Table 12.8-2 of ASCE 7-10 under the category of 'All other structural systems' and are 0.02 and 0.75, respectively. The term htot is the total height of the wall section. This value for the period has to be compared to TL, which is the long term period in order to determine which equation to use next. The long term period for the Boston area is 6 seconds. For the height of the case study building, the estimated period will never reach above 6 seconds, so the seismic response coefficient needed will be calculated as: CS= (2) sS And not to exceed: Cs,max - D1 (3) Where T is the fundamental period calculated in Equation 3, R is a response modification factor found in Table 12.14-1 of ASCE 7-10 and is assumed as 1.5 for the unreinforced masonry typology, and Iis an Importance Factor taken from Table 11.5-1 in ASCE 7-10 and is assumed as 1. Table 2 below shows the necessary values needed to calculate SDs and SDI. 30 Table 2 Ground motion and response spectrum expected building base shear. values necessary to deterinine Label Term Description Source Value Fa 0.214 1.2 ASCE 7-10 Figure 22-1 ASCE 7-10 Table 11.4-1 SMS 0.324 Sms=FaSs SDS 0.2292 Spectral Acceleration of MCE at Is period Site coefficient for Soil Class D Spectral Response Acceleration Is period Si F, 0.069 2.4 ASCE 7-10 Figure 22-2 ASCE 7-10 Table 11.4-2 SMI 0.167 Smi=FSi Design Earthquake Response Acceleration at Is period SDI 0.110 SDI=( 2/ 3 )SMI Spectral Acceleration of MCE at short periods Site coefficient for Soil Class D Spectral Response Acceleration for MCE at short periods Design Earthquake Response Acceleration at short periods S, SDS=( 2/ 3 )SMs Finally, the base shear can be calculated by: (4) Fb = CsW Where W is the weight of the building section in kips. This estimated base shear value will be useful for both the cantilever beam and finite element model to have a basis for comparison even though higher base shear values will be applied in these methods for pushover analysis. Cantilever Beam Method 5.2 A conservative method to estimating max P6 roof displacements and interstory drifts of an M6 - P6 - . h6 P5 unreinforced masonry building is to assume -5 - the building as a solid cantilever beam with a P4 --- m, fixed base boundary condition, visualized in Figure 13 and theorized in the textbook h3 Structural Motion Engineering by Jerome P- Connor. In this model, incremental lateral Sh, '- loads are applied to each floor as portions of the expected base shear dictated by ASCE 710 and in the shape of the fundamental mode. Equation 1 determines the proportional value of the base shear for the ith floor: Pi= nmq j=1j jj Fb Fb Fb Figure 13 The con tilever beam model lumps masses at the floor levels where portions of the base shear are applied. (5) 31 Where Fb is the base shear calculated from ASCE 7-10 provisions, mi is the lumped mass of the story, and P is the mode shape term for the ith or jth floor. The value of mi will be the weight of the interstory building portion. The value of Pi is used in Equation 2, which calculate deflection considering shear and bending: AROOF Zj (6) hi4Pi1 + Z.JPhi1 EA Where hi is the height of the ith floor, Pi is the proportional base shear calculated in Equation 1, E is the modulus of elasticity of masonry, I is the moment of inertia of the cross-section of the wall in the bending direction, and A is the shear area of the wall. Finite Element Model 5.3 The finite element program SAP2000 version 17.1.0 was used to model the four wall sections for two reasonable scenarios. The first scenario assumes that the facade is not connected well to the party wall due to mortar degradation and general wear. A broad example of this is in Figure 14. The second scenario considers portions of the party wall and floors to act with the facade. The distinctions between these scenarios will be explored further in Chapter 6. A few assumptions had to be made for the finite element model, just as for the cantilever beam model. Default material properties (Young's Modulus, Shear Modulus) from FEMA 356 and accepted values (Density, Poisson's Ratio, Thermal Coefficient) defined the inputted masonry material. Masonry values and the directional properties are unknown for the - is anisotropic, but since the material is developed from default Young's Modulus, Shear Modulus, Thermal Coefficient, and Poisson's Ratio, it was chosen to be isotropic in SAP2000. There are a few particulars about the model's geometry, section properties, and restraints that were assumed in order to Figure 14 Bad wall to wall connection approach accurate results: 1. To simulate a fixed foundation condition, moment restraints are modeled to line the bottom of the wall sections. 2. Perforations in the wall sections come from case study geometry. 32 3. Frame elements were given no section property, making them just constraints for the masonry panels. 4. The frame elements are chosen to allow load transfer from area objects. 5. Masonry panels are defined as a thick shell area section and is given the thickness of the wall from the case study. 6. Rectangular areas were drawn as piers and spandrels between the window perforations and were given the masonry area section and material properties. 7. During analysis, the masonry area sections automatically mesh into smaller sections to get more accurate deflections. 8. The mass source comes from the defined elements, not an applied dead load. Figure 15 shows the geometry of the four SAP2000 models done for the four wall sections for the first scenario of the facade acting alone apart from the party walls and floor. Pier - Worst Case Curved Wall Section Flat Wall Section Full Wall - Best Case Figure 15 The four wall sections make up four different SAP2000 models to analyze. Analysis cases are set up for the self-weight dead load, applied live load (proportional base shear values calculated for the cantilever beam model), and response spectrum earthquake load. Deflection values are read off at each floor level, as exemplified in Figure 16 and evaluated per the criteria in the next section. 33 Figure 16 An example of the data collection process for the SAP2000 models. Proportional base shear values are applied at the joints at floor level and deflection values are read from the joints on the right side of the model also at floor level. The second scenario that considers certain percentages of the party wall to act with the fagade is represented by the finite element model in Figure 17. Forces will be applied at the same level as in the first scenario but as a uniformly distributed line load. Only in-plane analysis will be studied in scenario 2 since this condition assumes exceptional fagade to party wall connections. Figure 17 SAP2000 models, from right, considering 10%, 50%, and 100% of the building acting together, will be also necessary to analyze. 34 Global Behavior Evaluation Criteria 5.4 The global performance of the four parameterized wall sections will be evaluated by their interstory drift ratios, which is the difference between the deflection of one story and the one below it, divided by the height between the points where the deflection was measured. Many codes provide interstory drift ratio limits to evaluate unreinforced masonry buildings and it is necessary to choose one that is right for these calculations and case study. There is a challenge in choosing the right code to go by since it is unclear as to where they measure their story drifts from. This thesis will measure drifts by measuring the deflection at the estimated location of the timber floor, subtracting the deflection at the timber floor below it, and dividing by the distance between the timber floors. It is suggested in the paper by Park, who also measure interstory drifts this way for low rise unreinforced masonry buildings, to use the interstory drift ratios provided by the FEMA 356 document. FEMA 356 has determined specific interstory drifts for three different performance levels including Immediate Occupancy (10), Life Safety (LS), and Collapse Prevention (CP). Table 3 summarizes the interstory drift limits. Table 3 FEAIA 356 interstorv drift limits for Collapse Prevention, Li/i Sa/e'ty, and Immediaie Occupancy periWrmance Ievels. Damage Levels Drift Ratio []1.00 0i 0.60 immediate Occupancy (10) 0.30 When the drift limit for 10 is met, there is expected to be minor cracking of veneer bricks, minor spalling at corner openings, and no observable out-of-plane offsets. When the drift limit for LS is met, there is expected to be extensive cracking, noticeable in-plane offsets, and minor out-of-plane offsets. Finally, when the drift limit for CP is met, there is also expected to be extensive cracking, but it is also expected that the brick face course and veneer may begin to peel back from the inner wythes of the building, and there is noticeable in-plane and out-of-plane offsets (Park, et al). It is important to note that even though 10 is defined by light damage, small brick failures can cause property damage and loss of life from falling brick pieces. The Back Bay case study interstory building heights are applied to the FEMA 356 drift ratios. Table 4 shows the distance that each story has to drift in order to be categorized in the three performance levels. The last line of the table shows the maximum interstory distance that the full 35 building can stand. These values are the main criteria that will evaluate the global performance of the four wall sections. Table 4 To violate the FEAIA 356 perfrnance levels, these are the interstorv distances that each loor is alloiwed to drult. Immediate Occupancy Damage Leves 10 jS 0.432 0.402 0.452 0.487 Basement [in] Floor 1 [in] Floor 2 [in] Floor 3 [in] 1.440 1.340 1.508 1.622 0.864 0.804 0.905 0.973 Floor 4 [in] 1.412 0.847 0.424 Roof [in] Full Building Height [in] 1.606 0.963 0.482 8.928 5.357 2.678 5.5 Local Behavior Evaluation Criteria In Plane This section will provide an overview of the equations used to calculate the shear capacities of the pier elements of the full fagade, which will act as the failure criteria of the in-plane four failure mechanisms (flexural rocking, bed joint sliding, toe crushing, and diagonal tension cracking). When these stress values, or yield stresses, are met within the pier elements, they will be considered to fail by the corresponding mechanism. Von Mises shear stresses taken from the finite element model will be compared to each of the pier yield stresses. A Von Mises stress is the combination of the three dimensional principle stresses, taken from Mohr's Circle, within a material to create an equivalent stress (CSI 2014). As a measure of distotional stresses within the material, it is assumed in this thesis that reading Von Mises shear stresses from the finite element model will be an adequate representation of the stresses within the masonry piers. By comparing the acquired Von Mises stresses to the masonry piers' shear capacity, it will be possible to determine if the piers will fail at a reasonable application of base shear. These results are in Chapter 6. Flexural rocking failure of the fagade piers is calculated in Equation 7: Vr = 0. 9 aPCE(L/heff) 36 (7) Where a is a factor that is 0.5 for a fixed-free cantilever and 1.0 for a fixed-free pier, PCE is the expected vertical axial compressive force acting upon the pier, L is the length of the wall, and heff is the effective height of the pier. The shear to cause bed-joint sliding in a facade pier is calculated in Equations 8 and 9: VbJs1 = Vbjs2 (8) VmeAn = VfrictionAn = 0 . 5 PCE (9) Where VbjsJ is bed joint sliding shear considering mortar bond and friction, Vbjs2 is bed joint sliding shear considering only friction, vine is the shear strength of the unreinforced masonry components assumed as the default value 27psi for bed joints in good condition per FEMA 273, and A, is the area of the net mortared section. Toe crushing in the fagade piers is calculated from the shear Equation 10: Vic = aPCE (L/heff)( - (10) fae/ 0. 7f'me) Where fae is the expected vertical axial compressive strength also known as PCE, and f'me is the expected masonry compressive strength. Finally, the shear to cause diagonal cracking in the facade piers will use Equation 11: Vdt = f'dtAnf3(1 + fac/f'dt)1/2 Wheref'dt is the diagonal tension strength and p is 0.67 for L/heff<0.67, L/heffwhen 0.67>L/he)Yl .0, and 1.0 when L/hegf>1 (FEMA 1998). Out of Plane The elements that will be evaluated for out-of-plane behavior are chimneys, full facade, and partial fagade. Masonry chimneys are considered to be the most vulnerable element of an unreinforced masonry building and damage to them are one of the ways that seismologists determine the Modified Mercalli Intensity (MMI) after an earthquake. Failure of chimneys are is crucial because bricks falling from such a height can damage other parts of the building and injure people below. The global out of plane behavior for the case study will be evaluated using the basic rocking rigid block method. b 0 Y" Figure 18 Rigid block free-body diagram (Dejong). 37 A facade or a chimney can be reimagined as Housner's rigid block theory. For the simplified rigid block calculations it is assumed that the connections of these blocks to the roof, other parts of the wall, the ground, etc. are flexible between two rigid bodies. At certain forces (ground acceleration, wind load), the block can then overcome its overturning moment at a critical angle of rotation and rock, or it can overcome friction at the base and slide. Figure 18 shows the free body diagram of a rigid block. By summing the moments about the point 0, a parameter to calculate the minimum ground acceleration to overturn the block can be determined: -gmin = - tan a 9 Where A is a dimensionless uplift parameter, Uginin (12) is the minimum ground acceleration for overturn, g is the force of gravity, and a is the critical angle. Rocking motion will be triggered if { is greater than tan(a). Therefore the inception of rocking in this case is completely dependent on the block's geometry (DeJong). In Chapter 6, the rigid block method will be applied to the case study chimney and fagade. Additionally, if there is no overturn of the present conditions, then the failure state will be calculated. As will be seen in Chapter 6, the rigid block theory is slightly inconclusive for chimney analysis. Therefore, the chimney will also be analyzed locally for shear failure instead of complete overturn. This method is suggested by Alan Darrell Ho, a student of Robert Whitman and also author of "Determination of Earthquake Intensities from Chimney Damage Reports". This paper analyzes the chimney capacities of those during the Cape Ann earthquake of 1755. There are a few assumptions when considering a simple analysis of shear failure. First, the bond stress at the failure location is zero, and second, the shear resistance is due to friction developing at the horizontal failure plane between the brick and mortar. The equation for shear failure is: V = PW (13) Where p is the coefficient of friction, assumed as 0.7 which is seen in literature, and W is the total weight of the chimney above the failure location. When applied forces exceed V, the friction force, the bricks will slide and be considered to fail (Ho 1979). The longer wall of the chimney will be analyzed with this method since its out of plane bending will be bending about the weak axis. 38 Another form of out of plane failure is when the brick wall between two floors cracks at the floor interfaces and in the middle of the masonry panel and bursts outward. There is a method in a paper presented by the University of Canterbury of calculating the minimum ground acceleration that would be necessary to crack this portion of a wall in the middle. The moment required to crack the unreinforced masonry wall is: (14) Mcr = Where t is the thickness of the wall and R is sum of the compressive force from the wall portion above and half of the wall at hand's weight. The next two equations are combined to simplify to Equation 17. 8 MCr win = ain = m (15) (16) Where win is the mass per unit area of wall surface, h, is the interstory wall height, m is the mass of the wall section, and ain is ground acceleration. 4Rt a=n = (17) The calculated acceleration is the minimum acceleration required to crack the wall mid-span (Priestley 1985). Results for this calculation are in Chapter 6. 39 6.0 Case Study Structural Analysis Results Structural assessment results of an unreinforced masonry case study building in the Back Bay neighborhood will be presented. The case study building, 37 Commonwealth Avenue, is one of several residences in all of the Back Bay that original structural drawings were acquired. This availability as well as the midblock location of the residence, and its similar building typology to most homes in the neighborhood, led to its selection. Before any analysis began, all available information about 37 Commonwealth was found. This included original structural drawings found at the Boston Athenmum, and long form building permits from the City of Boston, which provided information on building construction over the years and building materials. Site visits to similar buildings and 39 Commonwealth in the Back Bay led to the understanding of certain building components and connections. The residence of 37 Commonwealth is located in a block confined by Berkeley Street, Clarendon Street, Public Alley 423, and Commonwealth Avenue. Constructed in 1872 by architect N.J. Bradlee, who built 39 Commonwealth at the same time, this five story unreinforced masonry building is 74 feet from the bottom of the basement to the highest point of the roof, the facade is approximately 30 feet wide, and the party walls are approximately 75 feet long. The thickness of the external walls is 16 inches, the floors are made of wood, and the foundation is Figure 19 Front and rearfagades of 37 Commonwealth 40 granite. Figures 20 shows some of the original structural drawings of 37 Commonwealth and Figure 21 shows the wall brick course, vaulted basement flooring, and joist pocket holes of sister structure 39 Commonwealth. Figure 21 Original structural drawings of 37 Commonwealth (Boston Athenaeum N. Bradlee Collection). Figure 20 Counterclockwise from left, interior brick courses, vaulted brick basement floor, and joist pockets of 39 Commonwealth 41 6.1 Modelling Considerations It is assumed in the structural analysis of 37 Commonwealth that the building acts independently of its surrounding buildings. The aggregation of the housing block is predicted to stiffen the internal buildings under lateral loads and to not allow them to deflect as much as will be shown in this chapter. Since complex finite element models take a long time to run, the behavior of the unreinforced masonry buildings as an aggregate block is therefore out of the scope of this thesis for ease of calculation. Further, in order to explore how and if portions of the building act together, two behavior scenarios will be explored. The first scenario is to model the facade as if it was acting on its own, assuming that the wall to wall connection is in a bad condition and doesn't transfer lateral loads. Within the fagade analysis is a parametric study that will examine how different sections on the wall will behave if acting on their own due to possible masonry connection deterioration within the fagade. Four wall sections will be analyzed for their individual behavior including: the full facade (best case), a partial curved section of facade, a partial flat section of fagade, and a pier section through the windows (worst case). The 37 Commonwealth building has a complex fagade geometry and for calculations and will be simplified to act on one plane. The four parameters will be modeled similarly. A summary of this modeling simplification is in Figure 22. Pier - Worst Case Flat Wall Section Curved Wall Section Wall Plan View Simplification Used For Calculations Figure 22 Modeling simplificotions for scenario 1. 42 Full Wall - Best Case The second scenario is to model a certain percentage of the party wall and floor diaphragms as acting together with the fagade, assuming that the wall to wall Wall Plan connection is in excellent condition and is creating a box effect. This scenario is only applicable for the best case, full wall parameter in Figure 23 since the Simplification U other parameters don't have party wall connections. This application will not take into account the stiffness of the floor, just the weight. 6.2 Cantilever Beam Results The cantilever beam method described in Chapter 4 was applied to the four scenarios presented in the previous section. For the first scenario of the fagade Figure 23 Modeling simplification for scenario 2. acting apart from the party walls, of the four isolated wall sections only the pier section violated the FEMA 356 performance levels. The most conclusive results for the cantilever beam model come from graphing base shear vs. max roof displacement over total building height. The drift results are depicted in Figure 24. The FEMA 356 drift limits are represented on the graph as the vertical differently colored lines corresponding to their colors in Table 4. PUSHOVER CURVE (CANTILEVER BEAM METHOD) 140 0 I I I I I I I I I 120 0 I 100 0 k 800 -*-Full Wall Curved -Ar- Flat - Pier I (A LU I 60 0 (A La I 40 0 I I I I 20 0 0 0.000 0.002 0.010 0.008 0.006 0.004 MAX ROOF DISPLACEMENT/TOTAL HEIGHT [IN/IN] Figure 24 Pushover curve for the cantilever beam hand calculations. 43 0.012 0.014 The pier section, when acting on its own, will violate all of FEMA 356's drift ratio limits at approximately 172 kips for 10, 354 kips for LS, and 596 kips for CP. It is important to notice, however, that the magnitude of the base shear values on the y axis compared to the expected base shear values of the four parameters in Table 5, which were calculated from ASCE 7-10 provisions. Tai/c 5 Expecctcd base sharJor wall scctions in scenarioI Parameter and corresponldingi max ivaf dicmns ASCE 7 Expected Base Shear [kips] Max Roof Displacement [in] 40.74 17.60 20.71 7.50 0.004 0.010 0.013 0.112 Full Facade Curved Section Flat Section Pier Varying percentages of the party wall and floor load from zero to fifty percent were chosen to act in accordance with the fagade. To account for this, estimated floor weights were added to the weight of the masonry stories and the cross-sectional area and moment of inertia represented the 'C' shape. The whole building was represented as the one hundred percent value. The results of considering these varying percentages are in Table 6. Tailc 6 Expected base shcar bor wall sections in scepnario 2 and correspondingmax ra! dmspacmena. Percentage 10 20 ASCE 7 Expected Base Shear [kips] 87.23 119.47 Max Roof Displacement [in] 0.0021 0.0029 30 151.72 183.97 216.21 377.47 0.0036 0.0044 0.0052 0.0090 40 50 100 Max roof displacement for the full fagade and expected base shear increased as more of the building was taken into account. The interstory drifts in this scenario were nowhere near the 10 performance level and are not shown. Although the cantilever beam model represents the displacement behavior of the case study building and shows that the pier section in the first scenario, when acting on its own, will violate all of FEMA 356's performance levels, it is necessary to continue with the finite element results 44 in order to represent better the perforated quality of the fagade and the structural nature of the masonry. 6.3 Finite Element Model Results Scenario 1 Again, the building facade was modeled as four different sections, assuming that there is a possibility that portions of the wall can be acting on their own due to bad wall to wall connections. These four models are considered to be a worst case scenario situation because when percentages of the party walls are added in scenario 2, the interstory drifts are smaller. Overall, when force was applied to the floors of the four facade sections, proportional to increasing base shear values whose calculation were outlined earlier, the building deflected more, which is expected and inherent. The most important takeaway from the finite element analysis is determining at what base shear values the wall sections violate the interstory drift limits from FEMA 356. If this happens at realistic base shear values, the building will be considered to be at risk. For all of the following graphs, the expected base shear for the sections are represented by a colored single data point on each of the curves. PUSHOVER CURVE (SCENARIO 1) 450 I- 400 Full - 350 Curved Flat Pier 300 20 200 co 150 100 50 0 0.0000000 0.0020000 0.0080000 0.0060000 0.0040000 MAX ROOF DISPLACEMENT/TOTAL HEIGHT [IN/IN] Figure 25 Pushovercurve f-om SA P2000/brscenario 1. 45 0.0100000 Figure 25 begins with a broad observation of the sections' behavior. The max roof displacement at increasing values of base shear is graphed. This observation considers the interstory distance to be the full height of the building. The model was run until each wall section violated all of the interstory drift limits. Table 7 provides a detailed reading of Figure 25 and documents at what base shear each interstory drift limit will be met. Table 7 Base shear va/ue.s ncessarv ro exceed FEMA 356 perfiurmance Ieve/s for the uveral pushover curve. Base Shear at CP [kips] 1400 Full Facade 40.74 417 Base Shear at LS [kips] 860 Curved Section 17.60 41.8 83.2 138 Flat Section 20.71 44.7 88.6 147 Pier 7.50 1.8 3.5 6 Parameter ASCE 7 Expected Base Shear [kips] Base Shear at 10 [kips] If the pier is acting alone, it is likely that it will meet all of the drift limits. The other wall sections have expected base shear values that are fairly far off from the shear values needed to violate even the immediate occupancy drift limit. It is necessary to analyze what is happening at each floor rather than the global building drift since wall perforations and amount of compressive stress on the wall section can affect how much a story can drift. 46 BASEMENT INTERSTORY DRIFTS (SCENARIO 1) 600 500 400 300 IFull IU Wall <--Curved - 200 Flat Pier 100 0 0 0.2 0.6 0.4 1 0.8 1.2 1.4 1.6 1.8 2 INTERSTORY DRIFT [IN] Figure .6 Base shear vs. interstorv driftfor the basement level. Figure 26 shows the interstory drifts between the ground and the basement floor with increasing values of base shear. The base shear value that each drift limit will be met for the four wall sections of the basement is in Table 8. Table 8 Base shear valiues necessarv to exceed FEMA 356 perjbrmance levels f.r basement level. Base Shear at LS [kips] >2000 Base Shear at CP [kips] >2000 Full Facade 40.74 Base Shear at 10 [kips] 1200 Curved Section 17.60 190 377 620 Flat Section 20.71 143 321 550 Pier 7.50 6.2 13 21.9 Parameter ASCE 7 Expected Base Shear [kips] If the pier acts on its own, the shear values for the drift limits are reasonably close to the expected base shear for the expected earthquake event. 47 FLOOR 1 INTERSTORY DRIFTS (SCENARIO 1) 500 450 400 Full Wall Curved -Flat Pier --- 350 300 250 L 200 150 100 50 0 0 0.5 1 1.5 3 2.5 2 INTERSTORY DRIFT [IN] 3.5 4 Figure 27 Base shear vs. interstorv drift for the first floor. Figure 27 shows the interstory between the basement and first floor with increasing base shear. The drift limits are exceeded at much lower base shear values compared to the basement drift, which is expected since the basement is confined underground, it has no perforations, and is substantially heavier than the first floor masonry wall panel. Also, the basement has the compressive force of five stories above it whereas the first floor has only 4. The base shear value that each drift limit will be met for the four wall sections of the first floor is in Table 9. Table 9 Base shear values necessary to exceed FEAMA 356 per/brmance levels brfirst floor. ASCE 7 Expected Base Base Shear at Base Shear at Base Shear at 10 [kips] 420 LS [kips] 800 CP [kips] 1438 Curved Section 17.60 60 122 202 Flat Section 20.71 54 110 183 Pier 7.50 2.8 5 8.3 Parameter Full Facade Shear [kips] 40.74 Once again, if the pier acts on its own, the shear values for the drift limits are reasonably close to the expected base shear for the expected earthquake event. 48 FLOOR 2 INTERSTORY DRIFTS (SCENARIO 1) 500 450 400 I I 350 S300 L 250 Wall Curved Ln---Full V 200 -Flat Pier - 150 100 50 0 0 1 0.5 2 1.5 3 2.5 4 3.5 INTERSTORY DRIFTS [IN] Figure 28 Base shear vs. interstorV drift for the second floor Figure 28 shows the interstory drifts between the first and second floor with increasing base shear. Compared to the first floor, the drift limits have been exceeded at lower base shears, especially the full wall section which has previously been violating the drift limits at base shear values that would never realistically happen. The base shear value that each drift limit will be met for the four wall sections of the second floor is in Table 10. Table 10 Base shear values necessarY to exceed FEMA 356 perfbmzance levelsfbr second floor. Base Shear at LS [kips] 645 Base Shear at CP [kips] 1165 Full Facade 40.74 Base Shear at IO [kips] 340 Curved Section 17.60 38 81.1 130 Flat Section 20.71 41 85 141 Pier 7.50 2 3.8 6 Parameter ASCE 7 Expected Base Shear [kips] The flat and curved wall sections are approaching reasonable base shear values at which they violate the drift limits. The pier section is approaching incredibly low base shear values that it is considered to 'fail' at. 49 FLOOR 3 INTERSTORY DRIFTS (SCENARIO 1) 500 450 400 I-I 350 30 300II - Full Wall --- Curved - Flat -- Pier 250 CA 200 150 100 50 0 0.5 1 2 1.5 3 2.5 4 3.5 INTERSTORY DRIFT [IN] Figure29 Base shear is. interstory drift Jbr the third floor Figure 29 shows the interstory drifts between the second and third floor with increasing base shear. The results don't change much from the second floor for the full wall section, but the flat, pier, and curved sections were affected by the story change. The base shear value that each drift limit will be met for the four wall sections of the third floor is in Table 11. Table II Base shear values necessary to exceed FEMA 356 perbrmance levels for third floor. Base Shear at LS [kips] 770 Base Shear at CP [kips] 1280 Full Facade 40.74 Base Shear at 10 [kips] 360 Curved Section 17.60 33 70 113 Flat Section 20.71 36 74 120 Pier 7.50 0.6 3 5 Parameter ASCE 7 Expected Base Shear [kips] The curved and flat sections continue to have drift limit shear values that are closer to the expected base shear as higher floors are analyzed. The pier reaches immediate occupancy at a staggering 600 pounds of force. 50 FLOOR 4 INTERSTORY DRIFTS (SCENARIO 1) 500 450 Full Wall Curved Flat --Pier - 400 350 300 S250 Z200 150 100 50 0 0 0.5 1 1.5 3 2.5 2 INTERSTORY DRIFT [IN] 4 3.5 Figure 30 Base shear vs. interstory drift fbr the fourth floor Figure 30 shows the interstory drifts between the third and fourth floor with increasing base shear. These results are close to those for the third floor. The pier base shear for the drift limits did not decrease again but instead increased, albeit to values that are still incredibly low. The base shear value that each drift limit will be met for the four wall sections of the fourth floor is in Table 12. Table 12 Base shear values necessary to exceed FEMA 356 pertormance levels fbrJburthfloor. Base Shear at LS [kips] 745 Base Shear at CP [kips] 1255 Ful Facade 40.74 Base Shear at 10 [kips] 372 Curved Section 17.60 30.2 62 102 Flat Section 20.71 34.3 69.5 112 Pier 7.50 1.1 2.3 4 Parameter ASCE 7 Expected Base Shear [kips] The curved and flat sections continue to have drift limit base shears that are decreasing as floor height increases. It becomes more reasonable to say that these sections are likely to 'fail' to a certain degree closer to the roof which the pier section is likely to 'fail' to a certain degree at any floor. 51 ROOF INTERSTORY DRIFTS (SCENARIO 1) 500 450 400 - Full Wall 350 - Curved Flat Pier 300 250 200 150 100 50 0 0 0.5 1 2 1.5 2.5 3 3.5 4 INTERSTORY DRIFT [IN] Figure 31 Base shear vs. interstorv drift fbr the roof level. Figure 31 shows the interstory drifts between the roof and the fourth floor with increasing base shear. Compared to all floors below, the roof level has the largest interstory drifts at the lowest values of base shear. The full wall section still doesn't 'fail' at a reasonable value, but the pier section does especially. It is reasonable to predict that the flat and curved sections have the possibility of producing a low level of failure at the roof level. Again, it is important to remember that even very small pieces of wall that fail can cost a lot of money to repair and also injuries to people. The base shear value that each drift limit will be met for the four wall sections of the roof floor is in Table 13. Table 13 Base shear values necessary to exceed FEMA 356 performance levels for roof level. ASCE 7 Expected Base Shear [kips] 40.74 Base Shear at 10 [kips] 340 Base Shear at LS [kips] 680 Base Shear at CP [kips] 1175 Curved Section 17.60 33.6 67 102 Flat Section 20.71 30 60 113 Pier 7.50 0.7 1.2 4.1 Parameter Full Facade 52 Scenario 2 When considering portions of the party walls and floors to act in accordance with the fagade, the expected base shear is higher and the deflections are less. Three models were analyzed considering different percentages of the full building acting with the fagade including 10%, 50%, and the full 100%. It is less common in this scenario for the FEMA 356 interstory drifts to be surpassed. The results for this scenario will not go into detail as much as the first scenario did since most of the graphs are very similar and tell the same story about interstory drifts. In Figure 32 the overall behavior of the three percentages is shown when the interstory height is the full height of the building using the cantilever beam hand calculation method. It is observed that in this method, none of the interstory drift limits are even close to being met at reasonable or high base shear values. 2) PUSHOVER CURVE (CANTILEVER BEAM SCENARIO 3000 2500 2000 -100% 1500 -50% < 1000 500 0 0 0.00005 0.0002 0.00015 0.0001 IN/IN] HEIGHT MAX ROOF DISPLACEMENT/TOTAL 0.00025 Figure 32 Pushover curve from the cantileverbeam hand calculationsfbr scenario 2. Figure 33 displays the same type of information (base shear vs. interstory height/total height) but from the finite element model that is a more accurate representation of the structural behavior of brick masonry panels. The Immediate Occupancy drift limit is met, but for very highly unexpected base shear values. 53 PUSHOVER CURVE (SAP2000 SCENARIO 2) 3000 2500 2000 1500 LU -- 100% 50% -- 10% - < 1000 500 0 0.006 0.005 0.004 0.003 0.002 MAX ROOF DISPLACEMENT/TOTAL HEIGHT [IN/IN] 0.001 0 Figure 33 Pushover curve from SAP2000 /br scenario 2. Figure 34 begins the floor by floor analysis. Between the ground and the basement ceiling, the interstory drift limits are met for the 10% and 50% models at very high base shears. BASEMENT INTERSTORY DRIFTS (SCENARIO 2) 3000 2500 2000 u 1500 < 1000 -- 500 100% 50% -10% 0 0 0.05 0.1 0.15 0.3 0.25 0.2 INTERSTORY DRIFT [IN] 0.35 0.4 0.45 0.5 Figure 34 Base shear vs. interstory drift for the basement level. Figure 35 shows that slowly, as higher floors are analyzed, the interstory drifts get larger. In this case for the drift between the basement and the first floor, the Immediate Occupancy limit is met for the 10%, 50% and 100% models. The Life Safety limit would have been met if higher shear values were inputted, but since they are deemed unrealistic it is determined that the Life Safety limit won't be met. 54 FLOOR 1 INTERSTORY DRIFTS (SCENARIO 2) 3000 2500 2000 1500 LU 100% - < 1000 ca--50% -10% 500 0 0.2 0.1 0 0.3 0.6 0.5 0.4 INTERSTORY DRIFT [IN] 0.8 0.7 0.9 1 Figure 35 Base shear vs. interstor drift/fbr the firstfloor. The results for the drift between the first and second floor, in Figure 36, are very similar to the above results in Figure 35. However, the Life Safety limit is met at a lower value of base shear for the 10% model and all of the models surpass the Immediate Occupancy limit under a base shear of 2500 kips. FLOOR 2 INTERSTORY DRIFTS (SCENARIO 2) 3000 2500 I 2000 2 1500 < 1000 c---100% 500 -10% 0 0 0.1 0.2 0.3 0.5 0.4 0.6 0.7 0.8 0.9 1 INTERSTORY DRIFT [IN] Figure 36 Base shear vs. interstorv drift/brthe second floor As in scenario 1, the drifts between the second and third floors are not as much as the drifts between the first and second floors, which is due to a combination of vertical compressive stress and perforation amounts. Figure 37 displays this decrease in the interstory drifts. 55 FLOOR 3 INTERSTORY DRIFTS (SCENARIO 2) 3000 2500 _ 2000 1500 M --- < 1000 100% --- 50% -- 10% 500 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 INTERSTORY DRIFT [IN] Figure 37 Base shear vs. interstory drift/for the thirdfloor. The interstory drifts for the next story are larger, however, as seen in Figure 38. Even though this is true, the base shear values that it takes to violate the drift limits would be incredibly difficult to achieve in real life. FLOOR 4 INTERSTORY DRIFTS (SCENARIO 2) 3000 2500 0 2000 1500 < 1000 co---100% 050% -10% 500 0 0 0.1 0.2 0.3 0.4 0.5 0.6 INTERSTORY DRIFT [IN] 0.7 0.8 0.9 1 Figure 38 Base shear vs. inlerstory driftfor the fnirth floor. Unlike the drifts between the roof and the fourth floor for the first scenario where the fagade is acting on its own and the pier section fails at very low base shear values at the roof, the results for the second scenario suggest that at the roof level it is very unrealistic for the FEMA 356 limits to be broken. Figure 39 shows this. 56 ROOF INTERSTORY DRIFTS (SCENARIO 2) 3000 -100% 2500 a. --- 50% 10% 2000 1500 < 1000 500 0 0 Figure 39 Base shear vs. 0.2 0.6 0.4 0.8 1 INTERSTORY DRIFT [IN] interstorvdrift jbr the roof level. 6.4 Local Element Failure Results In-Plane Failure Results As introduced in Chapter 3, unreinforced masonry typically fails in four different ways. Three of these failure mechanisms (flexural rocking, bed-joint sliding, and toe crushing) are not really considered to be failures since they still allow vertical loads to transfer throughout the facade. The diagonal cracking failure mechanism however does not allow vertical loads to transfer after its inception and is considered to be a failure. The case study of 37 Commonwealth was analyzed to see which in-plane local failure mechanisms would occur in the faqade's pier elements. It is assumed that this building exhibits a weak pier, strong spandrel condition due to the pier and spandrel geometries and therefore the failures are expected to occur in the piers. 57 Figure 40 Most of the piers willfailfrom diagonal tension failure in the SAP2000 models. Bed joint sliding is observed in the full wall and rocking is observed in the curved wall section. The expected failure mechanisms of the facade piers using the equations in Chapter 5 are seen in Figure 40 and are determined from provisions in FEMA 306, which typically takes the smallest shear value from the equations to be the failing shear mechanism. In Figure 40, the green represents diagonal tension failure, red represents bed-joint sliding failure, and the blue represents rocking failure. These results rely heavily on pier length to effective height ratios, vertical compressive stress, and strength of mortar joints. Diagonal tension failure appears to be the most frequently occurring failure mechanism for many of the piers with small length to height ratios (L/heff). The larger piers, with length to height ratios closer to one are predicted to fail first from bed joint sliding. One pier is expected to fail from flexural rocking. Now that the shear capacities of the piers are known, it will be determined whether these shear values are met during the various applications of base shear loads. The Von Mises shear stresses observed in the full rear fagade will be compared to the shear capacities to determine at what value of base shear the wall elements will be expected to fail. 58 Results for this section will only focus on scenario 1, where the fagade is disconnected from the party walls and is acting on its own. This is because it is considered to be the worst case scenario. First the Von Mises shear stresses were found for the full fagade. In Figure 41, along the x axis are the pier numbers that were designated in Figure 40. The data points above the pier numbers represent the Von Mises shear stress values read from the finite element model caused by the application of the expected base shear and increasing base shear values. The black diamond shaped data point represents the shear stress capacity of the pier element. The red dotted line represents the compressive strength of the brick. When this value is exceeded the brick fails. VON MISES INTERNAL SHEAR STRESS OF PIERS - FULL WALL 1200 X 1000 0- . LU 800 COMPRESSIVE......... STRENGTH OF BRICK=78OPSI ................................................. ....... V ................................... x X 600 400 2 400x 0 X x 200 X x x x X 6 7 x X 13 14 x x Xxx X X X X X x 0 0 1 2 3 4 5 9 8 10 11 12 15 16 17 18 19 20 21 22 DESIGNATED PIER NUMBER APPLIED BASE SHEARS A Expected Base Shear=40.74 kips X50 kips X 100 kips X250 kips X500 kips *Shear Failure Figure 41 Local shearfJilure examination for fidl wall. The pier elements numbered 16 and 21 are the first to exceed their shear stress capacity and fail, around 175 kips of applied base shear. This is expected because the piers are compressed significantly as the building bends. At 250 kips base shear, piers 16 and 21 are still the only ones to have failed, but when that value is doubled to 500 kips, piers 12, 15, 17, 18, and 20 fail. By the time 1500 kips base shear is applied, all but piers 2, 4, and 6 have failed. Even though the piers are 59 considered to fail at these points, the applied base shears are still significantly higher than the expected base shear. The data points for 1000 and 1500 kips base shear were removed for graph clarity and can be found in the Appendix. Internal shear stresses for the flat wall section are in Figure 42. The piers were not renumbered and are the same as for the full wall section, just with some of the piers missing since they aren't included in the flat wall section. VON MISES INTERNAL SHEAR STRESS OF PIERS - FLAT SECTION 1000 900 X 800 COMPRESSIVE STRENGTH OF BRICK=780PSI 700 u-i x 600 u- x 500 U) L/) 400 z 0 300 x 200 x x I I 5 6 100 0 0 1 2 3 4 A x X X x X X X X A X X A 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 DESIGNATED PIER NUMBER APPLIED BASE SHEARS *Shear Failure X 10 kips A Expected Base Shear=21 kips X 50 kips X 60 kips X 80 kips X 90 kips 100 kips Figure 42 Local shear failure examination for flat wall section. The first pier element to fail is pier 19, which exceeds its shear stress capacity at the expected base shear value, just barely. It is the only pier to fail until the application of 100 kips base shear when piers 17 and 18 are seen to have failed. At 175 kips application, piers 13 and 14 have failed, though by interpolating from the data given in the Appendix, they fail somewhere between 100 and 175 kips. By the time 250 kips is applied, pier 9 has also failed. Even though the piers are considered to fail at these points, the applied base shears are still significantly higher than the expected base shear. The data points for 175 and 250 kips are not represented in Figure 42 but are found in the Appendix. 60 Internal Von Mises shear stresses for the curved wall section are in Figure 43. Just as with the flat wall section, the piers were not renumbered, some were simply excluded. VON MISES INTERNAL SHEAR STRESS OF PIERS - CURVED SECTION 1200.00 x X 1000.00 x V-) 800.00 COMPRESSIVE STRENGTH OF BRICK=780PSI xx VI) LU Ln x x x xX 600.00 x W x x x x x 0j x x 400.00 x x A 20 21 A 6x 200.00 x A A A + A 0 1 2 3 4 5 6 7 8 9 10 A A A 0.00 11 12 13 14 15 16 17 18 19 22 DESIGNATED PIER NUMBER APPLIED BASE SHEARS * Shear Failure A Expected Base Shear=17.6 kips X 50 kips X 60 kips X 80 kips X 90 kips X 100 kips Figure 43 Local shearfiilure examinationfor the curved wall section. The first pier failure, occurs between the application of the expected base shear and 50 kips, approximately halfway at 34 kips, which is about double the expected base shear. The piers to fail around this point are 11, 16, 19, and 21. It is not until 90 kips base shear that piers 14 and 20 fail. Once again, even though the piers are considered to fail at these points, the base shears that are causing pier failures are still significantly higher than the expected base shear. Last, internal Von Mises shear stresses for the long pier section of the wall (not to be confused with the pier element denotation), considered to be the worst case, are in Figure 44. 61 VON MISES INTERNAL SHEAR STRESS OF PIERS - LONG PIER 2000 X 1800 1600 1400 x Ln LU 1200 C: 3 X 1000 I (A Z 0 80 800 x COMPRESSIVE STRENGTH OF BRICK=780PSI ...... .......... X........................................ 0..................................................... x 600 X x 400 * X A 200 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 DESIGNATED PIER NUMBER APPLIED BASE SHEARS * Shear Failure X 10 kips X 15 kips X 20 kips X 40 kips A Expected Base Shear=7.5 kips Figure 44 Local shearfailure examinationJbr the long pier section of the fiiade. A pier element does not fail at the expected base shear of 7.5 kips, although pier 20 is very close, a mere 15 psi away. Increasing to applying 10 kips base shear, pier 20 definitely fails. At 15 kips, pier 15 is seen to have failed although it failed somewhere in-between 10 and 15 kips around 11 kips. By 20 kips, pier 11 fails and by 40 kips, pier 7 fails. The base shear values that causes shear failure within the elements is higher than the expected base shear, but not by much. Therefore it is possible that if the pier is acting on its own due to load paths and bad masonry connections, it is likely to fail locally except for the pier element in the roof level. 62 Out-of-Plane Failure Results The rigid block method is first applied to the full fagade. This situation assumes that the fagade is disconnected from the party walls and floors, as speculated in the aforementioned scenario 1. Table 14 shows the parameters required from Figure 18. Tab/c 14 Terns necded for rigid block analvis olidln/Igade. Height [ft] 74.23 Length Ift] 29.46 tw [ft] 1.33 a [radians] a [degrees] R W [lb] W [kips] Stabilizing Moment [kip-ft] 0.018 1.03 37.12 221311.12 221.31 147171.89 Table 15 applies Equation 12 to examine at which ground acceleration the block will overturn at. Table I5 Stability results for ./id afi ade rigid block calculation. REAR FACADE ing,min [g] 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 0.17 0.175 0.178 0.18 X 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.015 0.017 0.018 0.0181 0.0183 tan(a) Stability 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 0.018 STABLE STABLE STABLE STABLE STABLE STABLE STABLE STABLE STABLE STABLE STABLE STABLE STABLE OVERTURN OVERTURN 63 This method shows that for the full fagade to overturn it would take a ground acceleration of 0.1 78g, which is less than the force of gravity expected for the Back Bay. If the facade becomes cracks between the floors, the smaller portion can overturn as well. The interstory wall section between the roof and fourth floor is 13.4 feet high. Table 16 shows what it takes for this section to overturn. Table 6 Stability results ofroof wall. ROOF LEVEL FACADE fig,min [g] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.975 1 X 0.010 0.020 0.030 0.041 0.051 0.061 0.071 0.082 0.092 0.099 0.102 tan(a) Stability 0.099 0.099 0.099 0.099 0.099 0.099 0.099 0.099 0.099 0.099 0.099 STABLE STABLE STABLE STABLE STABLE STABLE STABLE STABLE STABLE OVERTURN OVERTURN The calculation suggests that it would take a little less than on e time the force of gravity to overturn the shorter wall. The smaller the angle of a, the harder it is for a wall to overturn. Therefore if this method is used for parapet analysis it will only show that a parapet will overturn at larger ground accelerations. It is acceptable then to also examine the seismic vulnerability of the interstory wall sections in terms of mortar strength and subsequent cracking. The chimneys will find a similar result if the actual thickness of the chimney is used, as seen in Figure 45. By using this method and assuming that the height of the chimney is 15 feet, the chimney is expected to overturn at 0.87g. But the chimney is hollow, so if it is speculated that the mortar to masonry connections are in a bad condition, it is possible that Figure 45 Case study chimney 64 the walls of the chimney can act alone. In this case, the thickness of the wall is one wythe of bricks (Ho 1979). The bricks in the case study are 3.63 inches thick each and the chimney is 15 feet high. Table 17 shows the results for this condition. Table 17 Stability analysis o/chinnev that is one ythC thick. CHIMNEY WEAK AXIS fir,min [g] 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.010194 0.012232 0.014271 0.01631 0.018349 0.020387 0.022426 tan(a) Stability 0.0201 0.0201 0.0201 0.0201 0.0201 0.0201 0.0201 STABLE STABLE STABLE STABLE STABLE OVERTURN OV RTURN If the chimney is one wythe thick, is expected to overturn around 0.22g, which is smaller than the also be expected acceleration of 0.2292g. As with the facade section, the out of plane failure will analyzed by estimating local shear failures as outlined in Chapter 5. Values of shear failure and corresponding ground acceleration in terms of g at different distances from the top of the chimney are in Table 18. Table 18 Shear capacities at dieflirent failure plane heighls. Distance From Top of Chimney Ift] 0.5 Shear Failure [lb] 160.62 Acceleration to Fail [g] 0.023 1.0 321.25 0.047 1.5 481.87 0.070 2.0 642.50 0.093 2.5 803.12 0.117 3.0 963.74 0.140 3.5 1124.37 0.163 4.0 1284.99 4.5 1445.61 0.210 5.0 10.0 15.0 1606.24 3212.48 4818.71 0.233 0.467 0.700 65 This analysis determined that the top third of the chimney will fail in shear if the maximum considered earthquake (0.2292g) occurs. The minimum ground acceleration to crack interstory fagade portions, whose method is presented by Priestley, are displayed in Table 19. Note that this behavior does not take into account wall perforations but the calculated interstory mass does. Table / 9 Aiinimuni ground accelerationto crack interstorv fagade portions. Wall Thickness [ft] Roof 1.33 Floor 4 1.33 Floor 3 1.33 Floor 2 1.33 Floor 1 1.33 Interstory Height [ft] Wall Length [ft] Interstory Mass [ft] R [ki s] 13.38 29.46 50.42 25.21 11.77 29.46 43.01 46.72 13.52 29.46 49.49 71.46 12.56 29.46 44.97 93.95 11.14 29.46 35.95 111.92 To middle of the wall between the fourth floor and roof will start cracking at a mere ground acceleration of 0.25g. The expected peak ground acceleration for the soil type in the Back Bay is 0.2292g. It is incredibly reasonable that during the maximum considered earthquake that this wall section can crack. Although wall cracking is not considered a failure because it can still transfer vertical loads, continued ground motion can cause the cracked section to start crushing at the floor interfaces from continuous rocking and eventually steps out of plane. In the event of the maximum considered earthquake, if this crack occurs and is not fixed, an aftershock or additional earthquake won't have to exert force to cause the cracking and brick crushing at the floor interfaces will begin immediately. 6.5 Analysis Model Conclusions There are a few major conclusions that can be made from the analysis model results: For the condition when the building's faqade to party wall connection is bad (scenario 1): * If the full wall is acting on its own, it is within the performance drift limits for the expected base shear and is not expected to fail globally. * If the curved and flat wall sections are acting on their own, they are within the performance drift limits for the expected base shear and are not expected to fail globally. 66 * If the long pier section is action on its own, it is expected to globally fail according to the performance drift limits for the expected base shear for the section. For the condition when the building's fagade to party wall connection is good and the walls act as one (scenario 2), the building is so stiff that none of the wall sections are expected to fail globally or locally. For local shear behavior, none of the stresses read from the piers in the finite element model surpass the shear capacities calculated for those elements. The closest this comes to happening is when the long pier wall section is considered to be acting on its own, but even still the capacity is not met. Overturn of the chimneys are expected at ground accelerations lower than the maximum considered earthquake. Out of plane shear failure and dislocation of the top third of the chimneys is guaranteed for the maximum considered earthquake. This is the biggest immediate threat to the Back Bay residences. Dislocated brick from the chimneys can plunge through roofs or fall to the street below, injuring people and property. 67 7.0 Conclusion A severe earthquake in Boston will happen. It is not a matter of if it will happen because history has proven that it is only a matter of time. If the Cape Ann earthquake of intensity VIII can happen once, it most definitely will happen again. The Back Bay buildings were built without seismic design and sit upon infilled land that will be susceptible to liquefaction during the next significant earthquake. This is a deadly combination for these historical unreinforced masonry buildings and it is naive to believe that the Back Bay is not at risk. The intent of this thesis was to gain understanding of the seismic performance of a typical unreinforced masonry building in Boston's Back Bay neighborhood. This was done by analyzing global and local behaviors in terms of accepted in-plane and out-of-plane failure mechanisms of unreinforced masonry walls from the application of the maximum considered earthquake for the Boston area. It is important to note that even though the results presented in this thesis are based on this earthquake scenario that does not mean that a larger earthquake cannot happen. Another important point is to realize that the results of this thesis do not take into account potential liquefaction ground failures. The influence from liquefaction may just be what the wall sections that are not expected to fail need to surpass performance levels. However, the results based on the current building without considering liquefaction or an earthquake larger than the expected one suggest that when good masonry to masonry connections exist, which means that the facade does not act on its own and the load paths are clear, under the maximum considered earthquake only minimal local failures may happen. This is evident when looking at the results of scenario 2. Contrarily, if cracks in the masonry and mortar degradation exist, load paths within the walls are unclear. The age of the masonry and mortar make this a real possibility if they are unmaintained. Even though the exterior wythe may appear in good condition, the interior wythes may not be. This situation can cause wall elements, such as the curved, flat, or long pier sections, to act on their own. The results show that if this is the case, damage under the maximum considered earthquake can be significant in the global and local spheres. This is evident when looking at the interstory results of the long pier section for scenario 1 and the results of local element shear failures. 68 Regardless of pre-existing conditions, certain intensities and complete overturn of chimney failures are guaranteed at ground accelerations below the maximum considered earthquake. These failures like this can be devastating since they can cause roof damage from impact, and severe injury to people below. Overall, the case study building is just on the edge of being considered vulnerable or not vulnerable. Some results don't have the scenarios violating the performance levels, but in some cases if the base shear was just a little higher, or the applied seismic force was slightly more intense the performance levels would be surpassed. Additionally, the performance levels are just guidelines and failures can happen before Immediate Occupancy, shear capacity, or overturning moment is met. 7.1 Areas of Future Work This study would benefit from a future analysis of how multiple buildings in the Back Bay respond to seismic activity together. The case study in this thesis was analyzed as acting alone, so the building-building interaction would be interesting for a few reasons. If the buildings are connected, they might act together as one stiff building or there might be differential movement which could cause shear failures. If the buildings are not connected, during seismic activity pounding may occur. Analyzing how this would affect the out of plane failure for walls at the end of the block would also be necessary to fully understanding a more accurate behavior of the neighborhood. Many of the calculation methods in this thesis that yielded conclusive results that were highly dependent on the facade geometry. The size and frequency of the perforations were very influential. A simple vulnerability of the buildings could then be analyzed in a parametric study of only the fagades. Future research in this subject should also focus on creating more detailed finite element models that better represent the nonlinear behavior of masonry. 69 Documentation References Adams, Charles Francis. "The Works of John Adams, Second President of the United States." 1856. Google Books. ASCE (2010). Minimum Design Loads for Buildings and Other Structures, ASCE 7-10. Reston, VA: ASCE. Print. ASCE/SEI (2010). Seismic Rehabilitation of Existing Buildings, ASCE/SEI Standard 41-10. Reston, VA: ASCE/SEI. Print. Bunting, Bainbridge (1967). Houses of Boston's Back Bay. Print. Buntrock, Rebecca (2010). Structural Performance of Early 2 0 th Century Masonry High Rise Buildings. Massachusetts Institute of Technology. Thesis. "Creating Land in Boston's Back Bay." Boston Geology. Web. CSI (2014) SAP2000 Version 17.1.1, Integrated software for structural analysis and design. Computers and Structures Inc., Berkeley, CA. DeJong, Matthew J. "Dynamically Equivalent Rocking Structures." Wiley Online Library. N.p., 10 Feb. 2014. Web. Ebel, John E. "Seismological Research Letters." The Cape Ann, Massachusetts Earthquake of 1755: A 250th Anniversary Perspective. Seismological Society of America, 2006. Web. FEMA (2000). Prestandard and Commentary for the Seismic Rehabilitation of Buildings. FEMA 356. Washington, D.C.: Federal Emergency Management Agency, 2000. Web. http://fema.gov/library FEMA (1998). Evaluation of Earthquake Damaged Concrete and Masonry Wall Buildings. FEMA 306. Washington, D.C.: Federal Emergency Management Agency. Web. http://fema.gov/library FEMA (1998). Promoting the Adoption and Enforcement of Seismic Building Codes: A Guidebook for State Earthquake and Mitigation Managers. FEMA 313. Washinton, D.C.: Federal Emergency Management Agency. Web. http://fema.gov/library FEMA (1997). NEHRP Guidelies for the Seismic Rehabiitation of Buildings. FEMA 273. Washington, D.C.: Federal Emergency Management Agency. Web. http://fema.gov/library FEMA E-74 Example 6.3.5.1 Unreinforced Masonry Parapets." FEMA, 24 June 2014. Web. FEMA (2009). Unreinforced Masonry Buildings and Earthquakes: Developing Successful Risk Reduction Programs. FEMA P-774. Washington, D.D.: Federal Emergency Management Agency, 2009. Web. http://fema.gov/library Friedman, D. (1995). HistoricalBuilding Construction:Design, Materials, and Technology. New York: W.W. Norton & Company. Print. 70 "Geology of Boston." Boston Geology. N.p., n.d. Web. 10 Feb. 2015. http://www.bostongeology.com/boston/geology/geology.htm. Hess, Richard L. (2008). Unreinforced Masonry (URM) Buildings. United States Geological Survey. "History of Earthquakes in New England." Underground Town Hall. Web. 30 Jan. 2015. Ho, Alan Darrell. (1979). Determination of Earthquake Intensities from Chimney Damage Reports. Massachusetts Institute of Technology. Thesis. Ingham, Jason. Seismic Assessment and Retrofit of Unreinforced Masonry Buildings. University of Auckland, New Zealand. PowerPoint presentation. Kafka, Alan L. "Why Does the Earth Quake in New England?" www2.bc.edu. Boston College, 15 Feb. 2014. Web. Korini, Oltion, and Bilgin, Huseyin. A new modeling approach in the pushover analysis of masonry structures. International Students Conference of Civil Engineering. Epoka University, Albania. 2012. "Massachusetts Earthquake History." USGS. United States Geological Survey. Web. 30 Jan. 2015. Maxwell, Kenneth. Lisbon 1755: The First 'Modern' Disaster (but If Modern, How Is It So?) Harvard University. Web. http://www.mod-langs.ox.ac.uk/files/windsor/5_maxwell.pdf Miller, Jeremy. "Boston's Earthquake Problem." Boston.com. The New York Times, 28 May 2006. Web. 31 Jan. 2015. Nordenson, Guy J P, and Glenn R. Bell. Seismic Design Requirements for Regions of Moderate Seismicity. www.iitk.ac.in. Web. Park, Joonam et al. Seismic Fragiit analysis of low-rise unreinforced masonry structures. Engineering Structures 31 (2009): 125-137. Web. Philosophical Transactions, Giving Some Account of the Present Undertakings, Studies, and Labours of the Ingenious, in Many Considerable Parts of the World." Google Books. Web. Priestley, M. J. N. "Seismic Behaviour of Unreinforced Masonry Walls." Bulletin of the New Zealand National Society for Earthquake Engineering 18.2 (1985): 191-205. Web. Whitman, Robert. Seismic Design Decision Analysis. American Society of Civil Engineers. Journal of the Structural Division 101 (1975): 1067-1084. Image References Boston Athenaeum. N. Bradlee Collection. Jan 2015. Bruneau, Michel. "Performance of Masonry Structures during the 1994 Northridge (Los Angeles) Earthquake." Canadian Journal of Civil Engineering 22 (1995): 378-402. University of Buffalo. Web. 71 Masonry Terms. Digital Image. http://www.tpub.com/engbas/7-32.htm Costa, Costa A., Antonio Arede, Andrea Penna, and Anibal Costa. "Free Rocking Response of a Regular Stone Masonry Wall Withequivalent Block Approa Ch: Experimental and Analytical Evaluation." Earthquake Engineering & Structural Dynamics 42 (2013): 2297319. Print. Figure 5(a):Javed, M., Khan, A. N., and Magenes, G., 2008. Performance of masonry structures during earthquake-2005 in Kashmir, Mehran University Research Journal of Engineering & Technology 27, 271-282. Figure 5(b) Bed-joint sliding. FEMA. Figure 5(c) Christchurch earthquake damage. Digital Image. FEMA. Figure 5(d) Christchurch earthquake damage. Digital Image. FEMA. Figure 6(a) Parapet damage. Barry Welliver. Figure 6(b) Christchurch earthquake damage. Digital Image. FEMA. Figure 6(c) Northridge earthquake damage. Michel Bruneau. Figure 7(a) "The Modified Mercalli Intensity Scale for Insurance Underwriting." The Washington Surveying and Rating Bureau. N.p., 12 June 2012. Web. Figure 7(b) Long Beach earthquake damage. Historical Society of Long Beach. Figure 7(c) Northridge earthquake damage. Michel Bruneau. Figure 7(d) Loma Prieta earthquake damage. Digital Image. EERI Reconnaissance Team. Figure 7(e) Napa Valley earthquake damage. Digital Image. ZFA Structural Engineers. Figure 7(f) Christchurch earthquake damage. Digital Image. USGS PAGER Team. Figure 9(a & b) National Information Service for Earthquake Engineering, University of California, Berkeley Figure 1 1(a & b) Bunting, Bainbridge. (1967) Houses of Boston's Back Bay. Print. All uncited images are the property of myself or Ornella Iuorio. 72 Appendix A Cantilever Beam Hand Calculations 73 Constants and Base Shear Calculation for Scenario 1 wall elements. **Scenario 2 done in the same manner, but with different Total Wall Weight Full Wall Section Constants Base Shear Calculation E 429 120 S161. Building Dimensions Curved Wall Section Base Shear Calculation Building Dimensions Total Height [ft] 74.23 05 -0.18 0145 17.595 74 Flat Wall Section Building Dimensions Base Shear Calculation 7ft} 13.583 142.50 Total Width [ft) 77 1 Total Height [ft] 74.23 1.5 Total .ength 0.506 0.02 0.75 0.18 0.145 20.706 Pier Wall Section Base Shear Calculation Building Dimensions Total Length [ft) Total Width ftl 4.33 77 Total Height {ft) 74.23 51.55 1 1.5 0.506 0.02 0.75 0.18 0.145 7.490 75 Response Spectrum By using the method outlined in Chapter 5, the response spectrum can be calculated. This spectrum represents the response for the zip code of 02116, which contains the Back Bay. It is also assumed that the soil is of Soil Class D, which represents infilled land and is also the default value used when no reliable information is provided. The spectrum below shows that the peak ground acceleration is 0.2292g. Response Spectrum 0.25 0.2 0.15 0 CU 0.1 0.05 0 0 1 2 3 5 4 Period 76 6 [s] 7 8 9 10 Full Wall Section 144 278.00 428.75 591.00 732.25 892.81 I 12 0.99 1.53 1.98 1.91 2.48 A An1l 0.0002 0.0004 0.0007 0.0008 0.0011 5656 5656 5656 5656 5656 5656 A AA0i11 0.00009 0.00022 0.00024 0.00012 0.00030 58898854 58898854 58898854 58898854 58898854 58898854 167 1.48 2.30 2.97 2.87 3.72 0.402 0.558 0.693 0.814 0.906 1.000 0.000165 0.000294 0.000628 0.000994 0.001180 0.001628 5b.b U.56 U.UUUU6 u.uuuu6 35.9 45.0 49.5 43.0 50.4 Base Shear 0.49 0.77 0.99 0.96 1.24 0.00010 0.00021 0.00033 0.00039 0.00054 0.00004 0.00011 0.00012 0.00006 0.00015 Max Roof Disp Max Roof Disp/Total H 0.00054 5 0.000165 2.23 0.000129 0.000334 0.000366 0.000186 0.000449 1.97 3.06 3.96 3.83 4.95 15 20 0.001086 0.001628 0.002171 1.21584E-06 1.82376E-06 2.43168E-06 10 2.79 2.46 3.83 4.94 4.78 6.19 0.000275 0.000490 0.001047 0.001656 0.001966 0.002714 0.000275 0.000214 0.000557 0.000609 0.000310 0.000748 5.58 4.93 7.65 9.89 9.57 12.38 0.000551 0.000979 0.002093 0.003312 0.003932 0.005428 0.000551 0.000428 0.001114 0.001219 0.000620 0.001496 _ _ i 6.07919E-07 11.16 9.85 15.31 19.78 19.13 24.77 25 50 100 0.002714 0.0054276 0.0108552 3.03959E-06 0.0000061 1.21584E-05 0.000220 0.000392 0.000837 0.001325 0.001573 0.002171 0.000220 0.000171 0.000446 0.000487 0.000248 0.000598 0.001102 0.001959 0.004186 0.006623 0.007864 0.010855 0.001102 0.000857 0.002228 0.002437 0.001241 0.002991 27.89 24.63 38.27 49.45 47.84 0.002754 0.004896 0.010466 0.016558 0.019660 0.002754 0.002142 0.005569 0.006093 0.003101 61.92 0.027138 0.007478 44.63 39.41 61.24 79.12 76.54 99.07 0.004407 0.007834 0.016745 0.026493 0.031456 0.043421 0.004407 0.003428 0.008911 0.009748 0.004962 0.011965 55.79 49.26 76.55 98.89 95.67 123.84 0.005508 0.004284 0.011139 0.012185 0.006203 0.014956 0.013220 0.010283 0.026733 500 250 400 0.0271379 0.0434206 0.0542758 4.86335E-05 6.07919E-05 3.03959E-05 0.005508 0.009793 0.020932 0.033117 0.039320 0.054276 89.26 78.82 122.47 0.008814 0.015669 0.033491 0.008814 0.006855 0.017822 111.58 98.53 153.09 0.011017 0.019586 0.041863 0.011017 0.008569 0.022277 133.89 118.23 183.71 0.013220 0.023503 0.050236 158.23 0.052987 0.019496 197.79 0.066234 0.024370 237.35 0.079480 0.029244 153.08 198.14 800 0.0868413 0.062911 0.086841 0.009924 0.023930 191.35 247.67 1000 0.1085516 0.078639 0.108552 0.012405 0.029913 229.62 297.20 1200 0.1302619 0.094367 0.130262 0.014887 0.035895 9.7267E-05 0.000145901 0.000121584 78 1 223.15 0.022034 ntrtr 0.022034 167.37 0.016525 nesry 0.016525 147.79 0.029379 0.012853 197.05 0.039172 0.017138 229.64 0.062795 0.033416 306.18 0.083726 0.044555 296.68 0.099350 0.036556 395.58 0.132467 0.048741 287.02 0.117959 0.018608 382.69 0.157278 0.024811 371.51 0.162827 0.044869 495.34 0.217103 0.059825 P9 1500 2000 0.1628274 0.000182376 0.2171032 0.000243168 Curved Wall Section 144 278.00 428.75 591.00 732.25 892.81 2668 2668 2668 2668 2668 2668 6180991 6180991 6180991 6180991 6180991 6180991 0.402 0.558 0.693 0.814 0.906 1.000 79 26.68 16.23 18.44 20.57 17.86 21.32 Base Shear Max Roof Disp 0.62 0.0003 U.UU0I 0.52 0.74 0.96 0.93 1.23 5 0.002853 0.0005 0.0010 0.0017 0.0020 0.0029 0.0008 0.0017 0.0023 0.0013 0.0029 Max Roof Disp/Total H 3.19604E-05 1.23 1.04 1.47 1.93 1.86 2.46 0.001 0.001 0.002 0.003 0.004 0.006 0.000619 0.000438 0.000959 0.001309 0.000759 0.001623 1.85 1.57 2.21 2.89 2.80 3.69 10 0.0009 0.0016 0.0030 0.0050 0.0061 0.0086 2.47 2.09 2.94 3.86 3.73 4.91 0.000928 0.000657 0.001439 0.001963 0.001138 0.002434 15 0.005707 0.008560 0.011414 9.58811E-06 1.27841E-05 0.002 0.003 0.005 0.008 0.010 0.014 0.001547 0.001095 0.002399 0.003272 0.001897 0.004057 11.11 9.39 13.25 17.36 16.77 22.11 0.001238 0.000876 0.001919 0.002618 0.001518 0.003246 20 6.39207E-06 3.09 2.61 3.68 4.82 4.66 6.14 0.0012 0.0021 0.0040 0.0067 0.0082 0.0114 0.005569 0.009511 0.018146 0.029926 0.036756 0.051362 0.005569 0.003942 0.008635 0.011780 0.006831 0.014606 21.61 18.27 25.77 33.75 32.62 42.99 25 90 175 0.014267 0.0513624 1.59802E-05 0.0000575 0.0998712 0.000111861 80 0.010830 0.018494 0.035284 0.058189 0.071470 0.099871 0.010830 0.007665 0.016790 0.022905 0.013282 0.028401 30.87 26.10 36.81 48.21 46.60 61.42 250 0.015471 0.026420 0.050405 0.083126 0.102101 0.142673 0.015471 0.010950 0.023985 0.032721 0.018974 0.040573 0.000511366 0.024753 0.042272 0.080648 0.133002 0.163361 0.228277 0.024753 0.017519 0.038376 0.052354 0.030358 0.064916 0.049506 0.084545 0.161297 0.266005 0.326722 0.456554 0.049506 0.035039 0.076752 0.104708 0.060717 0.129833 0.061883 0.105681 0.201621 0.332506 0.408402 0.570693 123.47 104.39 147.24 192.85 186.38 245.67 1000 0.5706928 0.000639207 74.08 62.63 88.35 115.71 111.83 147.40 600 0.037130 0.063409 0.120973 0.199504 0.245041 0.342416 0.037130 0.026279 0.057564 0.078531 0.045538 0.097374 0.074259 0.126817 0.241945 0.399007 0.490082 0.684831 0.074259 0.052558 0.115128 0.157062 0.091075 0.194749 0.3424157 0.000383524 0.2282771 0.000255683 0.1426732 0.000159802 98.78 83.51 117.79 154.28 149.10 196.53 800 0.4565543 49.39 41.75 58.90 77.14 74.55 98.27 400 0.061883 0.043798 0.095940 0.130885 0.075896 0.162291 148.17 125.26 176.69 231.42 223.66 294.80 1200 0.6848314 0.000767049 1 81 185.21 156.58 220.87 289.27 279.57 368.50 0.092824 0.158522 0.302432 0.498759 0.092824 0.065697 0.143910 0.196327 0.612603 0.856039 0.113844 0.243436 246.95 208.77 294.49 385.70 372.76 0.123766 0.211362 0.403242 0.665012 0.816804 0.123766 0.087596 0.191880 0.261769 0.151792 491.33 1.141386 0.324582 1500 2000 0.8560393 1.1413857 0.000958811 0.001278415 Flat Wall Section 144 2608 5773904 0.402 U.UU.L1J.U 0U.0.L13 2608 5773904 0.558 26.08 18.62 2.UI7 278.00 2.062 0.002018 0.000915 428.75 2608 5773904 0.693 24.34 3.35 0.004278 0.002261 591.00 732.25 2608 2608 5773904 5773904 0.814 0.906 25.37 22.52 4.10 4.05 0.007228 0.009010 0.002950 0.001781 892.81 2608 5773904 1.000 25.55 5.07 0.012563 0.003553 Base Shear Max Roof Disp Max Roof Disp/Total H 82 20.71 0.012563 1.40712E-05 1.004 0.996 1.62 1.98 1.96 2.45 0.000533 0.000974 0.002066 0.003490 0.004350 0.006066 0.000533 0.000442 0.001092 0.001424 0.000860 0.001716 1.51 1.49 2.43 2.97 2.93 3.67 0.000799 0.001462 0.003099 0.005235 0.006526 0.009099 0.000799 0.000663 0.001637 0.002136 0.001290 0.002574 2.01 1.99 3.23 3.96 3.91 4.90 15 20 0.006066 0.009099 0.012132 6.79442E-06 1.01916E-05 1.35888E-05 10 2.51 2.49 4.04 4.95 4.89 6.12 25 0.015165 1.6986E-05 0.001331 0.002436 0.005165 0.008726 0.010876 0.015165 0.001331 0.001104 0.002729 0.003561 0.002150 0.004289 9.03 8.96 14.55 17.81 17.60 22.04 0.004793 0.008769 0.018593 0.031412 0.039153 0.054595 0.004793 0.003976 0.009824 0.012819 0.007741 0.015442 10.04 9.96 16.17 19.79 19.55 24.49 90 100 0.0545953 0.0000611 0.0606615 6.79442E-05 83 0.001065 0.001949 0.004132 0.006980 0.008701 0.012132 0.005326 0.009744 0.020659 0.034902 0.043503 0.060661 0.001065 0.000884 0.002183 0.002849 0.001720 0.003432 0.005326 0.004418 0.010915 0.014243 0.008601 0.017158 806Lt7Z*O SHOEVO t7Z88TVO 9T60LT*O ztlozzs*o LTZEOT*O LE6LZL*O 968SOZ*O 696E9E*O 8t,6ZOT*O t's StIT9909'0 t'LE6LZL*O Z17t76L9000*0 EES18000,0 EZ69TT'O 08STLT*O ST0980*0 oEtzt7T*O t7ST60T*O OS*6TT ZT6E90*0 TZOT9z*o ZTt760Z'O 609TSO'O 8svssoo OOOT OOZT Z6'E6Z Z9,tEz SVLEZ t7O*t76T TTOESO'o ST9909*0 SEOSEt?*O OZ06tE'O 06S90Z'O t79ZLET'O E6*ttZ ZT8890*0 ZS*S6T t7t,6ETT'O L8*L6T EZEL80'0 OL'T9T 9LTtlt,0'0 9Et7L60'0 Tt7ESEO*o 6S'66 09zssoo Lt?*OZT ZT6E90'0 09ZESO*o WOOT Z6ZS8t7'0 8zo8t7E*O 9TZ6LZ*O ZLZS9T*O 6t76LLO*O 809Zt7O'O 809ZI70*0 ESSEt7sooo*o 9T6ZS817'0 008 S6*S6T Ttl*9sT oE*8sT 9E'6ZT L9*6L T E'08 .......... 2' 9S6TEO'O 9S6TEO*O 8stl9ztlz*o OOV L896E9E*O 009 LLLTLZOOO*O S99LOV000*0 ZE9890,0 96*9tT 90t,17EO*o TE*LTT ZL69SO*O ZL'STT ZO*L6 Z6t7S90*0 VS6EZT*O 9os9zo*o 9869TOOO'O 9Es9TSTo osz S68Zt,0'0 Lu Lv 9tl9ztz*o t7osTzoo 809SEO*o TZ'SL ST'6L tITOtILT*O 8096ET*O 9E9z8o*o 68ZLZO*O 89't,9 Z99Et,0'0 E8'6E 17L68EO*O OL9LTO'O SL'6S T9t78so*o t7s9TST*O 6SL80T'O SSZLSO*O Lt?9TSO'O i7t'OTTO*O STEETO*O 9T*ot7 tOETZO'O 170ETZO*O EZ'09 6SEt7ZO*O STEETTO EzT9 88*817 Lt7'617 EV017 0617Z OT*SZ 150.58 149.38 242.55 296.81 293.28 367.40 200.78 199.17 323.40 395.75 391.04 489.86 2000 0.079889 0.066264 0.163731 0.213645 0.129022 0.257370 0.079889 0.146154 0.309885 0.523530 0.652552 0.909922 1500 0.9099218 1.2132290 0.001019163 0.001358883 0.106519 0.194871 0.413180 0.698040 0.870069 1.213229 0.106519 0.088352 0.218308 0.284861 0.172029 0.343160 Pier Wall Section 144 831 187045 0.402 8.31 3.52 0.045057 0.045057 278.00 831 187045 0.558 7.74 4.55 0.092235 0.047178 428.75 831 187045 0.693 8.70 6.35 0.187196 0.094961 591.00 831 187045 0.814 9.37 8.03 0.333327 0.146131 732.25 831 187045 0.906 8.15 7.78 0.425668 0.092341 892.81 831 187045 1.000 9.27 9.77 0.598002 0.172335 Base Shear Max Roof Disp Max Roof Disp/Total H 85 40 0.598002 0.000669795 0.88 1.14 1.59 2.01 0.011264 0.023059 0.046799 0.083332 0.011264 0.011794 0.023740 0.036533 1.32 1.71 2.38 3.01 0.016896 0.034588 0.070198 0.124997 0.016896 0.017692 0.035610 0.054799 1.76 2.27 3.18 4.02 0.022528 0.046117 0.093598 0.166663 0.022528 0.023589 0.047480 0.073066 1.95 0.106417 0.023085 2.92 0.159625 0.034628 3.89 0.212834 0.046170 2.44 0.149501 0.043084 3.66 0.224251 0.064625 4.88 0.299001 0.086167 0.112642 10 15 0.149501 0.000167449 0.1911970 0.000214151 20 0.299001 0.000334898 1_1 2.20 0.028161 0.028161 4.40 0.056321 0.056321 8.79 0.112642 2.84 0.057647 0.029486 5.69 0.115294 0.058972 11.37 0.230587 0.117945 3.97 5.02 0.116997 0.208329 0.059350 0.091332 7.94 10.04 0.233994 0.416658 0.118701 0.182664 15.89 20.08 0.467989 0.833317 0.237402 0.365328 4.86 0.266042 0.057713 9.73 0.532084 0.115426 19.45 1.064169 0.230852 6.10 0.373751 0.107709 12.21 0.747503 0.215418 24.42 1.495005 0.430836 25 50 100 0.3737513 0.000418622 0.7475026 0.0008372 0.001674488 1.4950051 86 28.43 0.576468 39.72 50.19 48.63 61.04 1.169972 2.083291 2.660422 3.737513 0.471780 1 400 68.23 95.32 120.45 116.72 146.50 600 0.006697953 0.010046929 45.49 0.294862 0.593505 0.913319 0.577131 1.077091 63.55 80.30 77.81 97.67 250 0.922348 1 1.871955 3.333266 4.256675 5.980020 1.383522 1 2.807933 4.999899 6.385013 8.970031 3.7375128 0.00418622 1.126421 105.54 1.351705 136.46 2.767044 190.65 5.615866 70.36 0.901136 90.97 1.844696 127.10 3.743911 158.87 160.60 6.666532 200.75 8.33316524.0.979 155.63 8.513350 194.54 10.641688 233.44 12.770025 14.950051 293.01 17.940061 195.34 800 0.013395905 11.960041 87.95 0.450113.72 2.305870 244.17 1.1795 4.679889 1ooo 1200 0.016744881 0.020093858 87 0.707670 1 131.92 1.689631 175.90 2.252841 170.58 3.458806 227.43 4.611741 238.31 7.019833 317.74 9.359777 12.499748 15.962532 401.51 16.666330 389.07 21.283375 22.425077 488.35 29.900102 301.13 291.80 366.26 1500 2000 0.025117322 0.033489763 88 Appendix B SAP2000 Pushover Results Full Wall Section PD PC PB PA Basement 0.01675 0.01675 0.012666 0.012666 0.01466 0.01466 0.016647 0.016647 1 0.026658 0.009908 0.033311 0.020645 0.039981 0.025321 0.046623 0.029976 2 0.050222 0.023564 0.063212 0.029901 0.076241 0.03626 0.089213 0.04259 3 0.078647 0.028425 0.097891 0.034679 0.117195 0.040954 0.136414 0.047201 0.024746 0.103393 4 0.025298 0.128691 Roof Base Shear Max Roof Disp Max Roof Disp/Total H 0.128082 0.160103 0.030191 0.152848 0.035653 0.177505 0.041091 0.032021 0.191612 0.038764 0.222979 0.045474 5 10 15 20 0.128691 0.0001441 0.160103 0.0001793 0.191612 0.0002146 0.222979 0.0002497 P2 P1 PE P4 P3 0.018634 0.018634 0.028598 0.028598 0.048517 0.048517 0.108276 0.053265 0.034631 0.086572 0.057974 0.153164 0.104647 0.352943 0.687456 0.168035 0.168035 0.244667 0.552718 0.384683 0.334513 1.077612 0.524894 0.108276 0.102185 0.04892 0.167231 0.080659 0.297288 0.144124 0.155632 0.053447 0.252 0.084769 0.444686 0.147398 1.02274 0.335284 1.600774 0.523162 0.12832 1.314626 0.291886 2.056218 0.455444 0.153153 1.669686 0.35506 2.613178 0.55696 0.202161 0.046529 0.325798 0.073798 0.573006 0.25435 0.052189 0.411645 0.085847 0.726159 25 50 100 0.25435 0.0002849 0.4116450 0.0004611 0.7261590 0.0008133 400 250 1.6696860 2.6131780 0.0018701 0.0029269 __160 0.758045 11.351788 3963 2.638186 2.117989 4.033731 ___3.912837_ 5.022499 _____5.836999_ 7.491123 Soo 1000 1500 5.1290750 6.3870350 9.5278230 0.0057448 0.0071538 0.01067171 _ 3.142142 0.605984 _____0.605984 0.07590.407059 0.3732 03238 1.085427 PS P7 P6 Curved Wall Section PD PC PB PA 1 0.01527 0.047425 0.01527 0.032155 0.026734 0.092255 0.026734 0.065521 0.038239 0.137229 0.038239 0.09899 0.049675 0.181931 0.049675 0.132256 2 0.103799 0.056374 0.206522 0.114267 0.309532 0.172303 0.411931 0.23 3 0.179941 0.076142 0.353033 0.146511 0.526573 0.217041 0.699087 0.287156 4 0.250683 0.070742 0.492841 0.139808 0.73561 0.209037 0.976923 0.277836 Roof 0.332141 0.081458 0.653749 0.160908 0.976145 0.240535 1.296587 0.319664 Basement Base Shear Max Roof Disp Max Roof Disp/Total H 0.653749 15 0.976145 1.296587 0.000732235 0.001093336 0.001452249 5 10 0.332141 0.000372016 91 20 0.8 0.03 1.73147 0.7 0.233166 0.165609 0.450876 0.33241 0.927 6 0.2648 0.5461 .2885 102057 0.7781 2.055343 P4 P3 P2 Pi PE 0.577228 0.666131 0.577228 0.921249 2_24414_ 3.589374 16.086093 250 25.730408 400 __ 8.217 995 __5_136839_ 0.87066 0.35465 1.73147 0.7909 0.69196 0.796037 .6569713.850877 .46956 4.85042419.375421 6.440763 25 2.429107 3.225144 50 1.617969 3.225144 6.440763 16.086093 25.730408 0.003612339 0.00721401 0.018017311 0.028819475 0.346848 0.399055 1.218914 1.617969 0.001812214 PS 100 P6, 1.397997 1.838651 5.382759 7.175974 12.326418 16.434445 20.772802 27.694071 29.058722 38.741118 38.590123 51.448657 600 800 38.590123 51.448657 0.043223065 0.057625332 92 Flat Wall Section PD PC PB PA Basement 1 2 0.019119 0.05905 0.112824 0.019119 0.039931 0.053774 0.032249 0.10879 0.2155 0.032249 0.076541 0.10671 0.045305 0.158241 0.317563 0.045305 0.112936 0.159322 0.058417 0.207914 0.420109 0.058417 0.149497 0.212195 3 0.179134 0.06631 0.348715 0.133215 0.517255 0.199692 0.686644 0.266535 4 0.239854 0.06072 0.471354 0.122639 0.701406 0.184151 0.932662 0.246018 Roof 0.309076 0.211201 1.215027 0.282365 Base Shear Max Roof Disp Max Roof Disp/Total H 0.069222 0.140448 0.611802 0.912607 5 10 15 0.309076 0.000346182 0.611802 0.00068525 0.912607 0.00102217 0.001360897 P4 P3 P2 P1 PE 20 1.215027 0.66088 1.053772 0.07151 0.07151 0.137024 0.137024 0.267956 0.267956 0.66088 0.257510.522468 0.186 0.264958 0.505684 1.00166 0.733704 2.490084 !3.978386 1.034721 0.36866 0.529037 2.0584411 5.130651 8.202645 0.855675 1.163382 1.516698 0.333207 0.307707 0.353316 1.70666 2.31823 3.026807 0.671939 0.61157 0.708577 3.392334 4.626076 6.044549 8.466097 11.55279 15.101051 13.539552 18.477693 24.150 25 50 1.516698 3.026807 0.001698785 0.003390191 400 100 1250 0.01691401 0.006770228 93 0.02705016 Pier Wall Section PC PB Basement 0.32849 0.32849 0.660809 U.9911bb 1 2 1.122165 0.793675 2.25314 3.377458 7.267349 12.364982 3 4 Roof Base Shear Max Roof Disp Max Roof Disp/Total H 5 7.616062 10 15.278195 17.211867 22.897955 15 22.897955 0.008530409 0.017112421 0.025646972 Pi PE 1.652722 3.309447 5.629017 11.267506 12.110827 24.24001 20.605025 41.240212 28.681325 57.404229 38.156012 76.367 50 25 38.156012 76.367 0.042736837 0.08553525 94 1.322365 4.504698 9.692261 16.490284 22.953742 30.536252 20 30.536252 0.034202286 Appendix C In-Plane Failure Results Full Wall Section-Pier Element Capacities Pier 1 Pier 2 Pier 3 Pier 4 Pier 5 Rocking Rocking Rocking Rocking Rocking 0.5 a hen [in] 26.0 vme 2 27.6 An [in ] L [in] 36.8 L [in] 57.2 72.5 hen [in] 72.5 hen [in] 72.5 hef [in] 80.1 heff [in] Bed Joint Sliding 27.0 vte [psi] 1.2 2547.1 Vr [lb] Bed Joint Sliding 27.0 vte [psi] 26.0 vme 2 27.6 An [in ] 0.7 L/he 1017.2 Vr [lb] Bed Joint Sliding 27.0 vte [psi] 26.0 vme 2 27.6 An [in ] PCE [psi] 2 Vbjsi 718.7 Vbjsl Vbjs2 1656.1 Diagonal Tension Vbjs2 2436.2 VbIs2 1539.5 Diagonal Tension Vbjs2 Vdt f'dt 315.9 Vdt 13.0 1.0 Beta 417.8 f'me [Psi] Weight above [lbs] 0.7 Beta 291.6 Vdt 3.1 780.0 3312.3 4.6 fae [Psi] 780.0 N'e [Psi] Weight above [lbs] 4872.4 2.9 fae [PSI] 780.0 f'me [Psi] 26.0 2 1063.3 VbJs2 718.7 6082.7 Diagonal Tension 13.0 0.7 258.6 Vdt 27.6 An [in ] 13.0 f'dt 0.7 Beta 351.9 Vdt Toe Crushing 2.0 fae [Psi] 780.0 f'me [Psi] fae [Psi] 11.4 f'me [psi] 780.0 12165.4 Weight above [lbs] 3079.1 Weight above [lbs] 2126.6 Weight above [lbs] Area [in 2] 1065.6 Area [in 2] 1065.6 Vtc Min Vtc Min 4255.3 351.9 Area [in 2] 1065.6 Area [in 2] 1065.6 Area [in 2] VC Min 1300.5 315.9 Vc Min 2806.4 417.8 Vtc Min 1124.2 291.6 Diagonal Tension 27.0 vme Toe Crushing 1065.6 Diagonal Tension Bed Joint Sliding vte [psi] Vbjsl Beta 3911.6 Vr [Ib] 718.7 f'dt 0.7 L/heff Diagonal Tension Toe Crushing Toe Crushing Toe Crushing fae [Psi] 13.0 f'dt 27.6 An [in ] 718.7 13.0 0.8 26.0 vme Vbjsl Beta 485.2 Bed Joint Sliding 27.0 vte [psi] 718.7 Diagonal Tension 0.5 L/heff Vr [lb] Vbjsl f'dt 12165.4 PCE [psi] 53.2 72.5 1177.1 V, [lb] 2126.6 0.5 a 84.2 57.2 L/heff 0.5 a 3079.1 4872.4 0.8 L/heff 0.5 a PCE [psi] L [in] PCE [psi] L [in] 3312.3 PCE [psi] L [in] 0.5 a Diagonal Tension 537.2 258.6 Diagonal Tension Diagonal Tension 0.5 a PCE [psi] L [in] 17895.3 84.2 1w, [i] 80.1 1.1 L/heff 8464.0 Vr [Ib] -Rocking 0.5 a PCE 11308.9 53.2 [PSI] L [in] 80.1 hff [in] 0.7 L/heff 3380.2 Vr [Ib] vte [psi] vine 27.0 26.0 vte [psi] vme An [in2 ] 27.6 An [in 2 ] 27.6 718.7 -T8947.7 1.0 Beta 544.0 Vdt PS]780.0 Weight above [lbs] 17895.3 2 ] Area [in1 _____ m _ twf [in] 0.5 L/heff 1612.4 Vr [Ib] L [in] 91.8 0.6 L/heff 6000.3 Vr [lb] PCE 31457.1 84.2 [psi] L [in] 91.8 heff [in] 0.9 L/heff 12983.8 Vr [lb] Bed Joint Sliding Bed Joint Sliding vte [psi] vme 27.0 26.0 vte [psi] vme 27.0 26.0 vte [psi] vine 27.0 26.0 An [in2] 27.6 An [in 2] 27.6 An [in 2] 27.6 Vbjsl 718.7 Vbjsl 718.7 Vbjsl 718.7 Vbjsz 5654.4 Vbjs2 3905.3 Vbjsz 10692.4 Vbjs2 15728.6 13.0 f'dt 0.7 Beta 324.5 Vdt 13.0 f'dt 0.7 Beta 301.1 Vdt 13.0 f'dt 0.7 Beta 384.0 Vdt f'e[S]780.0 f'me [PSI] Weight above [lbs] Weight above [lbs] 11308.9 1.0 649.9 Vdt Toe Crushing 20.1 7.3 -fae [PSI] fae [PSI] 13.0 f'dt Beta Toe Crushing Toe Crushing fe[S]10.6 Diagonal Tension Diagonal Tension Diagonal Tension Toe Crushing 16.8 80.1 21384.9 57.2 0.5 a 718.7 Toe Crushing fae [PSI] PCE [PSI] hegt[n] Diagonal Tension 13.0 f'dt 7810.6 36.8 780.0 7810.6 f'e[S]780.0 f'e[S]780.0 Weight above [lbs] 2 29.5 fae [PSI] 21384.9 Weight above [lbs] Area [in 2] 1065.6 Area [in ] 1065.6 Area [in ] 1065.6 Area [in ] 9115.2 Vtc 3682.8 Vt Mi 1767.5 Vtc 6422.0 Vtc Diagonal Tension 544.0 324.5 Min Diagonal Tension 2 301.1 Diagonal Tension 97 384.0 Min Diagonal Tension 31457.1 2 1065.6 T ____ VbJsl Diagonal Tension f'e L [in] 0.5 a Bed Joint Sliding 27.0 26.0 Vbjs2 [psi] Bed Joint Sliding Bed Joint Sliding Vbjsl 0.5 a PCE Pier 10 Rocking Pier 9 Rocking Pier 8 Pier 7 Rocking Pier 6 Rocking 13646.4 649.9 Min Bed Joint Sliding 0.5 a L [in] Rocking Rocking Rocking PCE [psi] Pier 13 Pier 12 Pier 11 0.5 a 19879.3 53.2 PCE [psij 91.8 he [in] heo [in) 0.6 L/heff 5185.2 Vr [lb] Bed Joint Sliding 27.0 vte [psi] 26.0 vme 2 27.6 An [in ] 20478.3 36.8 L [in] 87.5 0.4 L/heff 3872.6 V, [Ib] Bed Joint Sliding 27.0 vte [psi] 26.0 vine 2 An [in ] 27.6 0.5 a PCE [psi] L [in] Pier 14 Pier 15 Rocking Rocking 0.5 a 31896.0 PCE [psi] 57.2 1L [in] 46919.0 84.2 hf [in] 87.5 87.5 htf [in] 0.7 L/heff 9394.8 Vr [Ib] Bed Joint Sliding 27.0 vte [psi] 26.0 vme 2 An [in ] 27.6 1.0 L/heff 20328.9 V, [Ib] Bed Joint Sliding 27.0 vte [psi] vine 2 An [in ] 0.5 29650.4 53.2 a PCE [psi] L [in] 87.5 heff [in] 0.6 L/heff 8118.5 Vr [Ib] Bed Joint Sliding 27.0 vte [psi] 26.0 vine 26.0 27.6 An [in 2] 27.6 Vblsl 718.7 Vbjsl 718.7 Vbjsl 718.7 Vbjsl 718.7 Vbjsi 718.7 Vbjs2 9939.6 Diagonal Tension Vbjs2 10239.2 Diagonal Tension Vbjs2 15948.0 Diagonal Tension Vbjs2 23459.5 Diagonal Tension VbJs2 14825.2 Diagonal Tension 13.0 f'dt 0.7 Beta 375.7 Vdt 13.0 f't 0.7 Beta 379.0 Vdt 0.7 Beta 437.5 Vt 18.7 fae [PSI] 780.0 f' me [PSI] 19879.3 fae [PSI] 19.2 f'me [PSI] 780.0 Weight above [lbs] 13.0 f'_t 1.0 Beta 752.6 Vdt 20478.3 29.9 fae [PSI] f'm 31896.0 Weight above [lbs] Area [in 2] 1065.6 Area [in 2] 1065.6 Area [in 2] 1065.6 Area [in2] Vtc Min 5564.5 375.7 Vtc Min 4151.5 379.0 Vtc Min 9866.4 437.5 Vtc Min Diagonal Tension Diagonal Tension Diagonal Tension 98 0.7 Beta V _ 426.6 _ Toe Crushing 44.0 fa. [psi] _f'me [PSI]_ PI78. Weight above [lbs] 13.0 f'_t Toe Crushing Toe Crushing Toe Crushing Toe Crushing weight above [lbs] 13.0 f'_t 780__ _f'me[PSI]_ _ _ 46919.0 1065.6 20766.2 718.7 Bed Joint Sliding 27.8 fae [PSI] Weight above [lbs] __ _ 780.0 29650.4 Area [in2] 1065.6 Vtc Min 8560.9 426.6 Diagonal Tension 0.5 a PCE [PSi] L [in] heff [in] L/heff Vr [lb] 13729.8 36.8 91.8 0.4 2473.4 Bed Joint Sliding 27.0 vte [psi] 26.0 vine An [in 2] 27.6 0.5 a PCE [psi] L [in] heff [in] L/heff Bed Joint Sliding 27.0 vte [psi] An [in 0.7 Beta 339.7 Toe Crushing Vdt fae [PSI] f'me [PSI] 12.9 780.0 27.6 13.0 0.7 Beta 478.0 Toe Crushing Vdt fae [PSI] f'me [psi] 38.2 780.0 13729.8 above [Ibs] 40754.9 5381.2 Vr [lb] Bed Joint Sliding 27.0 vte [psi] 26.0 vme 2 27.6 An [in ] 0.5 a L [in] Bed Joint Sliding 27.0 vte [psi] 0.7 407.4 Toe Crushing Vdt 24.2 fae [PSI] 780.0 f'me [PSI] 26.0 vme 2 27.6 An [in ] 13.0 0.7 Beta 564.1 Vdt 56.3 f'me [PSI] above [Ibs] 2 1065.6 Area [in2] 1065.6 Area [in ] 1065.6 Area [in ] Vtc Min 2683.4 339.7 Vtc Min 6975.5 478.0 Vtc Min 5713.9 407.4 Vtc Min Diagonal Tension Bed Joint Sliding 27.0 vte [psi] 59950.4 1065.6 18285.2 564.1 Diagonal Tension Diagonal Tension 99 26.0 vine 2 27.6 An [in ] 0.5 a L [in] Bed Joint Sliding 27.0 vte [psi] 2 472.3 Toe Crushing Vdt 27.6 An [in ] Vbjs2 0.7 26.0 vme 18942.8 Diagonal Tension 13.0 5530.7 Vr [lb] Vbjsl Beta 78.2 0.5 he [in] L/heff 718.7 f'dt 26166.0 36.8 PCE [psi] 718.7 13083.0 Diagonal Tension 13.0 f'dt 0.7 Beta 409.2 Crushing Toe Vdt fae [PSI] 35.6 fae [PSI] 24.6 f'me [PSI] 780.0 f'me [PSI] 780.0 Weight above [lbs] Area [in 2] Diagonal Tension 780.0 11594.5 Vr [lb] Toe Crushing fae [PSI] 78.2 0.7 hef [in] L/heff Vbjs2 Weight 25810.0 1 [in] 29975.2 Diagonal Tension f'dt 37885.5 53.2 PCE [psi] Vbjsl Weight 2 0.5 a 718.7 VbJs2 Beta 18347.2 Vr [Ib] 12905.0 Tension Diagonal 13.0 78.2 0.7 heff [in] L/heff Vbjsl f'dt 59950.4 53.2 PCE [psi] 718.7 Vbjs2 f'dt 78.2 0.5 heff [in] L/heff 20377.4 Diagonal Tension Weight Weight above [lbs] ] L [in] Vbjsl Vbjs2 13.0 2 25810.0 36.3 PCE [psi] 718.7 6864.9 Diagonal Tension f'dt 26.0 vine Vbjsi Vbjs2 78.2 0.4 6750.8 Vr [lb] 718.7 Vbjsl 40754.9 28.8 0.5 a Pier 21 Rocking Pier 20 Rocking Pier 19 Rocking Pier 18 Rocking Pier 17 Rocking Pier 16 Rocking above [lbs] 2 Area [in ] Vtc Min 37885.5 1065.6 12043.9 472.3 Diagonal Tension above [bs] 2 26166.0 Area [in ] 1065.6 Vtc Min 5868.8 409.2 Diagonal Tension Curved Wall Section-Pier Element Capacities 1018.5 17.6 PCE [psi] L [in] 3079.1 53.2 PCE [psi] L [in] 0.5 0.5 a 0.5 0.5 a a PCE [psi] L [in] Pier 7 Rocking Pier 6 Rocking Pier 4 Rocking Pier 3 Rocking Pier 2 Rocking 0.5 a 2126.6 36.8 PCE [psi] L [in] 3740.6 17.6 PCE [psi] L [in] 11308.9 53.2 80.1 h.n [in] 72.5 heff [inff [in]he 72.5 he [in] 80.1 he [in] L/heff 0.24 L/heff 0.7 0.51 L/heff 0.22 L/heff 111.3 Vr [lb] Bed Joint Sliding 27 vte [psi] 26 vme Vr [lb] 1017.2 Bed Joint Sliding 27 vte [psi] 26 vme 485.2 Vr [Ib] Bed Joint Sliding 27 vte [psi] 26 vme 369.8 V, [Ib] Bed Joint Sliding 27 vte [psi] 26 vme 0.66 3380.2 Vr [Ib] Bed Joint Sliding 27 vte [psi] 26 vme An [in 2] An [in 2 ] An [in 2 ] An [in2 ] An [in 2 ] 27.6 27.6 L/heff 27.6 27.6 27.6 Vbjsl 718.7 Vbjsl 718.7 Vbjsl 718.7 Vbjsl 718.7 Vbjsl 718.7 Vbjs2 509.2 Diagonal Tension Vbjs2 1539.5 Diagonal Tension Vbjs2 1063.3 Diagonal Tension Vbjs2 1870.3 Diagonal Tension Vbjs2 5654.4 Diagonal Tension f'dt Beta Vdt 13 0.67 f'dt 249.4 Vdt Beta 13 0.73 f'dt 291.6 Vdt Beta Toe Crushing Toe Crushing 13 0.67 f'dt 258.6 Vdt Beta Toe Crushing 13 1 fdt 404.9 Vdt 13 0.67 Beta 324.5 Toe Crushing Toe Crushing fae [Psi] 1.0 fae [Psi] 2.9 fae [Psi] 2.0 fae [Psi] 3.5 fae [Psi] 10.6 f'me [Psi] 780 f'me [Psi] 780 f'me [PSI] 780 f'me [PSI] 780 f'me [Psi] 780 Weight above [lbs] Area [in 2] 1018.5 1065.6 123.4 Vtc 111.3 Min Rocking 2126.6 Weight above [lbs] 3740.6 Weight above [lbs] 1065.6 Area [in2] 1065.6 Area [in 2] 1065.6 408.3 Vtc 3682.8 369.8 Min Weight above [lbs] Area [in 2] 3079.1 1065.6 Weight above [lbs] Area [in 2] Vtc 1124.2 Vtc 537.2 Vtc Min 291.6 Min 258.6 Min Diagonal Tension Diagonal Tension 100 Rocking 11308.9 324.5 Diagonal Tension 0.5 a 0.5 a 6575.4 0.5 a 7810.6 36.8 [PSI] L [in] lwf [in] 80.1 hff9 L/heff 0.46 L/heff 0.19 L/heff Vr [Ib] 567.3 Vr [lb] Vr 1612.4 [lb] PCE 17.6 L [in] i 0.5 a PCE [Psi] 20478.3 L [in] 17.6 91.8 he [in] 87.5 hf[in] 87.5 L/heff 0.20 Vr [lb] 888.2 0.6 5185.2 0.4 1/heff 3872.6 Vr [lb] Bed Joint Sliding Bed Joint Sliding 27 vte [psi] 27 vte [psi] 27 vte [psi] vne 26 vme 26 vme 26 vie 26 27.6 An [in 2 ] 27.6 An [in 2 ] 9807.3 36.8 vte [psi] 27.6 PCE [Psi] L [in] 27 An [in 2 ] 0.5 a 53.2 Bed Joint Sliding Bed Joint Sliding Bed Joint Slidin 19879.3 PCE [PsI] [PSI] L [in] PCE Pier 14 Rocking Pier 12 Rocking Pier 11 Rocking Pier 10 Rocking Pier 8 Rocking 2 27.6 An [in ] vte [psi] 27 vie 26 2 An [in ] 27.6 Vbjsl 718.7 Vbjsl 718.7 Vbjsl 718.7 Vbjsi 718.7 Vbjsl 718.7 Vbjs2 3905.3 VbJs2 3287.7 Vbjs2 9939.6 VbJs2 10239.2 Vbjs2 4903.7 13 f'dt 0.67 Beta 301.1 Vdt Toe Crushing dt 1 Diagonal Tension 13ft f't Diagonal Tension 13 f'dt 1 Beta 436.4 Vdt Crushing ____Toe Diagonal Tension 1 13 f'dt 0.67 Beta 375.7 Vdt 'tDiagonal Tension f'dt 0.67 Beta 379.0 Vdt Weight above Weight above f m [PI]780 f'e[S]780 f'me [psi] 780 19.2 20478.3 [lbs] Weight above [lbs] 7810.6 Weight above [Ibs] 6575.4 Weight above [lbs] Area [in 2] 1065.6 Area [in 2 ] 1065.6 Area [in 2] 1065.6 Area [in2 ] Vtc Min 1767.5 301.1 Vtc Min Vtc Min 5564.5 375.7 Vtc Mi 101 Toe Crushing 780 fae [PSI] Diagonal Tension 469.6 Vdt f'me [PSI] 18.7 Diagonal Tension 1 Beta f'e[S]780 fae [PSI] Diagonal Tension 13 9.2 fe[S]6.2 19879.3 1 fae [PSI] fae PSI]7.3 623.2 436.4 dtDiagonal Tension f'dt Toe Crushing Crushing ____Toe 1 13 1065.6 4151.5 379.0 Diagonal Tension [lbs] 9807.3 Area [in 2] 1065.6 Vtc Min 970.3 469.6 Diagonal Tension Rocking 0.5 a 0.5 0.5 a 13729.8 53.2 PCE [psi] L [in] 36.8 PCE [psi] L [in] he [in] 87.5 he [in] 91.8 hI[in] L/heff 0.61 L/heff 0.40 L/heff PCE [psi] L [in] Vr [lb] 29650.4 8118.5 Vr [Ib] Bed Joint Sliding vte [psi] vme An [in 2] Rocking Rocking Rocking a 2473.4 Vr [lb] Bed Joint Sliding 25810.0 0.5 36.3 78.2 hef [in) 0.5 5381.2 59950.4 Vr [lb] 0.5 a 17.6 PCE [psi] L [in] 78.2 he[in] L/heff Bed Joint Sliding Rocking Rocking a PCE [psi] L [in] Pier 21 Pier 20 Pier 19 Pier 18 Pier 16 Pier 15 0.2 6068.6 53.2 PCE [psi] L [in] 78.24 he [in] 78.2 L/heff 0.5 37885.5 0.7 L/heff Vr [lb] Bed Joint Sliding 0.5 a 11594.5 Vr [lb] Bed Joint Sliding 26166.0 36.8 5530.7 Bed Joint Sliding 27 26 vte [psi] vme 27 26 vte [psi] vine 27 26 vte [psi) vme 27 26 vte [psi] vine 27 26 27.6 An [in 2 ] 27.6 An [in 2] 27.6 An [in 2] 27.6 An [in2] 27.6 vte [psi] vme 27 26 An [in 2] 27.6 Vbjsl 718.7 Vbjsi 718.7 Vbjsl 718.7 Vbjsi 718.7 Vbjsl 718.7 Vbjsl 718.7 Vbjs2 14825.2 Vbjs2 6864.9 VbJs2 12905 Vbjs2 29975.2 Vbjs2 18942.8 Vbjs2 13083 Diagonal Tension Diagonal Tension 13 f'dt 0.67 Beta 426.6 Vdt Diagonal Tension 13_____ Beta f'dt 339.7 Vdt Toe Crushing 13 0.67 Beta Diagonal Tension 0.67 407.4 Vt Toe Crushing Diagonal Tension 13 f't 0.22 Beta 186.6 Vt Toe Crushing Diagonal Tension 13 f'dt 0.68 Beta 472.3 Vm Toe Crushing 13 f'dt 0.67 Beta 409.2 Vdt Toe Crushing Toe Crushing fae [PSI] 27.8 fae [PSI] 12.9 fae [PSI] 24.2 fae [PSI] 56.3 fae [PSI] 35.6 fae [PSI] 24.6 f'me [PSI] 780 f'me [PSI] 780 f' me [PSI] 780 f'me [PSI] 780 f'me [PSI] 780 f'me [PSI] 780 Weight above [lbs] 29650.4 Weight above [lbs] 13729.8 Weight above [lbs] 25810 Weight above [lbs] 59950.4 Weight above [lbs] Area [in 2] 1065.6 Area [in 2] 1065.6 Area [in 2] 1065.6 Area [in 2] 1065.6 Area [in 2] Vtc Min 8560.9 426.6 Vtc Min 2683.4 339.7 Vtc Min 5713.9 407.4 Vtc Min 6048.1 186.6 Vtc Min Diagonal Tension Diagonal Tension Diagonal Tension Diagonal Tension 102 37885.5 1065.6 12043.9 472.3 Diagonal Tension Weight above [lbs] 26166 Area [in 2] 1065.6 Vtc 5868.8 Min 409.2 Diagonal Tension Pier 1 Pier 2 Rocking Rocking 0.5 a PCE [psi] L [in] he [in] 3312.3 57.2 72.5 0.8 L/heff V, [lb] Bed Joint Sliding 1177.1 27.0 vte [psi] 26.0 vme 2 27.6 An [in ] 718.7 Vbjsl 1656.1 Vbjs2 a PCE [psi] L [in] hew [in] Vr [Ib] Bed Joint Sliding 0.8 315.9 Vdt 26.0 vme 2 718.7 VbJsl 1927.0 Vbjs2 Beta 406.2 Vdt 0.7 Vr [lb] Bed Joint Sliding 3.1 780.0 f'.. [psi] 27.0 vte [psi] 26.0 vme 2 27.6 An [in ] L [in] heff [in] 1.9 L/heff Vr [Ib] Bed Joint Sliding 3.6 780.0 PCE [psi] L [in] hew [in] Vr [Ib] Bed Joint Sliding 27.0 vte [psi] 27.0 vme 26.0 vme 26.0 2 27.6 An [in ] 2 718.7 Vbjsl Vbjs2 6082.7 13238.7 Diagonal Tension Vbjs2 t 351.9 Vdt Beta Vdt 11.4 780.0 f'me [P-si] 3312.3 1065.6 3853.9 1065.6 1300.5 315.9 Vtc Min 1758.9 406.2 Vtc Min 12165.4 1065.6 4255.3 351.9 Diagonal Tension 103 718.7 10692.4 Diagonal Tension 13.0 1.0 f'dt 613.1 Vdt 13.0 0.7 384.0 Beta Toe Crushin g Toe Crushing fae [Psi] 27.6 An [in ] Vbjs2 0.7 6000.3 vte [psi] Vbjsl 0 f'. Beta 0.6 L/heff 718.7 Weight above [lbs] Area [in 2] Diagonal Tension 22287.5 0.5 21384.9 57.2 91.8 a Vbjsi Weight above [Ibs] Area [in 2] Diagonal Tension PCE [psi] Toe Crushing fae [Psi] f'me [Psi] 3911.6 0.5 26477.4 124.6 66.6 a Diagonal Tension 13. 1.0 f'dt 57.2 80.1 L/heff Toe Crushingi _fae [PSI] Vtc Min 27.6 An [in ] Toe Crushing Weight above [Ibs] Area [in2] 27.0 12165.4 L [in] hew [in] Diagonal Tension 13.0 Beta 1593.6 vte [psi] Diagonal Tension f'dt 0.9 L/heff 0.5 a PCE [psi] Rocking Rocking Rocking 0.5 3853.9 66.6 72.5 Pier 9 Pier 6 Pier 5 24.8 fae [Psi] 780.0 f'me [Psi] 780.0 f'me [Psi] Weight above [lbs] Area [in2] 26477.4 1065.6 Weight above [lbs] Area [in 2] Vtc Min 23637.0 613.1 Vtc Min Diagonal Tension 20.1 fae [Psi] 21384.9 1065.6 6422.0 384.0 Diagonal Tension a PCE [psi] L [in] heff [n] 0.5 24881.8 66.6 91.8 0.7 L/heff Vr [Ib] Bed Joint 8123.2 Sliding a PCE [psi] L [in] he" [inl 0.5 31896.0 57.2 87.5 0.7 L/heff V, [lb] Bed Joint Sliding 9394.8 vte [psi] 27.0 vte [psi] 27.0 vme An [in 2 ] 26.0 27.6 718.7 12440.9 vme An [in2] 26.0 27.6 718.7 15948.0 Vbjsl Vbjs2 Vbjsl Vbjs2 Beta 13.0 1.0 13.0 f'dt 0.7 Beta 437.5 Vm 689.2 Vm 478.0 Vm 407.4 Vdt Toe Crushing Toe Crushing 29.9 23.4 fae [PSI] fae [PSI] 600._ 9 Vdt V_ f'me [PSI] Weight above [lbs] 780.0 24881.8 PCE [psi] L [in] heff iat] f'me [PSI] Weight above [lbs] 780.0 31896.0 Vr [Ib] Bed Joint Sliding 12718.6 PCE [psi] L [in] hei [in] 0.5 40754.9 28.8 78.2 0.4 L/heff Vr [lb] Bed Joint 6750.8 Sliding 27.0 vte [psi] 27.0 vme An [in 2 ] 26.0 27.6 718.7 18555.9 vme An [in 2] 26.0 27.6 718.7 20377.4 Vbjsl Vbjs2 Vbjsl Vbjs2 f'dt Beta 13.0 1.0 a PCE [psi] L [in] heff [n] 0.7 Beta 0.5 Vr [lb] Bed Joint Sliding vte [psi] vme An [in 2 ] Vbjsl Vbjs2 5381.2 a PCE [psi] L [in] heff [in] 0.3 Vr [Ib] Bed Joint 9361.5 Sliding 27.0 26.0 27.6 718.7 12905.0 13.0 f'dt 0.5 59950.4 27.2 78.2 L/heff vte [psi] vme An [in 2 ] Vbjsl Vbjs2 0.7 Beta 27.0 26.0 27.6 718.7 29975.2 Diagonal Tension Diagonal Tension 13.0 f'dt 0.5 25810.0 36.3 78.2 L/heff Diagonal Tension Diagonal Tension 13.0 f'dt 0.3 Beta 287.8 Toe Crushing 34.8 fae [PSI] f'me [PSI] Weight above [lbs] 1065.6 Area [in 2] 1065.6 Area [in2] Vtc Min 8639.7 600.9 Vtc Min 9866.4 437.5 Vtc Min Diagonal Tension 0.8 a vte [psi] Area [in 2] Diagonal Tension 0.5 37111.7 66.6 87.5 L/heff Diagonal Tension Diagonal Tension f'dt a Pier 19 Rocking Pier 18 Rocking Pier 17 Rocking Pier 14 Rocking Pier 13 Rocking Pier 10 Rocking 780.0 37111.7 1065.6 13230.3 689.2 Toe Crus ing 38.2 fae [PSI] f'me [PSI] Weight above [lbs] 780.0 40754.9 Toe Crushing 24.2 fae [PSI] f'me [PSI] Weight above [lbs] 780.0 25810.0 Toe Crushing 56.3 fae [PSI] f'e [PSI] Weight above [lbs] 780.0 59950.4 Area [in 2] 1065.6 Area [in 2] 1065.6 Area [in 2] 1065.6 Vtc Min 6975.5 478.0 Vtc Min 5713.9 407.4 Vtc Min 9329.9 287.8 Diagonal Tension Bed Joint Sliding 104 Diagonal Tension Diagonal Tension Von Mises Shear Stresses from SAP2000 Full Wall Max Von Mises Shear Stress [psil at various base shears -t 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 4 DQT7 agona enson )6'77 '1 95 A4 A 28 Diagonal Tension Diagonal Tension Diagonal Tension Diagonal Tension Diagonal Tension Bed Joint Sliding Diagonal Tension Diagonal Tension Diagonal Tension Bed Joint Sliding Diagonal Tension Diagonal Tension Diagonal Tension Diagonal Tension Diagonal Tension Diagonal Tension Diagonal Tension 1 154 13Q4A O 50kips 22.93 19.43 21.99 100 kips 28.2 26.23 28.3 250 kips 67.72 53.23 50.14 500 kips 145.23 102.24 97.52 34.82 45.69 55.4 70.57 52.82 74.03 94.98 120.43 67.37 102.83 139.33 180.23 80.14 107.33 161.02 175.63 226.39 29.49 53.81 70.32 88.38 38.8 86.31 122.19 157.38 39.49 119.35 179.16 241.65 74.23 85.96 176.42 233.25 321.67 61.55 115.84 118.38 144.12 99.35 138.79 206.84 269.11 174.57 201.49 239.27 242.42 242.18 350.66 1000 kips 300.28 202.15 185.37 1l44 . Pier 1 2 3 40.74 kips 22.08 18.5 20.9 Failure Shear 315.88 417.77 291.64 Failure Mechanism Diagonal Tension Diagonal Tension Diagonal Tension . Characterization 351.86 543.98 324.46 301.07 383.97 649.86 375.68 379.00 437.51 718.66 426.64 339.72 478.00 407.37 564.06 472.27 409.19 36.91 44.57 52.77 67.36 56.86 72.29 90.07 114.25 74.44 100.64 132.05 168.98 95.13 113.77 157.27 168.08 215.48 239.04 232.74 415.1 327.84 1500 ki s 297.68 Characterization Fail Failure . Pier Mechanism Shear Diagonal 315.9 1 5 6 Diagonal Tension Diagonal Tension Diagonal Tension 10 Diagonal Tension Diagonal 13 Tension Diagonal 60 kips 31.07 80 kips 38.41 90 kips 100 kips 175 kips 250 kips 42.31 46.29 79.38 117.88 20.0 22.81 32.85 36.71 44.74 48.87 53.05 85.25 118.08 I 351.9 42.5 39.69 43.11 47.46 59.09 69.75 80.61 162.25 243.96 613.1 47.3 51.05 80.64 90.96 111.61 121.96 123.81 209.93 287.56 384.0 68.3 61.92 64.08 86.25 130.67 152.93 175.17 342.14 600.9 78.5 89.21 155.15 176.68 219.72 241.29 262.83 424.49 437.5 92.7 79.17 118.99 156.36 231.24 268.77 306.28 292.5 400.84 404.48 441.82 273.8 383.02 464.42 ___________ I 689.2 113.0 146.05 255.21 478.0 110.1 61.48 210.29 407.4 176.0 202.5 278.84 Tension 565.15 ___ ___ ___ Tension Diagonal 19 406.3 Tension Diagonal 18 50 kips 28.12 Tension Diagonal 17 20.71 kips 21.13 Tension 9 14 10 kips 19.6 Tension Diagonal 2 Flat Wall Section Max Von Mises Shear Stress [psi] at various base shears 302.78 359.69 386.99 287.8 Curved Wall Section Characterization ailure F17.6 Failure Shear Mechanism . Pier Rocking 2 111.29 Max Von Mises Shea Stress [psi] at various base shears 175 100 90 80 60 50 23.07 26.19 33.13 47.69 55.15 62.67 109.67 61.24 66.39 105.4 Diagonal Tension 291.64 25.87 41.05 46.02 4 Diagonal Tension 258.57 45.56 75.25 84.82 104.13 113.89 123.68 197.58 6 Rocking 369.81 114.76 141.89 100-02 198.01 226.55 127.29 255.33 136.63 208.34 1178.31 7 Di onal Tension . 3 56.12 32446 49.13 65 34 8 Diagonal Tension 301.07 98.72 10 Diagonal Tension 436.35 31.47 11 12 Diagonal Tension Diagona1 Tension 375.68 242.59 379.00 109.93 14 jDiagonal Tension 469.60 27.19 15 Diagonal Tension 426.64 157.28 16 Diagonal Tension 339.72 245.3 19 Diagonal Tension 186.57 35.36 20 Diagonal Tension 472.27 199.59 21 Diagonal Tension 409.19 327.24 91.28 229.41 106 118.06 1 206.65 1 259.61 300.92 444.07 j 286.27 Long Pier Section Max Von Mises Shear Stress [psi] at various base shears g .~_ 7 1 Diagonal Tension_ , 11 1 Diagonal Tension , 15 20 Diagonal Tension Diagonal Tension Failure Shear 7.5 kips 29Q14 327 __ _ 324.46 375.68 426.64 472.27 98.39 197.2 330.5 457.64 10 kips + 3966 . , 31 Failure Mechanism IaTe-nsion% Dia , Pier 107 t 15 kips 20 kips 40 kips 50 kips 5346 67.34 100.32 150.77 . Characterization