TIMES and EQUIVALENT SYSTEMS II 6.1 for Concurrent
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Glossary: Definitions
Appendix B Glossary: Definitions The definitions given here apply to the terminology used throughout this book. Some of the terms may be defined differently by other authors; when this is the case, alternative terminology is noted. When two or more terms with identical or similar meaning are in general acceptance, they are given in the order of preference of the current writers. Allowable stress (working stress): If a member is so designed that the maximum stress as calculated for the expected conditions of service is less than some limiting value, the member will have a proper margin of security against damage or failure. This limiting value is the allowable stress subject to the material and condition of service in question. The allowable stress is made less than the damaging stress because of uncertainty as to the conditions of service, nonuniformity of material, and inaccuracy of the stress analysis (see Ref. 1). The margin between the allowable stress and the damaging stress may be reduced in proportion to the certainty with which the conditions of the service are known, the intrinsic reliability of the material, the accuracy with which the stress produced by the loading can be calculated, and the degree to which failure is unattended by danger or loss. (Compare with Damaging stress; Factor of safety; Factor of utilization; Margin of safety. See Refs. l–3.) Apparent elastic limit (useful limit point): The stress at which the rate of change of strain with respect to stress is 50% greater than at zero stress. It is more definitely determinable from the stress–strain diagram than is the proportional limit, and is useful for comparing materials of the same general class. -
Bending Stress
Bending Stress Sign convention The positive shear force and bending moments are as shown in the figure. Figure 40: Sign convention followed. Centroid of an area Scanned by CamScanner If the area can be divided into n parts then the distance Y¯ of the centroid from a point can be calculated using n ¯ Âi=1 Aiy¯i Y = n Âi=1 Ai where Ai = area of the ith part, y¯i = distance of the centroid of the ith part from that point. Second moment of area, or moment of inertia of area, or area moment of inertia, or second area moment For a rectangular section, moments of inertia of the cross-sectional area about axes x and y are 1 I = bh3 x 12 Figure 41: A rectangular section. 1 I = hb3 y 12 Scanned by CamScanner Parallel axis theorem This theorem is useful for calculating the moment of inertia about an axis parallel to either x or y. For example, we can use this theorem to calculate . Ix0 = + 2 Ix0 Ix Ad Bending stress Bending stress at any point in the cross-section is My s = − I where y is the perpendicular distance to the point from the centroidal axis and it is assumed +ve above the axis and -ve below the axis. This will result in +ve sign for bending tensile (T) stress and -ve sign for bending compressive (C) stress. Largest normal stress Largest normal stress M c M s = | |max · = | |max m I S where S = section modulus for the beam. For a rectangular section, the moment of inertia of the cross- 1 3 1 2 sectional area I = 12 bh , c = h/2, and S = I/c = 6 bh . -
Engineering Drawing (NSQF) - 1St Year Common for All Engineering Trades Under CTS (Not Applicable for Draughtsman Trade Group)
ENGINEERING DRAWING (NSQF) 1st Year Common for All Engineering Trades under CTS (Not applicable for Draughtsman Trade Group) (For all 1 Year & 2 Year Trades) DIRECTORATE GENERAL OF TRAINING MINISTRY OF SKILL DEVELOPMENT & ENTREPRENEURSHIP GOVERNMENT OF INDIA NATIONAL INSTRUCTIONAL MEDIA INSTITUTE, CHENNAI Post Box No. 3142, CTI Campus, Guindy, Chennai - 600 032 (i) Copyright Free. Under CC BY License. Engineering Drawing (NSQF) - 1st Year Common for all Engineering Trades under CTS (Not applicable for Draughtsman Trade Group) Developed & Published by National Instructional Media Institute Post Box No.3142 Guindy, Chennai - 32 INDIA Email: [email protected] Website: www.nimi.gov.in Printed by National Instructional Media Institute Chennai - 600 032 First Edition :November 2019 Rs.170/- © (ii) Copyright Free. Under CC BY License. FOREWORD The Government of India has set an ambitious target of imparting skills to 30 crores people, one out of every four Indians, by 2020 to help them secure jobs as part of the National Skills Development Policy. Industrial Training Institutes (ITIs) play a vital role in this process especially in terms of providing skilled manpower. Keeping this in mind, and for providing the current industry relevant skill training to Trainees, ITI syllabus has been recently updated with the help of comprising various stakeholder's viz. Industries, Entrepreneurs, Academicians and representatives from ITIs. The National Instructional Media Institute (NIMI), Chennai, has now come up with instructional material to suit the revised curriculum for Engineering Drawing 1st Year (For All 1 year & 2 year Trades) NSQF Commom for all engineering trades under CTS will help the trainees to get an international equivalency standard where their skill proficiency and competency will be duly recognized across the globe and this will also increase the scope of recognition of prior learning. -
Tessellation Techniques
Tessellation Techniques Stefanie Mandelbaum Jacqueline S. Guttman 118 Tunicflower Lane West Windsor, NJ 08550, USA E-mail [email protected] Abstract Tessellations are continuous repetitions of the same geometric or organic shape, a tile or tessera in Latin . Inspired by the abstract tessellations of the Alhambra, aided by a correspondence with Coxeter, Escher developed a method of transforming polygonal tessellations into representational tessellations of organic shapes. Mathematical Presentation Tessellations of polygons can be edge-to-edge and vertex-to-vertex (Fig. 1) or translated to be edge-to- edge but not vertex-to-vertex (Fig. 2). Figure 1: Quadrilaterals, edge-to-edge and vertex-to-vertex Figure 2: Quadrilaterals, translated Escher first determined which polygons could tessellate by themselves, e.g. triangles, quadrilaterals, regular hexagons (Figs. 3 and 4 ), and which could not, e.g. regular octagons and regular pentagons (Fig. 5). (A polygon is regular if all of its sides and interior angles are congruent.) Figure 3: Quadrilaterals Figure 4: Triangles Figure 5: Regular Pentagons At the vertex of a tessellation, the sum of the angles of the adjacent polygons is 360 degrees. So, for example, a regular hexagon can tessellate by itself because at the vertex of the tessellation the sum of the angles is 360 degrees, each of the three adjoining angles being 120 degrees (Fig. 6). Conversely, regular octagons cannot tessellate by themselves because each angle of a regular octagon is 135 degrees and 135 is not a factor of 360 (Fig. 7). 2(135)=270. 360-270=90. Therefore, the other polygon needed to complete the tessellation is a square. -
SOFTIMAGE|XSI Application Uses Jscript and Visual Basic Scripting Edition from Microsoft Corporation
SOFTIMAGE®|XSI® Version 4.0 Modeling & Deformations © 1999–2004 Avid Technology, Inc. All rights reserved. Avid, meta–clay, SOFTIMAGE, XSI, and the XSI logo are either registered trademarks or trademarks of Avid Technology, Inc. in the United States and/ or other countries. mental ray and mental images are registered trademarks of mental images GmbH & Co. KG in the U.S.A. and some other countries. mental ray Phenomenon, mental ray Phenomena, Phenomenon, Phenomena, Software Protection Manager, and SPM are trademarks or, in some countries, registered trademarks of mental images GmbH & Co. KG. All other trademarks contained herein are the property of their respective owners. This software includes the Python Release 2.3.2 software. The Python software is: Copyright © 2001, 2002, 2003 Python Software Foundation. All rights reserved. Copyright © 2000 BeOpen.com. All rights reserved. Copyright © 1995-2000 Corporation for National Research Initiatives. All rights reserved. Copyright © 1991-1995 Stichting Mathematisch Centrum. All rights reserved. The SOFTIMAGE|XSI application uses JScript and Visual Basic Scripting Edition from Microsoft Corporation. Activision is a registered trademark of Activision, Inc. © 1998 Activision, Inc. Battlezone is a trademark of and © 1998 Atari Interactive, Inc., a Hasbro company. All rights reserved. Licensed by Activision. This document is protected under copyright law. The contents of this document may not be copied or duplicated in any form, in whole or in part, without the express written permission of Avid Technology, Inc. This document is supplied as a guide for the Softimage product. Reasonable care has been taken in preparing the information it contains. However, this document may contain omissions, technical inaccuracies, or typographical errors. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Bending Moment & Shear Force
Strength of Materials Prof. M. S. Sivakumar Problem 1: Computation of Reactions Problem 2: Computation of Reactions Problem 3: Computation of Reactions Problem 4: Computation of forces and moments Problem 5: Bending Moment and Shear force Problem 6: Bending Moment Diagram Problem 7: Bending Moment and Shear force Problem 8: Bending Moment and Shear force Problem 9: Bending Moment and Shear force Problem 10: Bending Moment and Shear force Problem 11: Beams of Composite Cross Section Indian Institute of Technology Madras Strength of Materials Prof. M. S. Sivakumar Problem 1: Computation of Reactions Find the reactions at the supports for a simple beam as shown in the diagram. Weight of the beam is negligible. Figure: Concepts involved • Static Equilibrium equations Procedure Step 1: Draw the free body diagram for the beam. Step 2: Apply equilibrium equations In X direction ∑ FX = 0 ⇒ RAX = 0 In Y Direction ∑ FY = 0 Indian Institute of Technology Madras Strength of Materials Prof. M. S. Sivakumar ⇒ RAY+RBY – 100 –160 = 0 ⇒ RAY+RBY = 260 Moment about Z axis (Taking moment about axis pasing through A) ∑ MZ = 0 We get, ∑ MA = 0 ⇒ 0 + 250 N.m + 100*0.3 N.m + 120*0.4 N.m - RBY *0.5 N.m = 0 ⇒ RBY = 656 N (Upward) Substituting in Eq 5.1 we get ∑ MB = 0 ⇒ RAY * 0.5 + 250 - 100 * 0.2 – 120 * 0.1 = 0 ⇒ RAY = -436 (downwards) TOP Indian Institute of Technology Madras Strength of Materials Prof. M. S. Sivakumar Problem 2: Computation of Reactions Find the reactions for the partially loaded beam with a uniformly varying load shown in Figure. -
Shear Force and Bending Moment in Beams and Frames CHAPTER2
Shear Force and Bending Moment in Beams and Frames CHAPTER2 2.0 INTRODUCTION To bridge a gap in land on earth is most pressing problem for structural engineers. Slabs, beams or any combination of these are extensively used in bridging gaps. Construction of bridges, covering of door and window openings and covering of roof of enclosures are examples where beams and slabs are used. The space below a beam is available for other use. Structural actions of beams and slabs are slightly different. A beam collects fl oor load and transfers it longitudinally to the supports, which are usually provided either at both ends (beam action) or at one end only (cantilever action). Cantilevers are used in construction of balconies in homes and footpaths in bridges. A slab transfers fl oor load in both directions whereas a beam transfers fl oor load in only longitudinal direction. Therefore, a beam is a one-dimensional structural element and it can be represented by a line as shown in Fig. 2.1(b) in which arrow shows the direction of load transfer. Mechanical engineers extensively use cantilevers. The boom of a crane, the cutting blade of a bulldozer and wings of a fan are few examples of cantilevers. The span of beam or cantilever, which is defi ned as distance between adjacent supports, governs the complexity of its design. Shear force and bending moment diagrams quantify structural action of beams and cantilevers and are required for their rational design. Beams and cantilevers shown in Fig. 2.1 are known as fl exural elements. -
Glossary of Notations
108 GLOSSARY OF NOTATIONS A = Earthquake peak ground acceleration. IρM = Soil influence coefficient for moment. = A0 Cross-sectional area of the stream. K1, K2, = aB Barge bow damage depth. K3, and K4 = Scour coefficients that account for the nose AF = Annual failure rate. shape of the pier, the angle between the direction b = River channel width. of the flow and the direction of the pier, the BR = Vehicular braking force. streambed conditions, and the bed material size. = BRa Aberrancy base rate. Kp = Rankine coefficient. = = bx Bias of ¯x x/xn. KR = Pile flexibility factor, which gives the relative c = Wind analysis constant. stiffness of the pile and soil. C′=Response spectrum modeling parameter. L = Foundation depth. = CE Vehicular centrifugal force. Le = Effective depth of foundation (distance from = CF Cost of failure. ground level to point of fixity). = CH Hydrodynamic coefficient that accounts for the effect LL = Vehicular live load. of surrounding water on vessel collision forces. LOA = Overall length of vessel. = CI Initial cost for building bridge structure. LS = Live load surcharge. = Cp Wind pressure coefficient. max(x) = Maximum of all possible x values. = CR Creep. M = Moment capacity. = cap CT Expected total cost of building bridge structure. M = Moment capacity of column. = col CT Vehicular collision force. M = Design moment. = design CV Vessel collision force. n = Manning roughness coefficient. = D Diameter of pile or column. N = Number of vessels (or flotillas) of type i. = i DC Dead load of structural components and nonstructural PA = Probability of aberrancy. attachments. P = Nominal design force for ship collisions. DD = Downdrag. B P = Base wind pressure. -
Truss Action Consider a Loaded Beam of Rectangular Cross Section As Shown on the Next Page
TRUSSES Church Truss Vermont Timber Works Truss Action Consider a loaded beam of rectangular cross section as shown on the next page. When such a beam is loaded, it is subjected to three internal actions: an axial force (A), a shear force (V), and an internal bending moment (M) The internal bending moment is a direct result of the induced shear force. If this shear force is eliminated, the bending moment is also eliminated. The end result is a tensile or compressive axial force Since this loaded beam does deflect (bend), the top of the beam will be subjected to compressive forces while the bottom of the beam will be subjected to tensile forces The magnitudes of these forces and moments and their internal distribution depends upon several factors. The details of this are beyond the scope of a course in statics but will be developed in a Strength of Materials course. For now, simply accept these phenomena at face value 2 Truss Action F (a) A simply supported beam of rectangular cross section is shown with a point load at midspan. Consider a section of the beam between the two dotted lines F (b) Force 'F' induces an internal shear (V) and axial (A) force. The shear force causes an internal bending moment (M) M A V V A M 3 Truss Action But if one replaces the beam with a truss as shown below, all shear forces within the beam are translated to pure compressive and tensile forces in the truss members. Since there is no shear, there is no bending moment. -
Explicit Analysis of Large Transformation of a Timoshenko Beam : Post-Buckling Solution, Bifurcation and Catastrophes Marwan Hariz, Loïc Le Marrec, Jean Lerbet
Explicit analysis of large transformation of a Timoshenko beam : Post-buckling solution, bifurcation and catastrophes Marwan Hariz, Loïc Le Marrec, Jean Lerbet To cite this version: Marwan Hariz, Loïc Le Marrec, Jean Lerbet. Explicit analysis of large transformation of a Timoshenko beam : Post-buckling solution, bifurcation and catastrophes. Acta Mechanica, Springer Verlag, 2021, 10.1007/s00707-021-02993-8. hal-03306350 HAL Id: hal-03306350 https://hal.archives-ouvertes.fr/hal-03306350 Submitted on 29 Jul 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Explicit analysis of large transformation of a Timoshenko beam : Post-buckling solution, bifurcation and catastrophes Marwan Hariza,1, Lo¨ıc Le Marreca,2, Jean Lerbetb,3 aUniv Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France bUniversit´eParis-Saclay, CNRS, Univ Evry, Laboratoire de Math´ematiques et Mod´elisation d’Evry, 91037, Evry-Courcouronnes, France Abstract This paper exposes full analytical solutions of a plane, quasi-static but large transformation of a Timoshenko beam. The problem isfirst re-formulated in the form of a Cauchy initial value problem where load (force and moment) is prescribed at one-end and kinematics (translation, rotation) at the other one. -
7.4 the Elementary Beam Theory
Section 7.4 7.4 The Elementary Beam Theory In this section, problems involving long and slender beams are addressed. As with pressure vessels, the geometry of the beam, and the specific type of loading which will be considered, allows for approximations to be made to the full three-dimensional linear elastic stress-strain relations. The beam theory is used in the design and analysis of a wide range of structures, from buildings to bridges to the load-bearing bones of the human body. 7.4.1 The Beam The term beam has a very specific meaning in engineering mechanics: it is a component that is designed to support transverse loads, that is, loads that act perpendicular to the longitudinal axis of the beam, Fig. 7.4.1. The beam supports the load by bending only. Other mechanisms, for example twisting of the beam, are not allowed for in this theory. applied force applied pressure cross section roller support pin support Figure 7.4.1: A supported beam loaded by a force and a distribution of pressure It is convenient to show a two-dimensional cross-section of the three-dimensional beam together with the beam cross section, as in Fig. 7.4.1. The beam can be supported in various ways, for example by roller supports or pin supports (see section 2.3.3). The cross section of this beam happens to be rectangular but it can be any of many possible shapes. It will assumed that the beam has a longitudinal plane of symmetry, with the cross section symmetric about this plane, as shown in Fig.