Relationship of Structure and Stiffness in Laminated Bamboo Composites
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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. -
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 . -
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. -
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. -
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. -
Basic Mechanics
93 Chapter 6 Basic mechanics BASIC PRINCIPLES OF STATICS All objects on earth tend to accelerate toward the Statics is the branch of mechanics that deals with the centre of the earth due to gravitational attraction; hence equilibrium of stationary bodies under the action of the force of gravitation acting on a body with the mass forces. The other main branch – dynamics – deals with (m) is the product of the mass and the acceleration due moving bodies, such as parts of machines. to gravity (g), which has a magnitude of 9.81 m/s2: Static equilibrium F = mg = vρg A planar structural system is in a state of static equilibrium when the resultant of all forces and all where: moments is equal to zero, i.e. F = force (N) m = mass (kg) g = acceleration due to gravity (9.81m/s2) y ∑Fx = 0 ∑Fx = 0 ∑Fy = 0 ∑Ma = 0 v = volume (m³) ∑Fy = 0 or ∑Ma = 0 or ∑Ma = 0 or ∑Mb = 0 ρ = density (kg/m³) x ∑Ma = 0 ∑Mb = 0 ∑Mb = 0 ∑Mc = 0 Vector where F refers to forces and M refers to moments of Most forces have magnitude and direction and can be forces. shown as a vector. The point of application must also be specified. A vector is illustrated by a line, the length of Static determinacy which is proportional to the magnitude on a given scale, If a body is in equilibrium under the action of coplanar and an arrow that shows the direction of the force. forces, the statics equations above must apply. -
Formulas in Solid Mechanics
Formulas in Solid Mechanics Tore Dahlberg Solid Mechanics/IKP, Linköping University Linköping, Sweden This collection of formulas is intended for use by foreign students in the course TMHL61, Damage Mechanics and Life Analysis, as a complement to the textbook Dahlberg and Ekberg: Failure, Fracture, Fatigue - An Introduction, Studentlitteratur, Lund, Sweden, 2002. It may be use at examinations in this course. Contents Page 1. Definitions and notations 1 2. Stress, Strain, and Material Relations 2 3. Geometric Properties of Cross-Sectional Area 3 4. One-Dimensional Bodies (bars, axles, beams) 5 5. Bending of Beam Elementary Cases 11 6. Material Fatigue 14 7. Multi-Axial Stress States 17 8. Energy Methods the Castigliano Theorem 20 9. Stress Concentration 21 10. Material data 25 Version 03-09-18 1. Definitions and notations Definition of coordinate system and loadings on beam Mz qx( ) My Tz N Mx Ty x M N y Ty x T z A z M y M L z Loaded beam, length L, cross section A, and load q(x), with coordinate system (origin at the geometric centre of cross section) and positive section forces and moments: normal force N, shear forces Ty and Tz, torque Mx, and bending moments My, Mz Notations Quantity Symbol SI Unit Coordinate directions, with origin at geometric centre of x, y, z m cross-sectional area A σ 2 Normal stress in direction i (= x, y, z) i N/m τ 2 Shear stress in direction j on surface with normal direction i ij N/m ε Normal strain in direction i i τ γ Shear strain (corresponding to shear stress ij) ij rad Moment with respect to axis iM, Mi Nm Normal force N, P N (= kg m/s2) Shear force in direction i (= y, z) T, Ti N Load q(x) N/m Cross-sectional area A m2 Length L, L0 m Change of length δ m Displacement in direction xu, u(x), u(x,y)m Displacement in direction yv, v(x), v(x,y)m Beam deflection w(x)m 4 Second moment of area (i = y, z) I, Ii m Modulus of elasticity (Young’s modulus) E N/m2 Poisson’s ratio ν Shear modulus G N/m2 Bulk modulus K N/m2 Temperature coefficient α K− 1 1 2. -
Module 8 General Beam Theory
Module 8 General Beam Theory Learning Objectives • Generalize simple beam theory to three dimensions and general cross sections • Consider combined effects of bending, shear and torsion • Study the case of shell beams 8.1 Beams loaded by transverse loads in general direc- tions Readings: BC 6 So far we have considered beams of fairly simple cross sections (e.g. having symmetry planes which are orthogonal) and transverse loads acting on the planes of symmetry. Figure 8.1 shows examples of beams loaded on a plane which does not coincide with a plane of symmetry of its cross section. In this section, we will consider beams with cross section of arbitrary shape which are loaded on planes that do not in general coincide with symmetry planes (or as we will see later more precisely, with principal directions of inertia of the cross section). We will still adopt Euler-Bernoulli hypothesis, which implies that the kinematic assump- tions about the allowed deformation modes of the beam remain the same, see Section 7.1.1. The displacement field is still given by equations (7.4), whereas the strain field is given by equations (7.7). It should be noted that the origin of coordinates in the cross section is still unspecified. 8.1.1 Constitutive law for the cross section We will assume that the beam is made of linear elastic isotropic materials and use Hooke's law. Since the strain distribution is still bound by the sames constraints, the stress distri- bution will be as before: 177 178 MODULE 8. GENERAL BEAM THEORY M2 M2 e2 e2 M2 e2 V2 V2 V2 e3 e3 M3 V3 -
Statics - TAM 211
Statics - TAM 211 Lecture 32 April 9, 2018 Chap 10.1, 10.2, 10.4, 10.8 Announcements No class Wednesday April 11 No office hours for Prof. H-W on Wednesday April 11 Upcoming deadlines: Tuesday (4/10) PL HW 12 Thursday (4/12) WA 5 due Monday (4/16) Mastering Engineering Tutorial 14 Chapter 10: Moments of Inertia Goals and Objectives • Understand the term “moment” as used in this chapter • Determine and know the differences between • First/second moment of area • Moment of inertia for an area • Polar moment of inertia • Mass moment of inertia • Introduce the parallel-axis theorem. • Be able to compute the moments of inertia of composite areas. Applications Many structural members like beams and columns have cross sectional shapes like an I, H, C, etc.. Why do they usually not have solid rectangular, square, or circular cross sectional areas? What primary property of these members influences design decisions? Applications Many structural members are made of tubes rather than solid squares or rounds. Why? This section of the book covers some parameters of the cross sectional area that influence the designer’s selection. Recap: First moment of an area (centroid of an area) The first moment of the area A with respect to the x-axis is given by The first moment of the area A with respect to the y-axis is given by The centroid of the area A is defined as the point C of coordinates and , which satisfies the relation In the case of a composite area, we divide the area A into parts , , Terminology: the term moment in this module refers to the mathematical sense of different “measures” of an area or volume. -
153 Chapter 8
Chapter 8 - Area Moments of Inertia Reading: Chapter 8 - Pages 297 – 313 8-1 Introduction Area Moment of Inertia The second moment of area is also known as the moment of inertia of a shape. The second moment of area is a measure of the "efficiency" of a cross-sectional shape to resist bending caused by loading. The Symbol for Moment of Inertia is I The units of moment of inertia are length raised to the fourth power, such as, in4 or mm4 Moments of inertia of areas are used in calculating the stresses and deflections of beams, the torsion of shafts, and the buckling of columns. The location of the centroid of an area involves the quantity ΣxΔA, which represents the first moment of the area. The area moment of inertia involves the quantity Σx2ΔA, which represent the second moment of an area (because x is squared). The moment of inertia is always computed with respect to an axis; its value is greatly affected by the distribution of the area relative to the axis. Both beams have the same area and even the same shape. Beam 1 is stronger than Beam 2 because it has a larger second moment of inertia (I). Orientation can change the second moment of area (I). For a rectangle, where b is the breadth (horizontal) and h is the height (vertical) if the load is vertical i.e. gravity load If Beam 1 and Beam 2 are 2 in x 12 in, Beam 1 h=12 in Ix = 1/12 (2 in)(12in)3 = 288 in4 X X b =2 in Beam 2 X X h=2 in Ix = 1/12 (12 in)(2in)3 = 8 in4 b =12 in Under the same loading conditions, Beam 2 will bend before Beam 1.