# Bending of Beams with Unsymmetrical Sections

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Bending of Beams with Unsmmetrical Sections Assume that CZ is a neutral ais. C = centroid of section Hence, if > 0, da has negative stress. From the diagram below, we have:

2 δ = α and s = αρ and δ ε = = s ρ E σ = = κ E ρ if is the onl load, we have: σ da = 0 κ E da = 0 or da = 0 hence the neutral ais passes through the centroid C. A similar result holds for and the Y ais. oment equilibrium about the Z ais: and about the Y ais we have: σ da = 2 κ E da = = κ E σ da = κ E da = = κ E and, if CZ is a neutral ais, we will have moments such that: = Similarl, if CY is a neutral ais, we will have moments such that: =

3 However, if CZ is a principal ais, = 0. Therefore, if CZ is also the neutral ais, we have = 0, i.e. bending takes place in the XY plane just as for smmetrical bending. Therefore the plane of the bending moment is perpendicular to the neutral surface onl if the Y and Z aes are principal aes. Hence, we can tackle bending of beams of non smmetric cross section b: (1) finding the principal aes of the section (2) resolving moment into components in the principal ais directions (3) calculating stresses and deflections in each direction (4) superimpose stresses and deflections to get the final result Let Y and Z be the principal aes and let be the bending moment vector. Resolving into components with respect to the principal aes we get: ' = sinθ ' = cosθ f, are the principal moments of inertia σ = ' ' sin σ = ' ' ' ' ( θ ) ' cos( θ ) ' ' '

4 For the neutral ais, σ = 0 b definition, hence as the point (, ) lies on the neutral ais in this case, we have the neutral ais at angle β with respect to the principal ais CZ and hence ' tan β = tanθ ' Note that β 0 in general. The above method is most useful when the principal aes are known or can be found easil b calculation or inspection. The method is also useful for finding deflections (see below). t is also possible to calculate stresses with respect to a set of non principal aes. σ = ( + ) ( + ) The neutral ais is at an angle φ given b: tan φ = This method is useful if the principal aes are not easil found but the components, and of the inertia tensor can be readil determined. Deflections: Using the first method described above, deflections can be found easil b resolving the applied lateral forces into components parallel to the principal aes and separatel calculating the deflection components parallel to these aes. The total deflection at an point along the beam is then found b combining the components at that point into a resultant deflection vector. Note that the resulting deflection will be perpendicular to the neutral ais of the section at that point.

5 Rotation Transformations: φ f, are the coordinates of point P in the the sstem YZ shown above, then the coordinates of P in the sstem Y Z are: ' = cosφ sinφ ' = sinφ + cosφ This transformation is useful in finding the coordinates of points with respect to the principal aes of a section.

6 Problems on Unsmmetrical Beams 1. An angle section with equal legs is subject to a bending moment vector having its direction along the Z-Z direction as shown below. Calculate the maimum tensile stress σt and the maimum compressive stress σc if the angle is a L 663/4 steel section and = in.lb. ( σ t = 3450 psi : σ c = 3080 psi ). 2. An angle section with unequal legs is subjected to a bending oment having its direction along the Z-Z direction as shown below. Calculate the maimum tensile stress σt and the maimum compressive stress σc if the angle is a L 861 and = lb.in. (σt = 1840 psi : σ c = 1860 psi ). 3. Solve the previous problem for a L 741/2 section and = lb. in. (σ t = 2950 psi : σ c = 2930 psi ). See net page for section properties needed in these problems.

7 Section properties for structural steel angle sections. Weight Ais ZZ Ais YY Ais Y'Y' Designation per ft. Area ZZ r ZZ d YY r YY c r min tan α in. lb. in 2 in 4 in. in. in 4 in. in. in. L663/ L L741/ Aes ZZ and YY are centroidal aes parallel to the legs of the section. 2. Distances c and d are measured from the centroid to the outside surfaces of the legs. 3. Aes Y Y and Z Z are the principal centroidal aes. 4. The moment of inertia for ais Y Y is given b Y Y = Ar 2 min. 5. The moment of inertia for ais Z Z is given b Z Z = YY + ZZ Z Z. Y c Y Z Z Z C d Z Y α Y

### MECHANICS OF MATERIALS Plastic Deformations of Members With a Single Plane of Symmetry

Plastic Deformations of Members With a Single Plane of Smmetr Full plastic deformation of a beam with onl a vertical plane of smmetr. The neutral axis cannot be assumed to pass through the section centroid.

### Bending Stress and Strain

Bending Stress and Strain DEFLECTIONS OF BEAMS When a beam with a straight longitudinal ais is loaded by lateral forces, the ais is deformed into a curve, called the deflection curve of the beam. We will

### Shear Forces and Bending Moments

Shear Forces and ending oments lanar (-D) Structures: ll loads act in the same plane and all deflections occurs in the same plane (-y plane) ssociated with the shear forces and bending moments are normal

### (Refer Slide Time: 0:45)

Strength of Materials Prof S. K. Bhattachara Department of Civil Engineering Indian Institute of Technolog, Kharagpur Lecture - 5 Analsis of Stress - IV (Refer Slide Time: 0:45) Welcome to the 5th lesson

### Area Moments of Inertia by Integration

Area Moments of nertia ntegration Second moments or moments of inertia of an area with respect to the and aes, da da Evaluation of the integrals is simplified choosing da to e a thin strip parallel to

### CHAPTER MECHANICS OF MATERIALS

CHPTER 4 Pure MECHNCS OF MTERLS Bending Pure Bending Pure Bending Other Loading Tpes Smmetric Member in Pure Bending Bending Deformations Strain Due to Bending Stress Due to Bending Beam Section Properties

### Fy = P sin 50 + F cos = 0 Solving the two simultaneous equations for P and F,

ENGR0135 - Statics and Mechanics of Materials 1 (161) Homework # Solution Set 1. Summing forces in the y-direction allows one to determine the magnitude of F : Fy 1000 sin 60 800 sin 37 F sin 45 0 F 543.8689

### Over-Reinforced Concrete Beam Design ACI

Note: Over-reinforced typically means the tensile steel does not yield at failure. for this example, over-reinforced means the section is not tension controlled per ACI standards even though the steel

### Stress and Deformation Analysis. Representing Stresses on a Stress Element. Representing Stresses on a Stress Element con t

Stress and Deformation Analysis Material in this lecture was taken from chapter 3 of Representing Stresses on a Stress Element One main goals of stress analysis is to determine the point within a load-carrying

### y Area Moment Matrices 1 e y n θ n n y θ x e x The transformation equations for area moments due to a rotation of axes are:

' t Area Moment Matrices x' t 1 e n 1 t θ θ θ x e x t x n x n n The transformation equations for area moments due to a rotation of axes are: θ θ θ θ 2 2 x x ' xx cos + sin 2 x sin cos 2 2 ' ' xx sin +

### cos 2u - t xy sin 2u (Q.E.D.)

09 Solutions 46060 6/8/10 3:13 PM Page 619 010 Pearson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copyright laws as they currently 9 1. Prove that

### Reinforced Concrete Design SHEAR IN BEAMS

CHAPTER Reinforced Concrete Design Fifth Edition SHEAR IN BEAMS A. J. Clark School of Engineering Department of Civil and Environmental Engineering Part I Concrete Design and Analysis 4a FALL 2002 By Dr.

### STRESS TRANSFORMATION AND MOHR S CIRCLE

Chapter 5 STRESS TRANSFORMATION AND MOHR S CIRCLE 5.1 Stress Transformations and Mohr s Circle We have now shown that, in the absence of bod moments, there are si components of the stress tensor at a material

### REVIEW OVER VECTORS. A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example.

REVIEW OVER VECTORS I. Scalars & Vectors: A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example mass = 5 kg A vector is a quantity that can be described

### CHAPTER 7. E IGENVALUE ANALYSIS

1 CHAPTER 7. E IGENVALUE ANALYSIS Written by: Sophia Hassiotis, January, 2003 Last revision: March, 2015 7.1 Stress Tensor--Mohr s Circle for 2-D Stress Let s review the use of the Mohr s Circle in finding

### Bending of Thin-Walled Beams. Introduction

Introduction Beams are essential in aircraft construction Thin walled beams are often used to lower the weight of aircraft There is a need to determine that loads applied do not lead to beams having ecessive

### Change of Coordinates in Two Dimensions

CHAPTER 5 Change of Coordinates in Two Dimensions Suppose that E is an ellipse centered at the origin. If the major and minor aes are horiontal and vertical, as in figure 5., then the equation of the ellipse

### BEAMS: SHEAR AND MOMENT DIAGRAMS (GRAPHICAL)

LECTURE Third Edition BES: SHER ND OENT DIGRS (GRPHICL). J. Clark School of Engineering Department of Civil and Environmental Engineering 3 Chapter 5.3 by Dr. Ibrahim. ssakkaf SPRING 003 ENES 0 echanics

### Phillips Policy Rules 3.1 A simple textbook macrodynamic model

1 2 3 ( ) ( ) = ( ) + ( ) + ( ) ( ) ( ) ( ) [ ] &( ) = α ( ) ( ) α > 0 4 ( ) ( ) = ( ) 0 < < 1 ( ) = [ ] &( ) = α ( 1) ( ) + + ( ) 0 < < 1 + = 1 5 ( ) ( ) &( ) = β ( ( ) ( )) β > 0 ( ) ( ) ( ) β ( ) =

### MECHANICS OF MATERIALS

2009 The McGraw-Hill Companies, Inc. All rights reserved. Fifth SI Edition CHAPTER 4 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Pure Bending Lecture

### OUTCOME 1 - TUTORIAL 1 COMPLEX STRESS AND STRAIN

UNIT 6: STRNGTHS OF MATRIALS Unit code: K/60/409 QCF level: 5 Credit value: 5 OUTCOM - TUTORIAL COMPLX STRSS AND STRAIN. Be able to determine the behavioural characteristics of engineering components subjected

### P4 Stress and Strain Dr. A.B. Zavatsky MT07 Lecture 4 Stresses on Inclined Sections

4 Stress and Strain Dr. A.B. Zavatsky MT07 Lecture 4 Stresses on Inclined Sections Shear stress and shear strain. Equality of shear stresses on perpendicular planes. Hooke s law in shear. Normal and shear

### Mechanics of Materials Qualifying Exam Study Material

Mechanics of Materials Qualifying Exam Study Material The candidate is expected to have a thorough understanding of mechanics of materials topics. These topics are listed below for clarification. Not all

### Assignment 7 - Solutions Math 209 Fall 2008

Assignment 7 - Solutions Math 9 Fall 8. Sec. 5., eercise 8. Use polar coordinates to evaluate the double integral + y da, R where R is the region that lies to the left of the y-ais between the circles

### Chapter 2: Load, Stress and Strain

Chapter 2: Load, Stress and Strain The careful text- books measure (Let all who build beware!) The load, the shock, the pressure Material can bear. So when the buckled girder Lets down the grinding span,

### ASEN Structures. Stress in 3D. ASEN 3112 Lecture 1 Slide 1

ASEN 3112 - Structures Stress in 3D ASEN 3112 Lecture 1 Slide 1 ASEN 3112 - Structures Mechanical Stress in 3D: Concept Mechanical stress measures intensit level of internal forces in a material (solid

### Chapter 3: Electric Fields

3.1 The Electric Field In chapter 2, Coulomb s law of electrostatics was discussed and it was shown that, if a charge q2 is brought into the neighborhood of another charge q1, a force is eerted upon q2.

### Physics 133: tutorial week 5 Alternating currents, magnetic fields and transformers.

Physics 133: tutorial week 5 Alternating currents, magnetic fields and transformers. 61. In the wires connecting an electric clock to a wall socket, how many times a day does the alternating current reverse

### 2 Topics in 3D Geometry

2 Topics in 3D Geometry In two dimensional space, we can graph curves and lines. In three dimensional space, there is so much extra space that we can graph planes and surfaces in addition to lines and

### Math 21a Old Exam One Fall 2003 Solutions Spring, 2009

1 (a) Find the curvature κ(t) of the curve r(t) = cos t, sin t, t at the point corresponding to t = Hint: You ma use the two formulas for the curvature κ(t) = T (t) r (t) = r (t) r (t) r (t) 3 Solution:

### CHAPTER 6 MECHANICAL PROPERTIES OF METALS PROBLEM SOLUTIONS

CHAPTER 6 MECHANICAL PROPERTIES OF METALS PROBLEM SOLUTIONS Concepts of Stress and Strain 6.1 Using mechanics of materials principles (i.e., equations of mechanical equilibrium applied to a free-body diagram),

### 6. ISOMETRIES isometry central isometry translation Theorem 1: Proof:

6. ISOMETRIES 6.1. Isometries Fundamental to the theory of symmetry are the concepts of distance and angle. So we work within R n, considered as an inner-product space. This is the usual n- dimensional

### COMPLEX STRESS TUTORIAL 3 COMPLEX STRESS AND STRAIN

COMPLX STRSS TUTORIAL COMPLX STRSS AND STRAIN This tutorial is not part of the decel unit mechanical Principles but covers elements of the following sllabi. o Parts of the ngineering Council eam subject

### 3D Stress Components. From equilibrium principles: τ xy = τ yx, τ xz = τ zx, τ zy = τ yz. Normal Stresses. Shear Stresses

3D Stress Components From equilibrium principles:, z z, z z The most general state of stress at a point ma be represented b 6 components Normal Stresses Shear Stresses Normal stress () : the subscript

### Lecture 4: Basic Review of Stress and Strain, Mechanics of Beams

MECH 466 Microelectromechanical Sstems Universit of Victoria Dept. of Mechanical Engineering Lecture 4: Basic Review of Stress and Strain, Mechanics of Beams 1 Overview Compliant Mechanisms Basics of Mechanics

### 0.1 Linear Transformations

.1 Linear Transformations A function is a rule that assigns a value from a set B for each element in a set A. Notation: f : A B If the value b B is assigned to value a A, then write f(a) = b, b is called

### 2. STRESS, STRAIN, AND CONSTITUTIVE RELATIONS

. STRESS, STRAIN, AND CONSTITUTIVE RELATIONS Mechanics of materials is a branch of mechanics that develops relationships between the eternal loads applied to a deformable bod and the intensit of internal

### MATHEMATICS SPECIALIST ATAR COURSE FORMULA SHEET

MATHEMATICS SPECIALIST ATAR COURSE FORMULA SHEET 06 Copyright School Curriculum and Standards Authority, 06 This document apart from any third party copyright material contained in it may be freely copied,

### Structural Axial, Shear and Bending Moments

Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants

### Chapt. 10 Rotation of a Rigid Object About a Fixed Axis

Chapt. otation of a igid Object About a Fixed Axis. Angular Position, Velocity, and Acceleration Translation: motion along a straight line (circular motion?) otation: surround itself, spins rigid body:

### Triple integrals in Cartesian coordinates (Sect. 15.4) Review: Triple integrals in arbitrary domains.

Triple integrals in Cartesian coordinates (Sect. 5.4 Review: Triple integrals in arbitrar domains. s: Changing the order of integration. The average value of a function in a region in space. Triple integrals

### EQUILIBRIUM STRESS SYSTEMS

EQUILIBRIUM STRESS SYSTEMS Definition of stress The general definition of stress is: Stress = Force Area where the area is the cross-sectional area on which the force is acting. Consider the rectangular

### ORIENTATIONS OF LINES AND PLANES IN SPACE

GG303 Lab 1 9/10/03 1 ORIENTATIONS OF LINES AND PLANES IN SPACE I Main Topics A Definitions of points, lines, and planes B Geologic methods for describing lines and planes C Attitude symbols for geologic

### Determination of Base Stresses in Rectangular Footings under Biaxial Bending 1

Digest 011, December 011, 1519-155 Determination of Base Stresses in Rectangular Footings under Biaial Bending 1 Güna ÖZMEN* ABSTRACT All the footings of buildings in seismic regions are to be designed

### Chapter 2: Concurrent force systems. Department of Mechanical Engineering

Chapter : Concurrent force sstems Objectives To understand the basic characteristics of forces To understand the classification of force sstems To understand some force principles To know how to obtain

### STRAIN COMPATIBILITY FOR DESIGN OF PRESTRESSED SECTIONS

Structural Concrete Software System TN T178 strain_compatibility_bending 030904 STRAIN COMPATIBILITY FOR DESIGN OF PRESTRESSED SECTIONS Concrete sections are not always rectangular or simple T, with one

### MATH 118, LECTURES 14 & 15: POLAR AREAS

MATH 118, LECTURES 1 & 15: POLAR AREAS 1 Polar Areas We recall from Cartesian coordinates that we could calculate the area under the curve b taking Riemann sums. We divided the region into subregions,

### Announcements. Moment of a Force

Announcements Test observations Units Significant figures Position vectors Moment of a Force Today s Objectives Understand and define Moment Determine moments of a force in 2-D and 3-D cases Moment of

### SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 STRESS AND STRAIN

1.1 Stress & Strain SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 STRESS AND STRAIN Stress is the internal resistance offered by the body per unit area. Stress is represented as force per unit area. Typical

### 10 Space Truss and Space Frame Analysis

10 Space Truss and Space Frame Analysis 10.1 Introduction One dimensional models can be very accurate and very cost effective in the proper applications. For example, a hollow tube may require many thousands

### MECE 3400 Summer 2016 Homework #1 Forces and Moments as Vectors

ASSIGNED 1) Knowing that α = 40, determine the resultant of the three forces shown: 2) Two cables, AC and BC, are tied together at C and pulled by a force P, as shown. Knowing that P = 500 N, α = 60, and

### Orthogonal Matrices. u v = u v cos(θ) T (u) + T (v) = T (u + v). It s even easier to. If u and v are nonzero vectors then

Part 2. 1 Part 2. Orthogonal Matrices If u and v are nonzero vectors then u v = u v cos(θ) is 0 if and only if cos(θ) = 0, i.e., θ = 90. Hence, we say that two vectors u and v are perpendicular or orthogonal

### Bending Stress in Beams

936-73-600 Bending Stress in Beams Derive a relationship for bending stress in a beam: Basic Assumptions:. Deflections are very small with respect to the depth of the beam. Plane sections before bending

### Introduction to Plates

Chapter Introduction to Plates Plate is a flat surface having considerabl large dimensions as compared to its thickness. Common eamples of plates in civil engineering are. Slab in a building.. Base slab

### Introduction to Mechanical Behavior of Biological Materials

Introduction to Mechanical Behavior of Biological Materials Ozkaya and Nordin Chapter 7, pages 127-151 Chapter 8, pages 173-194 Outline Modes of loading Internal forces and moments Stiffness of a structure

### Lines and planes in space (Sect. 12.5) Review: Lines on a plane. Lines in space (Today). Planes in space (Next class). Equation of a line

Lines and planes in space (Sect. 2.5) Lines in space (Toda). Review: Lines on a plane. The equations of lines in space: Vector equation. arametric equation. Distance from a point to a line. lanes in space

### SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 STRESS AND STRAIN

SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 STRESS AND STRAIN 1.1 Stress & Strain Stress is the internal resistance offered by the body per unit area. Stress is represented as force per unit area. Typical

### Analysis of Stress and Strain

07Ch07.qd 9/7/08 1:18 PM Page 571 7 Analsis of Stress and Strain Plane Stress Problem 7.-1 An element in plane stress is subjected to stresses s 4750 psi, s 100 psi, and t 950 psi, as shown in the figure.

### Engineering Mathematics 233 Solutions: Double and triple integrals

Engineering Mathematics s: Double and triple integrals Double Integrals. Sketch the region in the -plane bounded b the curves and, and find its area. The region is bounded b the parabola and the straight

### 1. The Six Trigonometric Functions 1.1 Angles, Degrees, and Special Triangles 1.2 The Rectangular Coordinate System 1.3 Definition I: Trigonometric

1. The Si Trigonometric Functions 1.1 Angles, Degrees, and Special Triangles 1. The Rectangular Coordinate Sstem 1.3 Definition I: Trigonometric Functions 1.4 Introduction to Identities 1.5 More on Identities

### Unit M2.4 Stress and Strain Transformations

Unit M2.4 Stress and Strain Transformations Readings: CDL 4.5, 4.6, 4.7, 4.11, 4.12, 4.13 CDL 4.14, 4.15 16.001/002 -- Unified Engineering Department of Aeronautics and Astronautics Massachusetts Institute

### Shear Center in Thin-Walled Beams Lab

Shear Center in Thin-Walled Beams Lab Shear flow is developed in beams with thin-walled cross sections shear flow (q sx ): shear force per unit length along cross section q sx =τ sx t behaves much like

### Electrostatics. Ans.The particles 1 and 2 are negatively charged and particle 3 is positively charged.

Electrostatics [ Two marks each] Q1.An electric dipole with dipole moment 4 10 9 C m is aligned at 30 with the direction of a uniform electric field of magnitude 5 10 4 N C 1. Calculate the net force and

### superimposing the stresses and strains cause by each load acting separately

COMBINED LOADS In many structures the members are required to resist more than one kind of loading (combined loading). These can often be analyzed by superimposing the stresses and strains cause by each

### Section 10.6 Vectors in Space

36 Section 10.6 Vectors in Space Up to now, we have discussed vectors in two-dimensional plane. We now want to epand our ideas into three-dimensional space. Thus, each point in three-dimensional space

### Lesson 03: Kinematics

www.scimsacademy.com PHYSICS Lesson 3: Kinematics Translational motion (Part ) If you are not familiar with the basics of calculus and vectors, please read our freely available lessons on these topics,

### Geometric Objects and Transformations

Geometric Objects and ransformations How to represent basic geometric types, such as points and vectors? How to convert between various represenations? (through a linear transformation!) How to establish

### Section V.2: Magnitudes, Directions, and Components of Vectors

Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions

### Laboratory 2 Application of Trigonometry in Engineering

Name: Grade: /26 Section Number: Laboratory 2 Application of Trigonometry in Engineering 2.1 Laboratory Objective The objective of this laboratory is to learn basic trigonometric functions, conversion

### Module 3. Analysis of Statically Indeterminate Structures by the Displacement Method. Version 2 CE IIT, Kharagpur

odule 3 Analysis of Statically Indeterminate Structures by the Displacement ethod Lesson 21 The oment- Distribution ethod: rames with Sidesway Instructional Objectives After reading this chapter the student

### Chapter 24 Physical Pendulum

Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...

### Matrices in Statics and Mechanics

Matrices in Statics and Mechanics Casey Pearson 3/19/2012 Abstract The goal of this project is to show how linear algebra can be used to solve complex, multi-variable statics problems as well as illustrate

### Chapter 3 THE STATIC ASPECT OF SOLICITATION

Chapter 3 THE STATIC ASPECT OF SOLICITATION 3.1. ACTIONS Construction elements interact between them and with the environment. The consequence of this interaction defines the system of actions that subject

### Analysis of Stresses and Strains

Chapter 7 Analysis of Stresses and Strains 7.1 Introduction axial load = P / A torsional load in circular shaft = T / I p bending moment and shear force in beam = M y / I = V Q / I b in this chapter, we

### Deformation of Single Crystals

Deformation of Single Crystals When a single crystal is deformed under a tensile stress, it is observed that plastic deformation occurs by slip on well defined parallel crystal planes. Sections of the

### Announcements. Dry Friction

Announcements Dry Friction Today s Objectives Understand the characteristics of dry friction Draw a FBD including friction Solve problems involving friction Class Activities Applications Characteristics

### BEAMS: DEFORMATION BY SUPERPOSITION

ETURE EMS: EFORMTION Y SUPERPOSITION Third Edition. J. lark School of Engineering epartment of ivil and Environmental Engineering 19 hapter 9.7 9.8 b r. Ibrahim. ssakkaf SPRING 00 ENES 0 Mechanics of Materials

### Measuring the Earth s Diameter from a Sunset Photo

Measuring the Earth s Diameter from a Sunset Photo Robert J. Vanderbei Operations Research and Financial Engineering, Princeton University rvdb@princeton.edu ABSTRACT The Earth is not flat. We all know

### Conservation Equations in Fluid Flow

Conservation Equations in Fluid Flow Q. Choose the correct answer (i) Mathematical statement of Renolds transport theorem is given b dn (a). da d C t dn (b) da. d C t dn (c) d. da C t dn (d) d. da C t

Announcements 2-D Vector Addition Today s Objectives Understand the difference between scalars and vectors Resolve a 2-D vector into components Perform vector operations Class Activities Applications Scalar

TABLE OF CONTENTS MECHANICS MECHANICS 113 Terms and Definitions 114 Unit Systems 114 Gravity 116 Metric (SI) System 118 Force Systems 118 Scalar and Vector Quantities 118 Graphical Resolution of Forces

### BASICS ANGLES. θ x. Figure 1. Trigonometric measurement of angle θ. trigonometry, page 1 W. F. Long, 1989

TRIGONOMETRY BASICS Trigonometry deals with the relation between the magnitudes of the sides and angles of a triangle. In a typical trigonometry problem, two angles and a side of a triangle would be given

### Physics 107 HOMEWORK ASSIGNMENT #8

Physics 107 HOMEORK ASSIGMET #8 Cutnell & Johnson, 7 th edition Chapter 9: Problems 16, 22, 24, 66, 68 16 A lunch tray is being held in one hand, as the drawing illustrates. The mass of the tray itself

Week 8 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

### Introduction to Beam. Area Moments of Inertia, Deflection, and Volumes of Beams

Introduction to Beam Theory Area Moments of Inertia, Deflection, and Volumes of Beams Horizontal structural member used to support horizontal loads such as floors, roofs, and decks. Types of beam loads

### Areas and double integrals. (Sect. 15.3)

Areas and double integrals. (Sect. 5.) Areas of a region on a plane. Average value of a function. More eamples of double integrals. Areas of a region on a plane The area of a closed, bounded region on

### Standard Terminology for Vehicle Dynamics Simulations

Standard Terminology for Vehicle Dynamics Simulations Y v Z v X v Michael Sayers The University of Michigan Transportation Research Institute (UMTRI) February 22, 1996 Table of Contents 1. Introduction...1

### MOHR'S CIRCLE FOR MOMENT OF INERTIA. Wentbridge Viaduct Yorkshire, UK

MOHR'S CIRCLE FOR MOMENT OF INERTIA Wentbridge Viaduct Yorkshire, UK Asymmetrical Sections and Loads To this point, calculation of moment of inertia has been based upon the section being loaded symmetrically

### INVERSE TRIGONOMETRIC FUNCTIONS

Chapter INVERSE TRIGONOMETRIC FUNCTIONS Overview Inverse function Inverse of a function f eists, if the function is one-one and onto, ie, bijective Since trigonometric functions are many-one over their

### Three Body Problem for Constant Angular Velocity

Three Body Problem for Constant Angular Velocity P. Coulton Department of Mathematics Eastern Illinois University Charleston, Il 61920 Corresponding author, E-mail: cfprc@eiu.edu 1 Abstract We give the

### Center of Mass and Centroids :: Guidelines

Center of Mass and Centroids :: Guidelines Centroids of Lines, reas, and Volumes 1.Order of Element Selected for ntegration.continuit.discarding Higer Order Terms.Coice of Coordinates 5.Centroidal Coordinate

### R A = R B = = 3.6 kn. ΣF y = 3.6 V = 0 V = 3.6 kn. A similar calculation for any section through the beam at 3.7 < x < 7.

ENDNG STRESSES & SHER STRESSES N EMS (SSGNMENT SOLUTONS) Question 1: 89 mm 3 mm Parallam beam has a length of 7.4 m and supports a concentrated load of 7.2 kn, as illustrated below. Draw shear force and

### Lecture 8. Metrics. 8.1 Riemannian Geometry. Objectives: More on the metric and how it transforms. Reading: Hobson, 2.

Lecture 8 Metrics Objectives: More on the metric and how it transforms. Reading: Hobson, 2. 8.1 Riemannian Geometry The interval ds 2 = g αβ dx α dx β, is a quadratic function of the coordinate differentials.

### a a. θ = cos 1 a b ) b For non-zero vectors a and b, then the component of b along a is given as comp

Textbook Assignment 4 Your Name: LAST NAME, FIRST NAME (YOUR STUDENT ID: XXXX) Your Instructors Name: Prof. FIRST NAME LAST NAME YOUR SECTION: MATH 0300 XX Due Date: NAME OF DAY, MONTH DAY, YEAR. SECTION

### The Theory of Birefringence

Birefringence spectroscopy is the optical technique of measuring orientation in an optically anisotropic sample by measuring the retardation of polarized light passing through the sample. Birefringence

### PHYS101 Vectors Spring 2014

Vectors Before we start with the tutorials, we should state the following summar for the calculation of the angles (direction) Let us consider the following situations In general we can summarise If 1

### Higher Still Level Paper

Higher Still Level Paper 7. Given the cube with sides units and B is (,,). Also P and Q are the centres of face OCGD and CBFG. Z G y F (a) Coordinates og G are (,, ) D E P Q C B(,, ) (b) Position vectors

### MATERIALS AND MECHANICS OF BENDING

HAPTER Reinforced oncrete Design Fifth Edition MATERIALS AND MEHANIS OF BENDING A. J. lark School of Engineering Department of ivil and Environmental Engineering Part I oncrete Design and Analysis b FALL