Bending Stress
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Fluid Mechanics
MECHANICAL PRINCIPLES HNC/D MOMENTS OF AREA The concepts of first and second moments of area fundamental to several areas of engineering including solid mechanics and fluid mechanics. Students who are not familiar with this concept are advised to complete this tutorial before studying either of these areas. In this section you will do the following. Define the centre of area. Define and calculate 1st. moments of areas. Define and calculate 2nd moments of areas. Derive standard formulae. D.J.Dunn 1 1. CENTROIDS AND FIRST MOMENTS OF AREA A moment about a given axis is something multiplied by the distance from that axis measured at 90o to the axis. The moment of force is hence force times distance from an axis. The moment of mass is mass times distance from an axis. The moment of area is area times the distance from an axis. Fig.1 In the case of mass and area, the problem is deciding the distance since the mass and area are not concentrated at one point. The point at which we may assume the mass concentrated is called the centre of gravity. The point at which we assume the area concentrated is called the centroid. Think of area as a flat thin sheet and the centroid is then at the same place as the centre of gravity. You may think of this point as one where you could balance the thin sheet on a sharp point and it would not tip off in any direction. This section is mainly concerned with moments of area so we will start by considering a flat area at some distance from an axis as shown in Fig.1.2 Fig..2 The centroid is denoted G and its distance from the axis s-s is y. -
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. -
Relationship of Structure and Stiffness in Laminated Bamboo Composites
Construction and Building Materials 165 (2018) 241–246 Contents lists available at ScienceDirect Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat Technical note Relationship of structure and stiffness in laminated bamboo composites Matthew Penellum a, Bhavna Sharma b, Darshil U. Shah c, Robert M. Foster c,1, ⇑ Michael H. Ramage c, a Department of Engineering, University of Cambridge, United Kingdom b Department of Architecture and Civil Engineering, University of Bath, United Kingdom c Department of Architecture, University of Cambridge, United Kingdom article info abstract Article history: Laminated bamboo in structural applications has the potential to change the way buildings are con- Received 22 August 2017 structed. The fibrous microstructure of bamboo can be modelled as a fibre-reinforced composite. This Received in revised form 22 November 2017 study compares the results of a fibre volume fraction analysis with previous experimental beam bending Accepted 23 December 2017 results. The link between fibre volume fraction and bending stiffness shows that differences previously attributed to preservation treatment in fact arise due to strip thickness. Composite theory provides a basis for the development of future guidance for laminated bamboo, as validated here. Fibre volume frac- Keywords: tion analysis is an effective method for non-destructive evaluation of bamboo beam stiffness. Microstructure Ó 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http:// Mechanical properties Analytical modelling creativecommons.org/licenses/by/4.0/). Bamboo 1. Introduction produce a building material [5]. Structural applications are cur- rently limited by a lack of understanding of the properties. -
Centrally Loaded Columns
Ziemian c03.tex V1 - 10/15/2009 4:17pm Page 23 CHAPTER 3 CENTRALLY LOADED COLUMNS 3.1 INTRODUCTION The cornerstone of column theory is the Euler column, a mathematically straight, prismatic, pin-ended, centrally loaded1 strut that is slender enough to buckle without the stress at any point in the cross section exceeding the proportional limit of the material. The buckling load or critical load or bifurcation load (see Chapter 2 for a discussion of the significance of these terms) is defined as π 2EI P = (3.1) E L2 where E is the modulus of elasticity of the material, I is the second moment of area of the cross section about which buckling takes place, and L is the length of the column. The Euler load, P E , is a reference value to which the strengths of actual columns are often compared. If end conditions other than perfectly frictionless pins can be defined mathemat- ically, the critical load can be expressed by π 2EI P = (3.2) E (KL)2 where KL is an effective length defining the portion of the deflected shape between points of zero curvature (inflection points). In other words, KL is the length of an equivalent pin-ended column buckling at the same load as the end-restrained 1Centrally loaded implies that the axial load is applied through the centroidal axis of the member, thus producing no bending or twisting. 23 Ziemian c03.tex V1 - 10/15/2009 4:17pm Page 24 24 CENTRALLY LOADED COLUMNS column. For example, for columns in which one end of the member is prevented from translating with respect to the other end, K can take on values ranging from 0.5 to l.0, depending on the end restraint. -
Course Objectives Chapter 2 2. Hull Form and Geometry
COURSE OBJECTIVES CHAPTER 2 2. HULL FORM AND GEOMETRY 1. Be familiar with ship classifications 2. Explain the difference between aerostatic, hydrostatic, and hydrodynamic support 3. Be familiar with the following types of marine vehicles: displacement ships, catamarans, planing vessels, hydrofoil, hovercraft, SWATH, and submarines 4. Learn Archimedes’ Principle in qualitative and mathematical form 5. Calculate problems using Archimedes’ Principle 6. Read, interpret, and relate the Body Plan, Half-Breadth Plan, and Sheer Plan and identify the lines for each plan 7. Relate the information in a ship's lines plan to a Table of Offsets 8. Be familiar with the following hull form terminology: a. After Perpendicular (AP), Forward Perpendiculars (FP), and midships, b. Length Between Perpendiculars (LPP or LBP) and Length Overall (LOA) c. Keel (K), Depth (D), Draft (T), Mean Draft (Tm), Freeboard and Beam (B) d. Flare, Tumble home and Camber e. Centerline, Baseline and Offset 9. Define and compare the relationship between “centroid” and “center of mass” 10. State the significance and physical location of the center of buoyancy (B) and center of flotation (F); locate these points using LCB, VCB, TCB, TCF, and LCF st 11. Use Simpson’s 1 Rule to calculate the following (given a Table of Offsets): a. Waterplane Area (Awp or WPA) b. Sectional Area (Asect) c. Submerged Volume (∇S) d. Longitudinal Center of Flotation (LCF) 12. Read and use a ship's Curves of Form to find hydrostatic properties and be knowledgeable about each of the properties on the Curves of Form 13. Calculate trim given Taft and Tfwd and understand its physical meaning i 2.1 Introduction to Ships and Naval Engineering Ships are the single most expensive product a nation produces for defense, commerce, research, or nearly any other function. -
The Bending of Beams and the Second Moment of Area
University of Plymouth PEARL https://pearl.plymouth.ac.uk The Plymouth Student Scientist - Volume 06 - 2013 The Plymouth Student Scientist - Volume 6, No. 2 - 2013 2013 The bending of beams and the second moment of area Bailey, C. Bailey, C., Bull, T., and Lawrence, A. (2013) 'The bending of beams and the second moment of area', The Plymouth Student Scientist, 6(2), p. 328-339. http://hdl.handle.net/10026.1/14043 The Plymouth Student Scientist University of Plymouth All content in PEARL is protected by copyright law. Author manuscripts are made available in accordance with publisher policies. Please cite only the published version using the details provided on the item record or document. In the absence of an open licence (e.g. Creative Commons), permissions for further reuse of content should be sought from the publisher or author. The Plymouth Student Scientist, 2013, 6, (2), p. 328–339 The Bending of Beams and the Second Moment of Area Chris Bailey, Tim Bull and Aaron Lawrence Project Advisor: Tom Heinzl, School of Computing and Mathematics, Plymouth University, Drake Circus, Plymouth, PL4 8AA Abstract We present an overview of the laws governing the bending of beams and of beam theory. Particular emphasis is put on beam stiffness associated with different cross section shapes using the concept of the second moment of area. [328] The Plymouth Student Scientist, 2013, 6, (2), p. 328–339 1 Historical Introduction Beams are an integral part of everyday life, with beam theory involved in the develop- ment of many modern structures. Early applications of beam practice to large scale developments include the Eiffel Tower and the Ferris Wheel. -
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. -
Vibration of Axially-Loaded Structures
Part C: Beam-Columns Beams Basic Formulation The Temporal Solution The Spatial Solution Rayleigh-Ritz Analysis Rayleigh’s Quotient The Role of Initial Imperfections An Alternative Approach Higher Modes Rotating Beams Self-weight A hanging beam Experiments Thermal Loading Beam on an Elastic Foundation Elastically Restrained Supports Beams with Variable Cross-section A beam with a constant axial force In this section we develop the governing equation of motion for a thin, elastic, prismatic beam subject to a constant axial force: x w w L M 1 2 R(x,t) ∆ x x P EI, m P M +∆ M S 0 P w(x) Q(x,t) ∆ x F(x,t) ∆ x S + ∆ S ∆x/2 (a) (b) Beam schematic including an axial load. It has mass per unit length m, constant flexural rigidity EI , and subject to an axial load P. The length is L, the coordinate along the beam is x, and the lateral (transverse) deflection is w(x, t). The governing equation is ∂4w ∂2w ∂2w EI + P + m = F (x, t). (1) ∂x 4 ∂x 2 ∂t2 This linear partial differential equation can be solved using standard methods. We might expect the second-order ordinary differential equation in time to have oscillatory solutions (given positive values of flexural rigidity etc.), however, we anticipate the dependence of the form of the temporal solution will depend on the magnitude of the axial load. The Temporal Solution In order to be a little more specific, (before going on to consider the more general spatial response), let us suppose we have ends that are pinned (simply supported), i.e., the deflection (w) and bending moment (∂2w/∂2x) are zero at x = 0 and x = l. -
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. -
Design of Roadside Channels with Flexible Linings
Publication No. FHWA-NHI-05-114 September 2005 U.S. Department of Transportation Federal Highway Administration Hydraulic Engineering Circular No. 15, Third Edition Design of Roadside Channels with Flexible Linings National Highway Institute Technical Report Documentation Page 1. Report No. 2. Government Accession No. 3. Recipient's Catalog No. FHWA-NHI-05-114 HEC 15 4. Title and Subtitle 5. Report Date Design of Roadside Channels with Flexible Linings September 2005 Hydraulic Engineering Circular Number 15, Third Edition 6. Performing Organization Code 7. Author(s) 8. Performing Organization Report No. Roger T. Kilgore and George K. Cotton 9. Performing Organization Name and Address 10. Work Unit No. (TRAIS) Kilgore Consulting and Management 2963 Ash Street 11. Contract or Grant No. Denver, CO 80207 DTFH61-02-D-63009/T-63044 12. Sponsoring Agency Name and Address 13. Type of Report and Period Covered Federal Highway Administration Final Report (3rd Edition) National Highway Institute Office of Bridge Technology April 2004 – August 2005 4600 North Fairfax Drive 400 Seventh Street Suite 800 Room 3202 14. Sponsoring Agency Code Arlington, Virginia 22203 Washington D.C. 20590 15. Supplementary Notes Project Manager: Dan Ghere – FHWA Resource Center Technical Assistance: Jorge Pagan, Joe Krolak, Brian Beucler, Sterling Jones, Philip L. Thompson (consultant) 16. Abstract Flexible linings provide a means of stabilizing roadside channels. Flexible linings are able to conform to changes in channel shape while maintaining overall lining integrity. Long-term flexible linings such as riprap, gravel, or vegetation (reinforced with synthetic mats or unreinforced) are suitable for a range of hydraulic conditions. Unreinforced vegetation and many transitional and temporary linings are suited to hydraulic conditions with moderate shear stresses.