National Academies Press: OpenBook
Page i
Suggested Citation:"Front Matter." National Academies of Sciences, Engineering, and Medicine. 2018. Bridge Superstructure Tolerance to Total and Differential Foundation Movements. Washington, DC: The National Academies Press. doi: 10.17226/25041.
×
Page R1
Page ii
Suggested Citation:"Front Matter." National Academies of Sciences, Engineering, and Medicine. 2018. Bridge Superstructure Tolerance to Total and Differential Foundation Movements. Washington, DC: The National Academies Press. doi: 10.17226/25041.
×
Page R2
Page iii
Suggested Citation:"Front Matter." National Academies of Sciences, Engineering, and Medicine. 2018. Bridge Superstructure Tolerance to Total and Differential Foundation Movements. Washington, DC: The National Academies Press. doi: 10.17226/25041.
×
Page R3
Page iv
Suggested Citation:"Front Matter." National Academies of Sciences, Engineering, and Medicine. 2018. Bridge Superstructure Tolerance to Total and Differential Foundation Movements. Washington, DC: The National Academies Press. doi: 10.17226/25041.
×
Page R4
Page v
Suggested Citation:"Front Matter." National Academies of Sciences, Engineering, and Medicine. 2018. Bridge Superstructure Tolerance to Total and Differential Foundation Movements. Washington, DC: The National Academies Press. doi: 10.17226/25041.
×
Page R5
Page vi
Suggested Citation:"Front Matter." National Academies of Sciences, Engineering, and Medicine. 2018. Bridge Superstructure Tolerance to Total and Differential Foundation Movements. Washington, DC: The National Academies Press. doi: 10.17226/25041.
×
Page R6
Page vii
Suggested Citation:"Front Matter." National Academies of Sciences, Engineering, and Medicine. 2018. Bridge Superstructure Tolerance to Total and Differential Foundation Movements. Washington, DC: The National Academies Press. doi: 10.17226/25041.
×
Page R7
Page viii
Suggested Citation:"Front Matter." National Academies of Sciences, Engineering, and Medicine. 2018. Bridge Superstructure Tolerance to Total and Differential Foundation Movements. Washington, DC: The National Academies Press. doi: 10.17226/25041.
×
Page R8
Page ix
Suggested Citation:"Front Matter." National Academies of Sciences, Engineering, and Medicine. 2018. Bridge Superstructure Tolerance to Total and Differential Foundation Movements. Washington, DC: The National Academies Press. doi: 10.17226/25041.
×
Page R9
Page x
Suggested Citation:"Front Matter." National Academies of Sciences, Engineering, and Medicine. 2018. Bridge Superstructure Tolerance to Total and Differential Foundation Movements. Washington, DC: The National Academies Press. doi: 10.17226/25041.
×
Page R10
Page xi
Suggested Citation:"Front Matter." National Academies of Sciences, Engineering, and Medicine. 2018. Bridge Superstructure Tolerance to Total and Differential Foundation Movements. Washington, DC: The National Academies Press. doi: 10.17226/25041.
×
Page R11

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

NCHRP Web-Only Document 245: Bridge Superstructure Tolerance to Total and Differential Foundation Movements Franklin Moon Nick Romano David Masceri John Braley, Rutgers University Dennis R. Mertz University of Delaware Naresh Samtani NCS GeoResources, LLC Thomas Murphy Maria Lopez de Murphy Modjeski and Masters Contractor’s Final Report for NCHRP Project 12-103 Submitted December 2017 ACKNOWLEDGMENT This work was sponsored by the American Association of State Highway and Transportation Officials (AASHTO), in cooperation with the Federal Highway Administration, and was conducted in the National Cooperative Highway Research Program (NCHRP), which is administered by the Transportation Research Board (TRB) of the National Academies of Sciences, Engineering, and Medicine. COPYRIGHT INFORMATION Authors herein are responsible for the authenticity of their materials and for obtaining written permissions from publishers or persons who own the copyright to any previously published or copyrighted material used herein. Cooperative Research Programs (CRP) grants permission to reproduce material in this publication for classroom and not-for-profit purposes. Permission is given with the understanding that none of the material will be used to imply TRB, AASHTO, FAA, FHWA, FMCSA, FRA, FTA, Office of the Assistant Secretary for Research and Technology, PHMSA, or TDC endorsement of a particular product, method, or practice. It is expected that those reproducing the material in this document for educational and not-for-profit uses will give appropriate acknowledgment of the source of any reprinted or reproduced material. For other uses of the material, request permission from CRP. DISCLAIMER The opinions and conclusions expressed or implied in this report are those of the researchers who performed the research. They are not necessarily those of the Transportation Research Board; the National Academies of Sciences, Engineering, and Medicine; or the program sponsors. The information contained in this document was taken directly from the submission of the author(s). This material has not been edited by TRB.

The National Academy of Sciences was established in 1863 by an Act of Congress, signed by President Lincoln, as a private, non- governmental institution to advise the nation on issues related to science and technology. Members are elected by their peers for outstanding contributions to research. Dr. Marcia McNutt is president. The National Academy of Engineering was established in 1964 under the charter of the National Academy of Sciences to bring the practices of engineering to advising the nation. Members are elected by their peers for extraordinary contributions to engineering. Dr. C. D. Mote, Jr., is president. The National Academy of Medicine (formerly the Institute of Medicine) was established in 1970 under the charter of the National Academy of Sciences to advise the nation on medical and health issues. Members are elected by their peers for distinguished contributions to medicine and health. Dr. Victor J. Dzau is president. The three Academies work together as the National Academies of Sciences, Engineering, and Medicine to provide independent, objective analysis and advice to the nation and conduct other activities to solve complex problems and inform public policy decisions. The National Academies also encourage education and research, recognize outstanding contributions to knowledge, and increase public understanding in matters of science, engineering, and medicine. Learn more about the National Academies of Sciences, Engineering, and Medicine at www.national-academies.org. The Transportation Research Board is one of seven major programs of the National Academies of Sciences, Engineering, and Medicine. The mission of the Transportation Research Board is to increase the benefits that transportation contributes to society by providing leadership in transportation innovation and progress through research and information exchange, conducted within a setting that is objective, interdisciplinary, and multimodal. The Board’s varied committees, task forces, and panels annually engage about 7,000 engineers, scientists, and other transportation researchers and practitioners from the public and private sectors and academia, all of whom contribute their expertise in the public interest. The program is supported by state transportation departments, federal agencies including the component administrations of the U.S. Department of Transportation, and other organizations and individuals interested in the development of transportation. Learn more about the Transportation Research Board at www.TRB.org.

NCHRP Project 12-103 iv In Memoriam Professor Dennis R. Mertz, co-principal investigator for NCHRP Project 12-103, died Friday, August 12, 2016. Although Dennis was not able to review the final version of this report, he contributed significantly to the formulation of the approach and the execution of the research. In addition to his role on Project 12-103, Dennis was a friend and mentor to several members of NCHRP 12-103 research team over the last few decades. His deep expertise in bridge performance and design, and his ability to convey complex issues in understandable terms is unparalleled in the field of bridge engineering. He will be missed.

NCHRP Project 12-103 v Abstract The purpose of this research was to develop a comprehensive understanding of the levels of support movements that bridges may tolerate before exceeding strength or service limit states. This research was devised to satisfy the following objectives: 1) Develop analytical procedures to objectively determine the acceptable levels of bridge foundation movements based upon superstructure tolerance considering AASHTO LRFD strength and service limit states (Phase II). 2) Propose revisions to the AASHTO LRFD Bridge Design Specifications that provide rational guidance for foundation movement limits that shall include vertical and rotational movements (Phase III and IV). Phase II focused on satisfying Objective 1 through (a) estimating tolerable support movements for common bridge types, (b) identifying the parameters that influence the level of tolerable support movement, and (c) developing simplified expressions for estimating tolerable support movement. In addition, this research examined the current AASHTO LRFD criteria, which was developed under previous specifications (i.e. AASHTO Standard Specifications), in light of the results obtained through (a). For the estimation of tolerable support movements, this research examined the following four types of support movements: • Longitudinal-differential (LD) movements – This type of support movement occurs when all bearing locations at a single support undergo the same amount of vertical translation. The tolerable level of this type of support movement may be limited by Strength and Service limit states or ride-ability criteria. • Transverse-differential (TD) movements – This type of support movement occurs when a single support undergoes a vertical translation that varies linearly in the transverse direction. The tolerable level of this type of support movement may be limited by Strength and Service limit states or ride-ability criteria. • Horizontal movement – This type of support movement occurs when a support moves horizontally in the longitudinal direction of the superstructure. The tolerable level of this type of

NCHRP Project 12-103 vi support movement is limited by the displacement capacity of the movement systems (joints and bearings). • Total (uniform) movements – This type of support movement occurs when all supports of a bridge undergo an equal vertical translation. The tolerable level of this type of support movement is limited by either clearance or ride-ability criteria. To accomplish the goals for Phase II, a large number of steel and pre-stressed (PS) concrete multi-girder bridge configurations and parameters were examined through a multivariate parametric study. This study employed an automated member sizing algorithm as well as automated three-dimensional (3D) finite element (FE) model construction, simulation, and results extraction approaches to permit a large sample population (or “suite”) of bridges to be examined. Tolerable support movements were calculated for each sample and the results were compared to the current AASHTO LRFD guidance when applicable. The guidance was deemed conservative if the observed level of tolerable support movement was found to be greater than what is suggested by AASHTO LRFD. In contrast, the guidance was deemed unconservative if the observed level of tolerable support movement was found to be less than what is suggested by AASHTO LRFD. Through this research four mechanisms were identified as influencing the tolerance to support movement for the bridge types examined. First, the AASHTO LRFD live load distribution factors used within the common single-line-girder (SLG) model have been found to be inherently conservative for certain bridge configurations. This conservatism allows certain bridges to exhibit a higher level of tolerance to LD and TD support movements. Second, the common tributary width approach to estimating dead load force effects can underestimate the level of these effects for exterior girders. As a result, some elements may be under-sized for dead load force effects, which can reduce (or completely eliminate) the tolerance of a bridge to support movement. Third, sections where the design is controlled by fatigue limit states exhibit higher levels of tolerance than other sections (since fatigue is only concerned with cyclic loading). This occurs in the positive moment region of steel multi-girder bridges, but does not influence PS concrete multi-girder bridges as the design of these bridges is not controlled by fatigue limit states. Fourth, bridges with higher skew angles have less tolerance to support movement, especially for steel bridges with skew angles greater than 45⁰. The results from the study of steel multi-girder bridges indicated that the current AASHTO LRFD criterion is conservative in certain situations and unconservative in others. For two-span continuous steel multi- girder bridges with skew angles less than 20⁰, the current AASHTO LRFD criterion was found to be

NCHRP Project 12-103 vii conservative for LD support movements. However, for larger skew angles and for three-span continuous steel multi-girder bridges, this criterion was found to be unconservative for approximately 20% of the notional bridges examined. Through analysis of the results and identification of relationships between various parameters and tolerable support movement, an alternative expression, which is a function of both girder spacing and span length, was developed. This expression is included within a draft ballot for the modification of the AASHTO LRFD. The results from the study of PS concrete multi-girder bridges indicated that this bridge type has relatively little tolerance to support movements (both LD and TD support movements). For two- and three-span continuous PS concrete bridges, the current AASHTO LRFD criterion was found to be unconservative for the entire population studied. Tolerance for these bridges was controlled by the Service III limit state when the movement induces tension in the PS concrete girder (e.g. a movement occurring at the pier). This is contrary to what was observed for steel bridges. The difference between steel and PS concrete bridges was traced to the fatigue limit state, which influences the design of steel multi-girder bridges in the positive moment region. This mechanism that provides additional conservatism in the positive bending section of steel bridges is not present with PS concrete bridges. The lack of this mechanism for PS concrete bridges renders them far more sensitive to both LD and TD support movements that occur at the pier. After analyzing the results, simplified expressions were developed to estimate the tolerable support movement of PS concrete bridges for both the Service III and Strength I limit states. These expressions were included within a draft ballot for the modification of the AASHTO LRFD. In addition to the primary bridge types of steel and PS concrete multi-girder bridges, this research conducted a limited parametric study on three secondary bridge types: open and closed steel box girder bridges, and multi-cell concrete box bridges. The goal of this secondary study was to investigate whether the observations made in the primary study could be extended to these secondary bridges types. The results from this study showed that multi-cell concrete box bridges follow the same trends as their multi-girder counterparts. In the case of steel box girder bridges, the results indicated that for flexural limit states they performed similar to multi-girder bridges. For shear limit states however, the steel box girder bridges were more sensitive to support movements in that they exhibited tolerable support movements approximately 10% smaller than their multi-girder counterparts.

NCHRP Project 12-103 viii This research also identified functionality limitations on tolerable support movements considering both rideability and clearance issues. In addition, a general procedure for calculating the tolerable support movement of bridges that fall outside the applicable range of the simplified expressions is presented.

NCHRP Project 12-103 ix Contents In Memoriam ............................................................................................................................................... iv Abstract ......................................................................................................................................................... v 1 Introduction .......................................................................................................................................... 1 1.1 Motivation and Research Objectives ............................................................................................ 1 1.2 Project Organization ..................................................................................................................... 4 1.3 Research Summary ....................................................................................................................... 5 1.4 Report Outline ............................................................................................................................. 10 2 Current AASHTO LRFD Guidance Related Tolerable Support Movement .......................................... 12 2.1 Summary of Current AASHTO LRFD Guidance ............................................................................ 12 2.2 Discussion of Comparison with AASHTO LRFD Guidance ........................................................... 13 3 Definition and Construction of 3D FE Bridge Suites ........................................................................... 14 3.1 Sampling of Parameters to Define a Bridge Suite (Task 2.1.1) ................................................... 14 3.2 Automated LRFD (Task 2.1.2) ...................................................................................................... 17 3.2.1 Approximation of Demands Using Single Line Girder (SLG) Model .................................... 17 3.2.2 Automated Sizing of Steel Members .................................................................................. 18 3.2.3 Automated Sizing Pre-Stressed Concrete Members ........................................................... 21 3.2.4 Software Validation ............................................................................................................. 25 3.3 Automated 3D FE Modeling of Bridge Suite (Task 2.1.3) ............................................................ 26 3.3.1 Model Form ......................................................................................................................... 27 3.3.2 Modeling the Superstructure .............................................................................................. 28 3.3.3 Defining Boundary Conditions at the Supports .................................................................. 29 4 Estimation of Maximum Tolerable Support Movement (Task 2.2) .................................................... 30 4.1 Definition of Longitudinal- and Transverse-Differential Support Movements ........................... 31 4.2 Simulation of Longitudinal- and Transverse-Differential Support Movements .......................... 32 4.3 Simulation of Dead Load ............................................................................................................. 33 4.3.1 Initial Dead Load (DC1) ....................................................................................................... 33 4.3.2 Superimposed Dead Load (DC2) ......................................................................................... 34 4.4 Simulation of Live Load ............................................................................................................... 34 4.5 Results Extraction ........................................................................................................................ 36 4.5.1 Response Regions of Interest .............................................................................................. 37 4.5.2 Moment on Composite Sections ......................................................................................... 38 4.5.3 Moment on Non-Composite Sections ................................................................................. 39

NCHRP Project 12-103 x 4.5.4 Shear in Girders ................................................................................................................... 39 4.6 Results Convergence ................................................................................................................... 40 5 Steel Multi-Girder Bridges .................................................................................................................. 43 5.1 Tolerable Support Movement Influences ................................................................................... 43 5.1.1 Live Load Distribution Factors ............................................................................................. 44 5.1.2 Load Distribution in Highly Skewed Bridges ....................................................................... 45 5.1.3 Elements Governed by the Fatigue Limit State .................................................................. 46 5.2 Methods for Identifying Tolerable Support Movement Influences ............................................ 47 5.3 Simple Span Steel Bridges ........................................................................................................... 48 5.3.1 Strength I Shear Tolerance to TD Support Movements ...................................................... 48 5.4 Two-Span Continuous Steel Bridges ........................................................................................... 50 5.4.1 Controlling Limit States ....................................................................................................... 50 5.4.2 Strength I Flexure Tolerance to LD Movements Occurring at the Abutment ..................... 52 5.4.3 Strength I Shear Tolerance to LD Movements Occurring at the Abutment ....................... 57 5.4.4 Service II Tolerance to LD Movements Occurring at the Abutment ................................... 58 5.4.5 Strength I Flexure Tolerance to TD Movements Occurring at the Abutment..................... 59 5.4.6 Strength I Shear Tolerance to TD Movements Occurring at the Abutment ....................... 63 5.4.7 Service II Tolerance to TD Movements Occurring at the Abutment ................................... 68 5.4.8 Strength I Flexure Tolerance to LD Movements Occurring at the Pier ............................... 69 5.4.9 Strength I Shear Tolerance to LD Movements Occurring at the Pier .................................. 72 5.4.10 Service II Tolerance to LD Movements Occurring at the Pier ............................................. 76 5.4.11 Strength I Flexure Tolerance to TD Movements Occurring at the Pier ............................... 77 5.4.12 Strength I Shear Tolerance to TD Movements Occurring at the Pier ................................. 83 5.4.13 Service II Tolerance to TD Movements Occurring at the Pier ............................................. 88 5.5 Three-Span Continuous Steel Bridges ......................................................................................... 89 5.5.1 Controlling Limit States ....................................................................................................... 89 5.5.2 Strength I Flexure Tolerance to LD Movements Occurring at the Abutment ..................... 92 5.5.3 Strength I Shear Tolerance to LD Movements Occurring at the Abutment ....................... 97 5.5.4 Service II Tolerance to LD Movements Occurring at the Abutment ................................. 100 5.5.5 Strength I Flexure Tolerance to TD Movements Occurring at the Abutment................... 101 5.5.6 Strength I Shear Tolerance to TD Movements Occurring at the Abutment ..................... 107 5.5.7 Service II Tolerance to TD Movements Occurring at the Abutment ................................. 111 5.5.8 Strength I Flexure Tolerance to LD Movements Occurring at the Pier ............................. 112

NCHRP Project 12-103 xi 5.5.9 Strength I Shear Tolerance to LD Movements Occurring at the Pier ................................ 115 5.5.10 Service II Tolerance to LD Movements Occurring at the Pier ........................................... 120 5.5.11 Strength I Flexure Tolerance to TD Movements Occurring at the Pier ............................. 121 5.5.12 Strength I Shear Tolerance to TD Movements Occurring at the Pier ............................... 126 5.5.13 Service II Tolerance to TD Movements Occurring at the Pier ........................................... 132 5.6 Summary of Results .................................................................................................................. 133 6 Pre-Stressed Concrete Multi-Girder Bridges..................................................................................... 141 6.1 Tolerable Support Movement Influences ................................................................................. 141 6.1.1 Live Load Distribution Factors ........................................................................................... 141 6.1.2 Load Distribution in Highly Skewed Bridges ..................................................................... 142 6.1.3 Uncoupling of Stiffness and Strength in Pre-Stressed Concrete Bridges .......................... 143 6.1.4 Elements Governed by the Fatigue Limit State ................................................................ 143 6.2 Methods for Identifying Tolerable Support Movement Influences .......................................... 143 6.3 Simple-Span Pre-Stressed Concrete Bridges ............................................................................. 144 6.3.1 Strength I Shear Tolerance to TD Movements .................................................................. 145 6.3.2 Service Tolerance to TD Movements ................................................................................ 148 6.4 Two-Span Continuous Pre-Stressed Concrete Bridges ............................................................. 153 6.4.1 Controlling Limit State ...................................................................................................... 153 6.4.2 Service Tolerance to LD Movements Occurring at the Abutment .................................... 155 6.4.3 Strength I Flexure Tolerance to LD Movements Occurring at the Abutment ................... 156 6.4.4 Strength I Shear Tolerance to LD Movements Occurring at the Abutment ..................... 158 6.4.5 Service Tolerance to TD Movements Occurring at the Abutment .................................... 162 6.4.6 Strength I Flexure Tolerance to TD Movements Occurring at the Abutment................... 167 6.4.7 Strength I Shear Tolerance to TD Movements Occurring at the Abutment ..................... 168 6.4.8 Service Tolerance to LD Movements Occurring at the Pier .............................................. 171 6.4.9 Strength I Flexure Tolerance to LD Movements Occurring at the Pier ............................. 171 6.4.10 Strength I Shear Tolerance to LD Movements Occurring at the Pier ................................ 172 6.4.11 Service Tolerance to TD Movements Occurring at the Pier .............................................. 173 6.4.12 Strength I Flexure Tolerance to TD Movements Occurring at the Pier ............................. 174 6.4.13 Strength I Shear Tolerance to TD Movements Occurring at the Pier ............................... 175 6.5 Three-Span Continuous Pre-Stressed Concrete Bridges ........................................................... 175 6.5.1 Controlling Limit State ...................................................................................................... 175 6.5.2 Service Tolerance to LD Movements Occurring at the Abutment .................................... 178

NCHRP Project 12-103 xii 6.5.3 Strength I Flexure Tolerance to LD Movements Occurring at the Abutment ................... 179 6.5.4 Strength I Shear Tolerance to LD Movements Occurring at the Abutment ..................... 180 6.5.5 Service Tolerance to TD Movements Occurring at the Abutment .................................... 184 6.5.6 Strength I Flexure Tolerance to TD Movements Occurring at the Abutment................... 186 6.5.7 Strength I Shear Tolerance to TD Movements Occurring at the Abutment ..................... 186 6.5.8 Service Tolerance to LD Movements Occurring at the Pier .............................................. 189 6.5.9 Strength I Flexure Tolerance to LD Movements Occurring at the Pier ............................. 190 6.5.10 Strength I Shear Tolerance to LD Movements Occurring at the Pier ................................ 191 6.5.11 Service Tolerance to TD Movements Occurring at the Pier .............................................. 192 6.5.12 Strength I Flexure Tolerance to TD Movements Occurring at the Pier ............................. 193 6.5.13 Strength I Shear Tolerance to TD Movements Occurring at the Pier ............................... 194 6.6 Summary of Results .................................................................................................................. 195 7 Spot Checking of Secondary Bridge Types (Task 2.4) ....................................................................... 202 7.1 Approach to Modeling, Simulation, and Evaluation of Secondary Bridges .............................. 203 7.2 Closed and Open Steel Boxes .................................................................................................... 205 7.3 Pre-Stressed Concrete Boxes .................................................................................................... 208 7.4 Conclusions ............................................................................................................................... 210 8 Functional Limitations on Tolerable Support Movements (Task 2.4) ............................................... 210 8.1 Background on Rideability Limitations to Tolerable Support Movement ................................ 210 8.2 Recommendations for Rideability Limitations to Tolerable Support Movements ................... 212 8.3 Clearance Limitations on Tolerable Support Movement .......................................................... 213 9 Framework for Estimating Maximum Tolerable Support Movement .............................................. 214 10 Conclusions & Recommendations ................................................................................................ 217 10.1 Tolerable Support Movements for Multi-Girder Bridges ......................................................... 217 10.1.1 Steel Multi-Girder Bridges................................................................................................. 219 10.1.2 Pre-Stressed Concrete Multi-Girder Bridges ..................................................................... 222 10.2 Spot Checking of Secondary Bridges ......................................................................................... 225 10.3 Functionality Limits on Tolerable Support Movement ............................................................. 226 10.4 Proposed Expressions for Estimating Tolerable Support Movements ..................................... 226 10.5 Proposed Revisions to AASHTO LRFD Bridge Design Specifications ......................................... 230 References ................................................................................................................................................ 235 Appendices A through F ............................................................................................................................ A-1

Next: 1 Introduction »
Bridge Superstructure Tolerance to Total and Differential Foundation Movements Get This Book
×
MyNAP members save 10% online.
Login or Register to save!
Download Free PDF

TRB's National Highway Cooperative Research Program (NCHRP) Web-Only Document 245: Bridge Superstructure Tolerance to Total and Differential Foundation Movements develops an understanding of the levels of support movements that bridges may tolerate before exceeding strength or service limit states. This research explores analytical procedures to objectively determine the acceptable levels of bridge foundation movements based upon superstructure tolerance considering American Association of State Highway and Transportation Officials (AASHTO) Load and Resistance Factor Design (LRFD) strength and service limit states (Phase II). The report also proposes revisions to the AASHTO LRFD Bridge Design Specifications that provide rational guidance for foundation movement limits that shall include vertical and rotational movements (Phase III and IV).

  1. ×

    Welcome to OpenBook!

    You're looking at OpenBook, NAP.edu's online reading room since 1999. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.

    Do you want to take a quick tour of the OpenBook's features?

    No Thanks Take a Tour »
  2. ×

    Show this book's table of contents, where you can jump to any chapter by name.

    « Back Next »
  3. ×

    ...or use these buttons to go back to the previous chapter or skip to the next one.

    « Back Next »
  4. ×

    Jump up to the previous page or down to the next one. Also, you can type in a page number and press Enter to go directly to that page in the book.

    « Back Next »
  5. ×

    To search the entire text of this book, type in your search term here and press Enter.

    « Back Next »
  6. ×

    Share a link to this book page on your preferred social network or via email.

    « Back Next »
  7. ×

    View our suggested citation for this chapter.

    « Back Next »
  8. ×

    Ready to take your reading offline? Click here to buy this book in print or download it as a free PDF, if available.

    « Back Next »
Stay Connected!