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10-1 CHAPTER 10 DEFORMATION 10.1 Stress-Strain Diagrams And
EN380 Naval Materials Science and Engineering Course Notes, U.S. Naval Academy CHAPTER 10 DEFORMATION 10.1 Stress-Strain Diagrams and Material Behavior 10.2 Material Characteristics 10.3 Elastic-Plastic Response of Metals 10.4 True stress and strain measures 10.5 Yielding of a Ductile Metal under a General Stress State - Mises Yield Condition. 10.6 Maximum shear stress condition 10.7 Creep Consider the bar in figure 1 subjected to a simple tension loading F. Figure 1: Bar in Tension Engineering Stress () is the quotient of load (F) and area (A). The units of stress are normally pounds per square inch (psi). = F A where: is the stress (psi) F is the force that is loading the object (lb) A is the cross sectional area of the object (in2) When stress is applied to a material, the material will deform. Elongation is defined as the difference between loaded and unloaded length ∆푙 = L - Lo where: ∆푙 is the elongation (ft) L is the loaded length of the cable (ft) Lo is the unloaded (original) length of the cable (ft) 10-1 EN380 Naval Materials Science and Engineering Course Notes, U.S. Naval Academy Strain is the concept used to compare the elongation of a material to its original, undeformed length. Strain () is the quotient of elongation (e) and original length (L0). Engineering Strain has no units but is often given the units of in/in or ft/ft. ∆푙 휀 = 퐿 where: is the strain in the cable (ft/ft) ∆푙 is the elongation (ft) Lo is the unloaded (original) length of the cable (ft) Example Find the strain in a 75 foot cable experiencing an elongation of one inch. -
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 . -
Guide to Rheological Nomenclature: Measurements in Ceramic Particulate Systems
NfST Nisr National institute of Standards and Technology Technology Administration, U.S. Department of Commerce NIST Special Publication 946 Guide to Rheological Nomenclature: Measurements in Ceramic Particulate Systems Vincent A. Hackley and Chiara F. Ferraris rhe National Institute of Standards and Technology was established in 1988 by Congress to "assist industry in the development of technology . needed to improve product quality, to modernize manufacturing processes, to ensure product reliability . and to facilitate rapid commercialization ... of products based on new scientific discoveries." NIST, originally founded as the National Bureau of Standards in 1901, works to strengthen U.S. industry's competitiveness; advance science and engineering; and improve public health, safety, and the environment. One of the agency's basic functions is to develop, maintain, and retain custody of the national standards of measurement, and provide the means and methods for comparing standards used in science, engineering, manufacturing, commerce, industry, and education with the standards adopted or recognized by the Federal Government. As an agency of the U.S. Commerce Department's Technology Administration, NIST conducts basic and applied research in the physical sciences and engineering, and develops measurement techniques, test methods, standards, and related services. The Institute does generic and precompetitive work on new and advanced technologies. NIST's research facilities are located at Gaithersburg, MD 20899, and at Boulder, CO 80303. -
Remarks on the Current Theory of Shear Strength of Variable Depth Beams A
28 The Open Civil Engineering Journal, 2009, 3, 28-33 Open Access Remarks on the Current Theory of Shear Strength of Variable Depth Beams A. Paglietti* and G. Carta Department of Structural Engineering, University of Cagliari, Italy Abstract: Though still in use today, the method of the effective shearing force to evaluate the maximum shear stress in variable depth beams does not stand close scrutiny. It can lead to overestimating the shearing strength of these beams, al- though it is suggested as a viable procedure by many otherwise excellent codes of practice worldwide. This paper should help to put the record straight. It should warn the practitioner against the general inadequacy of such a method and prompt the drafters of structural concrete codes of practice into acting to eliminate this erroneous though persisting designing pro- cedure. Key Words: Variable depth beams, Effective shearing force, Tapered beams, Building codes. INTRODUCTION to a sloped side of the beam. Accordingly, the component of such a force which is normal to the beam axis is supposed to The three simply supported beams shown in Fig. (1) all increase or decrease the internal shear needed to equilibrate have the same span and carry the same load. Their shape is the shearing force acting at the considered cross section. This different, though. Beam (a) is a constant depth beam, while is what is stated, almost literally, in Sect. R 11.1.1.2 of the beams (b) and (c) are just two different instances of a vari- American Concrete Institute Code [1] and represents the able depth beam. -
Navier-Stokes-Equation
Math 613 * Fall 2018 * Victor Matveev Derivation of the Navier-Stokes Equation 1. Relationship between force (stress), stress tensor, and strain: Consider any sub-volume inside the fluid, with variable unit normal n to the surface of this sub-volume. Definition: Force per area at each point along the surface of this sub-volume is called the stress vector T. When fluid is not in motion, T is pointing parallel to the outward normal n, and its magnitude equals pressure p: T = p n. However, if there is shear flow, the two are not parallel to each other, so we need a marix (a tensor), called the stress-tensor , to express the force direction relative to the normal direction, defined as follows: T Tn or Tnkjjk As we will see below, σ is a symmetric matrix, so we can also write Tn or Tnkkjj The difference in directions of T and n is due to the non-diagonal “deviatoric” part of the stress tensor, jk, which makes the force deviate from the normal: jkp jk jk where p is the usual (scalar) pressure From general considerations, it is clear that the only source of such “skew” / ”deviatoric” force in fluid is the shear component of the flow, described by the shear (non-diagonal) part of the “strain rate” tensor e kj: 2 1 jk2ee jk mm jk where euujk j k k j (strain rate tensro) 3 2 Note: the funny construct 2/3 guarantees that the part of proportional to has a zero trace. The two terms above represent the most general (and the only possible) mathematical expression that depends on first-order velocity derivatives and is invariant under coordinate transformations like rotations. -
Using Lenterra Shear Stress Sensors to Measure Viscosity
Application Note: Using Lenterra Shear Stress Sensors to Measure Viscosity Guidelines for using Lenterra’s shear stress sensors for in-line, real-time measurement of viscosity in pipes, thin channels, and high-shear mixers. Shear Stress and Viscosity Shear stress is a force that acts on an object that is directed parallel to its surface. Lenterra’s RealShear™ sensors directly measure the wall shear stress caused by flowing or mixing fluids. As an example, when fluids pass between a rotor and a stator in a high-shear mixer, shear stress is experienced by the fluid and the surfaces that it is in contact with. A RealShear™ sensor can be mounted on the stator to measure this shear stress, as a means to monitor mixing processes or facilitate scale-up. Shear stress and viscosity are interrelated through the shear rate (velocity gradient) of a fluid: ∂u τ = µ = γµ & . ∂y Here τ is the shear stress, γ˙ is the shear rate, µ is the dynamic viscosity, u is the velocity component of the fluid tangential to the wall, and y is the distance from the wall. When the viscosity of a fluid is not a function of shear rate or shear stress, that fluid is described as “Newtonian.” In non-Newtonian fluids the viscosity of a fluid depends on the shear rate or stress (or in some cases the duration of stress). Certain non-Newtonian fluids behave as Newtonian fluids at high shear rates and can be described by the equation above. For others, the viscosity can be expressed with certain models. -
Equation of Motion for Viscous Fluids
1 2.25 Equation of Motion for Viscous Fluids Ain A. Sonin Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 2001 (8th edition) Contents 1. Surface Stress …………………………………………………………. 2 2. The Stress Tensor ……………………………………………………… 3 3. Symmetry of the Stress Tensor …………………………………………8 4. Equation of Motion in terms of the Stress Tensor ………………………11 5. Stress Tensor for Newtonian Fluids …………………………………… 13 The shear stresses and ordinary viscosity …………………………. 14 The normal stresses ……………………………………………….. 15 General form of the stress tensor; the second viscosity …………… 20 6. The Navier-Stokes Equation …………………………………………… 25 7. Boundary Conditions ………………………………………………….. 26 Appendix A: Viscous Flow Equations in Cylindrical Coordinates ………… 28 ã Ain A. Sonin 2001 2 1 Surface Stress So far we have been dealing with quantities like density and velocity, which at a given instant have specific values at every point in the fluid or other continuously distributed material. The density (rv ,t) is a scalar field in the sense that it has a scalar value at every point, while the velocity v (rv ,t) is a vector field, since it has a direction as well as a magnitude at every point. Fig. 1: A surface element at a point in a continuum. The surface stress is a more complicated type of quantity. The reason for this is that one cannot talk of the stress at a point without first defining the particular surface through v that point on which the stress acts. A small fluid surface element centered at the point r is defined by its area A (the prefix indicates an infinitesimal quantity) and by its outward v v unit normal vector n . -
Review of Fluid Mechanics Terminology
CBE 6333, R. Levicky 1 Review of Fluid Mechanics Terminology The Continuum Hypothesis: We will regard macroscopic behavior of fluids as if the fluids are perfectly continuous in structure. In reality, the matter of a fluid is divided into fluid molecules, and at sufficiently small (molecular and atomic) length scales fluids cannot be viewed as continuous. However, since we will only consider situations dealing with fluid properties and structure over distances much greater than the average spacing between fluid molecules, we will regard a fluid as a continuous medium whose properties (density, pressure etc.) vary from point to point in a continuous way. For the problems that we will be interested in, the microscopic details of fluid structure will not be needed and the continuum approximation will be appropriate. However, there are situations when molecular level details are important; for instance when the dimensions of a channel that the fluid is flowing through become comparable to the mean free paths of the fluid molecules or to the molecule size. In such instances, the continuum hypothesis does not apply. Fluid : a substance that will deform continuously in response to a shear stress no matter how small the stress may be. Shear Stress : Force per unit area that is exerted parallel to the surface on which it acts. 2 Shear stress units: Force/Area, ex. N/m . Usual symbols: σij , τij (i ≠j). Example 1: shear stress between a block and a surface: Example 2: A simplified picture of the shear stress between two laminas (layers) in a flowing liquid. The top layer moves relative to the bottom one by a velocity ∆V, and collision interactions between the molecules of the two layers give rise to shear stress. -
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. -
Enghandbook.Pdf
785.392.3017 FAX 785.392.2845 Box 232, Exit 49 G.L. Huyett Expy Minneapolis, KS 67467 ENGINEERING HANDBOOK TECHNICAL INFORMATION STEELMAKING Basic descriptions of making carbon, alloy, stainless, and tool steel p. 4. METALS & ALLOYS Carbon grades, types, and numbering systems; glossary p. 13. Identification factors and composition standards p. 27. CHEMICAL CONTENT This document and the information contained herein is not Quenching, hardening, and other thermal modifications p. 30. HEAT TREATMENT a design standard, design guide or otherwise, but is here TESTING THE HARDNESS OF METALS Types and comparisons; glossary p. 34. solely for the convenience of our customers. For more Comparisons of ductility, stresses; glossary p.41. design assistance MECHANICAL PROPERTIES OF METAL contact our plant or consult the Machinery G.L. Huyett’s distinct capabilities; glossary p. 53. Handbook, published MANUFACTURING PROCESSES by Industrial Press Inc., New York. COATING, PLATING & THE COLORING OF METALS Finishes p. 81. CONVERSION CHARTS Imperial and metric p. 84. 1 TABLE OF CONTENTS Introduction 3 Steelmaking 4 Metals and Alloys 13 Designations for Chemical Content 27 Designations for Heat Treatment 30 Testing the Hardness of Metals 34 Mechanical Properties of Metal 41 Manufacturing Processes 53 Manufacturing Glossary 57 Conversion Coating, Plating, and the Coloring of Metals 81 Conversion Charts 84 Links and Related Sites 89 Index 90 Box 232 • Exit 49 G.L. Huyett Expressway • Minneapolis, Kansas 67467 785-392-3017 • Fax 785-392-2845 • [email protected] • www.huyett.com INTRODUCTION & ACKNOWLEDGMENTS This document was created based on research and experience of Huyett staff. Invaluable technical information, including statistical data contained in the tables, is from the 26th Edition Machinery Handbook, copyrighted and published in 2000 by Industrial Press, Inc.