Reinforced Concrete Design
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F(P,Q) = Ffeic(R+TQ)
674 MATHEMA TICS: N. H. McCOY PR.OC. N. A. S. ON THE FUNCTION IN QUANTUM MECHANICS WHICH CORRESPONDS TO A GIVEN FUNCTION IN CLASSICAL MECHANICS By NEAL H. MCCoy DEPARTMENT OF MATHEMATICS, SMrTH COLLEGE Communicated October 11, 1932 Let f(p,q) denote a function of the canonical variables p,q of classical mechanics. In making the transition to quantum mechanics, the variables p,q are represented by Hermitian operators P,Q which, satisfy the com- mutation rule PQ- Qp = 'y1 (1) where Y = h/2ri and 1 indicates the unit operator. From group theoretic considerations, Weyll has obtained the following general rule for carrying a function over from classical to quantum mechanics. Express f(p,q) as a Fourier integral, f(p,q) = f feiC(r+TQ) r(,r)d(rdT (2) (or in any other way as a linear combination of the functions eW(ffP+T)). Then the function F(P,Q) in quantum mechanics which corresponds to f(p,q) is given by F(P,Q)- ff ei(0P+7Q) t(cr,r)dodr. (3) It is the purpose of this note to obtain an explicit expression for F(P,Q), and although we confine our statements to the case in which f(p,q) is a polynomial, the results remain formally correct for infinite series. Any polynomial G(P,Q) may, by means of relation (1), be written in a form in which all of the Q-factors occur on the left in each term. This form of the function G(P,Q) will be denoted by GQ(P,Q). -
Working Load to Break Load: Safety Factors in Composite Yacht Structures
Working Load to Break Load: Safety Factors in Composite Yacht Structures Dr. M. Hobbs* and Mr. L. McEwen* ABSTRACT The loads imposed on yacht structures fall broadly into two categories: the distributed forces imposed by the action of the wind and waves on the shell of the yacht, and the concentrated loads imposed by the rig and keel to their attachment points on the structure. This paper examines the nature of the latter set of loads and offers a methodology for the structural design based on those loadings. The loads imposed on a rig attachment point vary continuously while the yacht is sailing. Designers frequently quote "working load", "safe working load", "maximum load" or "break load" for a rigging attachment, but the relationship of this value to the varying load is not always clear. A set of nomenclature is presented to describe clearly the different load states from the maximum "steady-state" value, through the "peak, dynamic" value to the eventual break load of the fitting and of the composite structure. Having defined the loads, the structure must be designed to carry them with sufficient stiffness, strength and stability. Inherent in structural engineering is the need for safety factors to account for variations in load, material strength, geometry tolerances and other uncertainties. A rational approach to the inclusion of safety factors to account for these effects is presented. This approach allows the partial safety factors to be modified to suit the choice of material, the nature of the load and the structure and the method of analysis. Where more than one load acts on an area of the structure, combined load cases must be developed that model realistically the worst case scenario. -
Alkali-Silica Reactivity: an Overview of Research
SHRP-C-342 Alkali-Silica Reactivity: An Overview of Research Richard Helmuth Construction Technology Laboratories, Inc. With contributions by: David Stark Construction Technology Laboratories, Inc. Sidney Diamond Purdue University Micheline Moranville-Regourd Ecole Normale Superieure de Cachan Strategic Highway Research Program National Research Council Washington, DC 1993 Publication No. SHRP-C-342 ISBN 0-30cL05602-0 Contract C-202 Product No. 2010 Program Manager: Don M. Harriott Project Maxtager: Inam Jawed Program AIea Secretary: Carina Hreib Copyeditor: Katharyn L. Bine Brosseau May 1993 key words: additives aggregate alkali-silica reaction cracking expansion portland cement concrete standards Strategic Highway Research Program 2101 Consti!ution Avenue N.W. Washington, DC 20418 (202) 334-3774 The publicat:Lon of this report does not necessarily indicate approval or endorsement by the National Academy of Sciences, the United States Government, or the American Association of State Highway and Transportation Officials or its member states of the findings, opinions, conclusions, or recommendations either inferred or specifically expressed herein. ©1993 National Academy of Sciences 1.5M/NAP/593 Acknowledgments The research described herein was supported by the Strategic Highway Research Program (SHRP). SHRP is a unit of the National Research Council that was authorized by section 128 of the Surface Transportation and Uniform Relocation Assistance Act of 1987. This document has been written as a product of Strategic Highway Research Program (SHRP) Contract SHRP-87-C-202, "Eliminating or Minimizing Alkali-Silica Reactivity." The prime contractor for this project is Construction Technology Laboratories, with Purdue University, and Ecole Normale Superieure de Cachan, as subcontractors. Fundamental studies were initiated in Task A. -
Reinforced Concrete Failure Mechanisms
Reinforced Concrete Failure Mechanisms Best Practices in Dam and Levee Safety Risk Analyses Part E – Concrete Structures Chapter E-2 Last modified June 2017, Presented July 2019 Reinforced Concrete Failure Mechanisms OUTLINE: • Types of Structures • Spillway Piers • Navigation Lock Walls • Floodwalls • Slabs • Buttresses • Factors influencing strength and stability of reinforced concrete sections • National code requirements in the context of risk • Considerations when determining risk analysis failure probabilities based on structural analysis results • Typical event tree of the progression of failure 2 Reinforced Concrete Failure Mechanisms OBJECTIVES: • Get a broad overview of potential failure modes for different kinds of reinforced concrete structures • Understand the mechanisms that affect reinforced concrete failures • Understand how to construct an event tree to represent reinforced concrete failures • Understand how to estimate event probabilities and probability of breach Reinforced Concrete Failure Mechanisms SUMMARY OF KEY CONCEPTS: • Significant uncertainty for reinforced concrete failure mechanisms under seismic loading due to limited case histories • Concrete and reinforcement material properties can be determined with confidence for dams and floodwalls. • Type and duration of loading is important to understand – consider both static and earthquake loading • Ductile and Brittle Failure mechanisms • Seismic reinforcement details have changed dramatically over the past few decades; older concrete hydraulic structures may be -
THE ZEROS of QUASI-ANALYTIC FUNCTIONS1 (3) H(V) = — F U-2 Log
THE ZEROS OF QUASI-ANALYTICFUNCTIONS1 ARTHUR O. GARDER, JR. 1. Statement of the theorem. Definition 1. C(Mn, k) is the class of functions f(x) ior which there exist positive constants A and k and a sequence (Af„)"_0 such that (1) |/(n)(x)| ^AknM„, - oo < x< oo,» = 0,l,2, • • • . Definition 2. C(Mn, k) is a quasi-analytic class if 0 is the only element / of C(Mn, k) ior which there exists a point x0 at which /<">(*„)=0 for « = 0, 1, 2, ••• . We introduce the functions (2) T(u) = max u"(M„)_l, 0 ^ m < oo, and (3) H(v) = — f u-2 log T(u)du. T J l The following property of H(v) characterizes quasi-analytic classes. Theorem 3 [5, 78 ].2 Let lim,,.,*, M„/n= °o.A necessary and sufficient condition that C(Mn, k) be a quasi-analytic class is that (4) lim H(v) = oo, where H(v) is defined by (2) and (3). Definition 4. Let v(x) be a function satisfying (i) v(x) is continuously differentiable for 0^x< oo, (ii) v(0) = l, v'(x)^0, 0^x<oo, (iii) xv'(x)/v(x)=0 (log x)-1, x—>oo. We shall say that/(x)GC* if and only if f(x)EC(Ml, k), where M* = n\[v(n)]n and v(n) satisfies the above conditions. Definition 5. Let Z(u) be a real-valued function which is less Received by the editors November 29, 1954 and, in revised form, February 24, 1955. 1 Presented to the Graduate School of Washington University in partial fulfill- ment of the requirements for the degree of Doctor of Philosophy. -
CONDITIONAL EXPECTATION Definition 1. Let (Ω,F,P)
CONDITIONAL EXPECTATION 1. CONDITIONAL EXPECTATION: L2 THEORY ¡ Definition 1. Let (,F ,P) be a probability space and let G be a σ algebra contained in F . For ¡ any real random variable X L2(,F ,P), define E(X G ) to be the orthogonal projection of X 2 j onto the closed subspace L2(,G ,P). This definition may seem a bit strange at first, as it seems not to have any connection with the naive definition of conditional probability that you may have learned in elementary prob- ability. However, there is a compelling rationale for Definition 1: the orthogonal projection E(X G ) minimizes the expected squared difference E(X Y )2 among all random variables Y j ¡ 2 L2(,G ,P), so in a sense it is the best predictor of X based on the information in G . It may be helpful to consider the special case where the σ algebra G is generated by a single random ¡ variable Y , i.e., G σ(Y ). In this case, every G measurable random variable is a Borel function Æ ¡ of Y (exercise!), so E(X G ) is the unique Borel function h(Y ) (up to sets of probability zero) that j minimizes E(X h(Y ))2. The following exercise indicates that the special case where G σ(Y ) ¡ Æ for some real-valued random variable Y is in fact very general. Exercise 1. Show that if G is countably generated (that is, there is some countable collection of set B G such that G is the smallest σ algebra containing all of the sets B ) then there is a j 2 ¡ j G measurable real random variable Y such that G σ(Y ). -
A Computational Study of the Shear Behavior of Reinforced Concrete Beams Affected from Alkali–Silica Reactivity Damage
materials Article A Computational Study of the Shear Behavior of Reinforced Concrete Beams Affected from Alkali–Silica Reactivity Damage Bora Gencturk 1,* , Hadi Aryan 1, Mohammad Hanifehzadeh 1, Clotilde Chambreuil 2 and Jianqiang Wei 3 1 Sonny Astani Department of Civil and Environmental Engineering, University of Southern California, 3620 S. Vermont Avenue, KAP 210, Los Angeles, CA 90089-2531, USA; [email protected] (H.A.); [email protected] (M.H.) 2 LMT—Laboratoire de Mécanique et Technologie, University Paris-Saclay, ENS Paris-Saclay, CNRS, 91190 Gif-sur-Yvette, France; [email protected] 3 Department of Civil and Environmental Engineering, University of Massachusetts, Lowell, MA 01854-5104, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-(213)-821-1036; Fax: +1-(213)-744-1426 Abstract: In this study, an investigation of the shear behavior of full-scale reinforced concrete (RC) beams affected from alkali–silica reactivity damage is presented. A detailed finite element model (FEM) was developed and validated with data obtained from the experiments using several metrics, including a force–deformation curve, rebar strains, and crack maps and width. The validated FEM was used in a parametric study to investigate the potential impact of alkali–silica reactivity (ASR) degradation on the shear capacity of the beam. Degradations of concrete mechanical properties were correlated with ASR expansion using material test data and implemented in the FEM for Citation: Gencturk, B.; Aryan, H.; different expansions. The finite element (FE) analysis provided a better understanding of the failure Hanifehzadeh, M.; Chambreuil, C.; mechanism of ASR-affected RC beam and degradation in the capacity as a function of the ASR Wei, J. -
How to Make Concrete More Sustainable Harald Justnes1
Journal of Advanced Concrete Technology Vol. 13, 147-154, March 2015 / Copyright © 2015 Japan Concrete Institute 147 Scientific paper How to Make Concrete More Sustainable Harald Justnes1 A selected paper of ICCS13, Tokyo 2013. Received 12 November 2013, accepted 16 February 2015 doi:10.3151/jact.13.147 Abstract Production of cement is ranking 3rd in causes of man-made carbon dioxide emissions world-wide. Thus, in order to make concrete more sustainable one may work along one or more of the following routes; 1) Replacing cement in con- crete with larger amounts of supplementary cementing materials (SCMs) than usual, 2) Replacing cement in concrete with combinations of SCMs leading to synergic reactions enhancing strength, 3) Producing leaner concrete with less cement per cubic meter utilizing plasticizers and 4) Making concrete with local aggregate susceptible to alkali silica reaction (ASR) by using cement replacements, thus avoiding long transport of non-reactive aggregate. 1 Introduction SCMs, also uncommon ones like calcined marl 2. Replacing cement in concrete with combinations of The cement industry world-wide is calculated to bring SCMs leading to synergic reactions enhancing about 5-8% of the total global anthropogenic carbon strength dioxide (CO2) emissions. The general estimate is about 3. Producing leaner concrete with less cement per cubic 1 tonne of CO2 emission per tonne clinker produced, if meter utilizing plasticizers. fossil fuel is used and no measures are taken to reduce it. 4. Making concrete with local aggregate susceptible to The 3rd rank is not because cement is such a bad mate- alkali silica reaction (ASR) by using cement re- rial with respect to CO2 emissions, but owing to the fact placements, thus avoiding long transport of non- that it is so widely used to construct the infrastructure reactive aggregate and buildings of modern society as we know it. -
ARIC Surveillance Variable Dictionary Incident Stroke
ARIC Surveillance Variable Dictionary Incident Stroke April 2002 j:\aric\admin\vardict\stroke_inc ARIC Surveillance Variable Dictionary – Incidenta Stroke Table of Contents for Year YYb Alphabetical Index .....................................................................................................................................ii 1. Classification Variable COMPDIAG (Computer Diagnosis) ............................................................................................... 1 COMP_DX (Computer Diagnosis – Format Value) ....................................................................... 2 FINALDX (Final Diagnosis) .......................................................................................................... 3 FINAL_DX (Final Diagnosis – Format Value) .............................................................................. 4 2. Event Time/Type & Eligibility Variable DISDATE (Discharge/Death Date) ................................................................................................5 EVENTYPE (Event Type) .............................................................................................................. 6 YEAR (Event Year) ........................................................................................................................ 7 SK_ELIG2 (Stoke Eligibility) ........................................................................................................ 8 3. Incidence* Stroke Variable 3.1 Incident Event CENSORYY (Censoring Date) ..................................................................................................... -
Proposal for Generation Panel for Latin Script Label Generation Ruleset for the Root Zone
Generation Panel for Latin Script Label Generation Ruleset for the Root Zone Proposal for Generation Panel for Latin Script Label Generation Ruleset for the Root Zone Table of Contents 1. General Information 2 1.1 Use of Latin Script characters in domain names 3 1.2 Target Script for the Proposed Generation Panel 4 1.2.1 Diacritics 5 1.3 Countries with significant user communities using Latin script 6 2. Proposed Initial Composition of the Panel and Relationship with Past Work or Working Groups 7 3. Work Plan 13 3.1 Suggested Timeline with Significant Milestones 13 3.2 Sources for funding travel and logistics 16 3.3 Need for ICANN provided advisors 17 4. References 17 1 Generation Panel for Latin Script Label Generation Ruleset for the Root Zone 1. General Information The Latin script1 or Roman script is a major writing system of the world today, and the most widely used in terms of number of languages and number of speakers, with circa 70% of the world’s readers and writers making use of this script2 (Wikipedia). Historically, it is derived from the Greek alphabet, as is the Cyrillic script. The Greek alphabet is in turn derived from the Phoenician alphabet which dates to the mid-11th century BC and is itself based on older scripts. This explains why Latin, Cyrillic and Greek share some letters, which may become relevant to the ruleset in the form of cross-script variants. The Latin alphabet itself originated in Italy in the 7th Century BC. The original alphabet contained 21 upper case only letters: A, B, C, D, E, F, Z, H, I, K, L, M, N, O, P, Q, R, S, T, V and X. -
Formwork.Pdf
Formwork INTRODUCTION ............................................................................................................................................................ 3 DESIGN REQUIREMENTS ............................................................................................................................................... 4 FACTORS RELATING TO FORMWORK ........................................................................................................................... 6 HIGH WALL FORMWORK .......................................................................................................................................... 6 Timber Framed Formwork. ....................................................................................................................................... 7 SHEATHING ............................................................................................................................................................... 8 WALL TYING SYSTEMS .............................................................................................................................................. 8 CONSTRUCTION JOINTS ............................................................................................................................................ 9 SLIPFORM ............................................................................................................................................................... 10 CLIMBING OR JUMP FORMS .................................................................................................................................. -
Makes Sense Only If P(E) = 0. Discrete Distribution Function: M(Ω)
Math 161 0 - Probability, Fall Semester 2012-2013 Dan Abramovich Conditional probabilities P (F jE) := P (F \ E)=P (E). Makes sense only if P (E) =6 0. Discrete distribution function: m(!) = P (f!g), so 8m(!) < ! 2 E m(!jE) := P (E) :0 ! 62 E: The effect is to replace Ω by E! Continuous: 8 f(!) < ! 2 E f(!jE) := P (E) :0 ! 62 E: 1 2 Example 1: cast a die. E = f! ≥ 4g, F = f! = 6g. P (F jE) = (1=6)=(1=2) = 1=3: P (EjF ) = 1. Example 2: two urns, 2B+3W, 1B+1W P (IjB) =? (2=10)=(2=10 + 1=4) \Bayes probabilities" 3 Example: Monty Hall problem. Say player chooses door 1 always (see book for removing this) draw table 13: 1/6; 12:1/6 23: 1/3 32: 1/6 P (C =6 1jM = 3) = (1=3)=(1=3 + 1=6) = 2=3. cgG 1/6 cGg 1/6 gcG 1/3 gCg 0 ggC 0 gGc 1/3 4 independent events P (F jE) = P (F ) Claim: if nonzero, implies P (F jE) = P (F ) Proof: the definition says P (F jE) = P (E \F )=P (E) so P (F jE)P (E) = P (E \ F ): Now P (EjF ) = P (E\F )=P (F ) = P (F jE)P (E)=P (F ) = P (E) by as- sumption. Can do the same for a collection of events: Ai are independent if P (Ai1∩· · ·∩Aik) = P (Ai1) ··· P (Aik) for distinct ij. Example: A1 first coin heads, A2 second coin heads, B =even number of heads.