MECHANICS OF MATERIALS
|
|
- Frederica Hood
- 6 years ago
- Views:
Transcription
1 00 The McGraw-Hill Companies, Inc. All rights reserved. Third E CHAPTER 7 Transformations MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Teas Tech Universit of Stress and Strain
2 Transformations of Stress and Strain Introduction Transformation of Plane Stress Principal Stresses Maimum Shearing Stress Eample 7.01 Sample Problem 7.1 Mohr s Circle for Plane Stress Eample 7.0 Sample Problem 7. General State of Stress Application of Mohr s Circle to the Three- Dimensional Analsis of Stress Yield Criteria for Ductile Materials Under Plane Stress Fracture Criteria for Brittle Materials Under Plane Stress Stresses in Thin-Walled Pressure Vessels 00 The McGraw-Hill Companies, Inc. All rights reserved. 7 -
3 Introduction The most general state of stress at a point ma be represented b 6 components,,,, z z, z normal stresses shearing stresses (Note:, z z, z Same state of stress is represented b a different set of components if aes are rotated. The first part of the chapter is concerned with how the components of stress are transformed under a rotation of the coordinate aes. The second part of the chapter is devoted to a similar analsis of the transformation of the components of strain. z ) 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-3
4 Introduction Plane Stress - state of stress in which two faces of the cubic element are free of stress. For the illustrated eample, the state of stress is defined b,, 0. and z z z State of plane stress occurs in a thin plate subjected to forces acting in the midplane of the plate. State of plane stress also occurs on the free surface of a structural element or machine component, i.e., at an point of the surface not subjected to an eternal force. 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-4
5 Transformation of Plane Stress Consider the cons for equilibrium of a prismatic element with faces perpendicular to the,, and aes. 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-5
6 Transformation of Plane Stress The equations ma be rewritten to ield 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-6
7 Principal Stresses 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-7
8 Principal Stresses The previous equations are combined to ield parametric equations for a circle, where ave R ave R Principal stresses occur on the principal planes of stress with zero shearing stresses. ma,min tan p Note : defines two angles separated b 90 o 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-8
9 Maimum Shearing Stress Maimum shearing stress occurs for ave ma R tan s Note :defines two angles separated b 90 offset from ave p b 45 o o and 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-9
10 Eample 7.01 For the state of plane stress shown, determine (a) the principal panes, (b) the principal stresses, (c) the maimum shearing stress and the corresponding normal stress. SOLUTION: Find the element orientation for the principal stresses from tan p Determine the principal stresses from ma,min Calculate the maimum shearing stress with ma 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-10
11 Eample MPa 10MPa 40MPa SOLUTION: Find the element orientation for the principal stresses from 40 tan p p p 53.1, , Determine the principal stresses from ma,min ma min 70MPa 30MPa 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-11
12 Eample 7.01 Calculate the maimum shearing stress with ma MPa 10MPa 40MPa ma 50MPa s 45 s p 18.4, The corresponding normal stress is ave MPa 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-1
13 Sample Problem 7.1 A single horizontal force P of 150 lb magnitude is applied to end D of lever ABD. Determine (a) the normal and shearing stresses on an element at point H having sides parallel to the and aes, (b) the principal planes and principal stresses at the point H. SOLUTION: Determine an equivalent force-couple sstem at the center of the transverse section passing through H. Evaluate the normal and shearing stresses at H. Determine the principal planes and calculate the principal stresses. 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-13
14 Sample Problem 7.1 SOLUTION: Determine an equivalent force-couple sstem at the center of the transverse section passing through H. T M P 150lb 150lb18in.7 kip 150lb10in 1.5kip in Evaluate the normal and shearing stresses at H. Mc I Tc J in 1.5kip in 0.6in in 4.7 kip in 0.6in 1 0.6in ksi 7.96ksi 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-14
15 Sample Problem 7.1 Determine the principal planes and calculate the principal stresses. tan p p p 61.0, , ma,min ma min ksi 4.68ksi The McGraw-Hill Companies, Inc. All rights reserved. 7-15
16 Mohr s Circle for Plane Stress With the phsical significance of Mohr s circle for plane stress established, it ma be applied with simple geometric considerations. Critical values are estimated graphicall or calculated. For a known state of plane stress,, plot the points X and Y and construct the circle centered at C. ave R The principal stresses are obtained at A and B. ma,min tan p ave R The direction of rotation of O to Oa is the same as CX to CA. 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-16
17 Mohr s Circle for Plane Stress With Mohr s circle uniquel defined, the state of stress at other aes orientations ma be depicted. For the state of stress at an angle with respect to the aes, construct a new diameter X Y at an angle with respect to XY. Normal and shear stresses are obtained from the coordinates X Y. 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-17
18 Mohr s Circle for Plane Stress Mohr s circle for centric aial loading: P, 0 A P A Mohr s circle for torsional loading: Tc Tc 0 0 J J 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-18
19 Eample 7.0 For the state of plane stress shown, (a) construct Mohr s circle, determine (b) the principal planes, (c) the principal stresses, (d) the maimum shearing stress and the corresponding normal stress. 00 The McGraw-Hill Companies, Inc. All rights reserved. SOLUTION: Construction of Mohr s circle ave CF MPa R CX FX 40MPa MPa 0MPa 7-19
20 Eample 7.0 Principal planes and stresses ma OA OC CA 0 50 ma 70MPa OB OC BC 0 50 min min 30MPa tan p p p FX CP The McGraw-Hill Companies, Inc. All rights reserved. 7-0
21 Eample 7.0 Maimum shear stress s p 45 ma R ave s ma 50 MPa 0 MPa 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-1
22 Sample Problem 7. For the state of stress shown, determine (a) the principal planes and the principal stresses, (b) the stress components eerted on the element obtained b rotating the given element counterclockwise through 30 degrees. SOLUTION: Construct Mohr s circle 100 ave R 60 80MPa CF FX MPa 00 The McGraw-Hill Companies, Inc. All rights reserved. 7 -
23 Sample Problem 7. Principal planes and stresses tan p p p XF CF clockwise ma OA OC CA 80 5 ma 13MPa ma OA OC 80 5 min 8 MPa BC 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-3
24 Sample Problem 7. Stress components after rotation b 30 o Points X and Y on Mohr s circle that correspond to stress components on the rotated element are obtained b rotating XY counterclockwise through The McGraw-Hill Companies, Inc. All rights reserved OK OC KC 80 5cos5.6 OL OC CL 80 5cos5.6 KX 5sin MPa 111.6MPa 41.3MPa 7-4
25 Stresses in Thin-Walled Pressure Vessels 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-5
26 Stresses in Thin-Walled Pressure Vessels 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-6
27 Stresses in Thin-Walled Pressure Vessels Clindrical vessel with principal stresses 1 = hoop stress = longitudinal stress Hoop stress: F z pr t 0 pr t t p r 1 Longitudinal stress: F rt p r 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-7
28 Stresses in Thin-Walled Pressure Vessels Spherical pressure vessel: 1 pr t Mohr s circle for in-plane transformations reduces to a point 1 ma(in -plane) constant 0 Maimum out-of-plane shearing stress ma 1 1 pr 4t 00 The McGraw-Hill Companies, Inc. All rights reserved. 7-8
PROBLEM Solve Probs. 7.5 and 7.9, using Mohr s circle.
PROBLEM 7.1 Solve Probs. 7.5 and 7.9, using Mohr s circle. PROBLEM 7.5 through 7.8 For the given state of stress, determine (a) the principal planes, (b) the principal stresses. PROBLEM 7.9 through 7.12
More informationArgand diagrams 2E. circle centre (0, 0), radius 6 equation: circle centre (0, 0), radius equation: circle centre (3, 0), radius 2
Argand diagrams E 1 a z 6 circle centre (0, 0), radius 6 equation: y y 6 6 b z 10 circle centre (0, 0), radius 10 equation: y 10 y 100 c z circle centre (, 0), radius equation: ( ) y ( ) y d z i z ( i)
More informationAC : APPLICATIONS OF SOLIDWORKS IN TEACHING COURSES OF STATICS AND STRENGTH OF MATERIALS
AC 2012-3232: APPLICATIONS OF SOLIDWORKS IN TEACHING COURSES OF STATICS AND STRENGTH OF MATERIALS Dr. Xiaobin Le P.E., Wentworth Institute of Technology Xiaobin Le is Assistant Professor, Ph.D., P.Eng.,
More information4. Two Dimensional Transformations
4. Two Dimensional Transformations CS362 Introduction to Computer Graphics Helena Wong, 2 In man applications, changes in orientations, sizes, and shapes are accomplished with geometric transformations
More informationTwo Dimensional Viewing
Two Dimensional Viewing Dr. S.M. Malaek Assistant: M. Younesi Two Dimensional Viewing Basic Interactive Programming Basic Interactive Programming User controls contents, structure, and appearance of objects
More informationES 230 Strengths Intro to Finite Element Modeling & Analysis Homework Assignment 5
ES 230 Strengths Intro to Finite Element Modeling & Analysis Homework Assignment 5 The GREAT Combined Load Problem Moving on to Principal Stresses and von Mises Failure Criteria Open up your ANSYS model
More informationAmbar Mitra. Concept Map Manual
Ambar Mitra Concept Map Manual Purpose: This manual contains the instructions for using the Concept Map software. We will show the features by solving three problems: (i) rosette and maximum, shear strain
More informationOP-029. INTERPRETATION OF CIU TEST
INTERPRETATION OF CIU TEST Page 1 of 7 WORK INSTRUCTIONS FOR ENGINEERS KYW Compiled by : Checked by LCH : TYC Approved by : OP-09. INTERPRETATION OF CIU TEST INTERPRETATION OF CIU TEST Page of 7 9. INTERPRETATION
More informationTo graph the point (r, θ), simply go out r units along the initial ray, then rotate through the angle θ. The point (1, 5π 6. ) is graphed below:
Polar Coordinates Any point in the plane can be described by the Cartesian coordinates (x, y), where x and y are measured along the corresponding axes. However, this is not the only way to represent points
More informationOn the Kinematics of Undulator Girder Motion
LCLS-TN-09-02 On the Kinematics of Undulator Girder Motion J. Welch March 23, 2010 The theor of rigid bod kinematics is used to derive equations that govern the control and measurement of the position
More informationModeling with CMU Mini-FEA Program
Modeling with CMU Mini-FEA Program Introduction Finite element analsis (FEA) allows ou analze the stresses and displacements in a bod when forces are applied. FEA determines the stresses and displacements
More informationMAE 323: Lab 7. Instructions. Pressure Vessel Alex Grishin MAE 323 Lab Instructions 1
Instructions MAE 323 Lab Instructions 1 Problem Definition Determine how different element types perform for modeling a cylindrical pressure vessel over a wide range of r/t ratios, and how the hoop stress
More informationJim Lambers MAT 169 Fall Semester Lecture 33 Notes
Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 33 Notes These notes correspond to Section 9.3 in the text. Polar Coordinates Throughout this course, we have denoted a point in the plane by an ordered
More information[3] Rigid Body Analysis
[3] Rigid Body Analysis Page 1 of 53 [3] Rigid Body Analysis [3.1] Equilibrium of a Rigid Body [3.2] Equations of Equilibrium [3.3] Equilibrium in 3-D [3.4] Simple Trusses [3.5] The Method of Joints [3.6]
More informationKINEMATICS STUDY AND WORKING SIMULATION OF THE SELF- ERECTION MECHANISM OF A SELF-ERECTING TOWER CRANE, USING NUMERICAL AND ANALYTICAL METHODS
The rd International Conference on Computational Mechanics and Virtual Engineering COMEC 9 9 OCTOBER 9, Brasov, Romania KINEMATICS STUY AN WORKING SIMULATION OF THE SELF- ERECTION MECHANISM OF A SELF-ERECTING
More informationis a plane curve and the equations are parametric equations for the curve, with parameter t.
MATH 2412 Sections 6.3, 6.4, and 6.5 Parametric Equations and Polar Coordinates. Plane Curves and Parametric Equations Suppose t is contained in some interval I of the real numbers, and = f( t), = gt (
More information8.6 Three-Dimensional Cartesian Coordinate System
SECTION 8.6 Three-Dimensional Cartesian Coordinate Sstem 69 What ou ll learn about Three-Dimensional Cartesian Coordinates Distance and Midpoint Formulas Equation of a Sphere Planes and Other Surfaces
More informationLesson 5: Perpendicular Lines
: Perpendicular Lines Learning Target I can generalize the criterion for perpendicularity of two segments that meet at a point I can determine if two segments are perpendicular and write the equation of
More informationCS F-07 Objects in 2D 1
CS420-2010F-07 Objects in 2D 1 07-0: Representing Polgons We want to represent a simple polgon Triangle, rectangle, square, etc Assume for the moment our game onl uses these simple shapes No curves for
More information[ ] [ ] Orthogonal Transformation of Cartesian Coordinates in 2D & 3D. φ = cos 1 1/ φ = tan 1 [ 2 /1]
Orthogonal Transformation of Cartesian Coordinates in 2D & 3D A vector is specified b its coordinates, so it is defined relative to a reference frame. The same vector will have different coordinates in
More informationANSYS Workbench Guide
ANSYS Workbench Guide Introduction This document serves as a step-by-step guide for conducting a Finite Element Analysis (FEA) using ANSYS Workbench. It will cover the use of the simulation package through
More informationGuidelines for proper use of Plate elements
Guidelines for proper use of Plate elements In structural analysis using finite element method, the analysis model is created by dividing the entire structure into finite elements. This procedure is known
More informationSection 13.5: Equations of Lines and Planes. 1 Objectives. 2 Assignments. 3 Lecture Notes
Section 13.5: Equations of Lines and Planes 1 Objectives 1. Find vector, symmetric, or parametric equations for a line in space given two points on the line, given a point on the line and a vector parallel
More informationD DAVID PUBLISHING. Finite Element Analysis of Polygon Shaped Shell Roof. 1. Introduction. Attia Mousa 1 and Hesham El Naggar 2
Journal of Civil Engineering and Architecture 11 (17) 4-4 doi: 1.1765/194-759/17.5. D DAVID PUBLISHING Finite Element Analsis of Polgon Shaped Shell Roof Attia Mousa 1 and Hesham El Naggar 1. Professor
More informationLecture 3.4 Differential Equation Based Schemes
Lecture 3.4 Differential Equation Based Schemes 1 Differential Equation Based Schemes As stated in the previous lecture, another important and most widel used method of structured mesh generation is based
More informationPre-Calculus Guided Notes: Chapter 10 Conics. A circle is
Name: Pre-Calculus Guided Notes: Chapter 10 Conics Section Circles A circle is _ Example 1 Write an equation for the circle with center (3, ) and radius 5. To do this, we ll need the x1 y y1 distance formula:
More information16 SW Simulation design resources
16 SW Simulation design resources 16.1 Introduction This is simply a restatement of the SW Simulation online design scenarios tutorial with a little more visual detail supplied on the various menu picks
More informationFocusing properties of spherical and parabolic mirrors
Physics 5B Winter 008 Focusing properties of spherical and parabolic mirrors 1 General considerations Consider a curved mirror surface that is constructed as follows Start with a curve, denoted by y()
More informationQuarter Symmetry Tank Stress (Draft 4 Oct 24 06)
Quarter Symmetry Tank Stress (Draft 4 Oct 24 06) Introduction You need to carry out the stress analysis of an outdoor water tank. Since it has quarter symmetry you start by building only one-fourth of
More information: LEARNING THE VIRTUAL WORK METHOD IN STATICS: WHAT IS A COMPATIBLE VIRTUAL DISPLACEMENT?
2006-823: LERNING THE VIRTUL WORK METHOD IN STTICS: WHT IS COMPTIBLE VIRTUL DISPLCEMENT? Ing-Chang Jong, Universit of rkansas Ing-Chang Jong serves as Professor of Mechanical Engineering at the Universit
More informationStructural performance simulation using finite element and regression analyses
Structural performance simulation using finite element and regression analyses M. El-Sayed 1, M. Edghill 1 & J. Housner 2 1 Department of Mechanical Engineering, Kettering University, USA 2 NASA Langley
More informationRevision of the SolidWorks Variable Pressure Simulation Tutorial J.E. Akin, Rice University, Mechanical Engineering. Introduction
Revision of the SolidWorks Variable Pressure Simulation Tutorial J.E. Akin, Rice University, Mechanical Engineering Introduction A SolidWorks simulation tutorial is just intended to illustrate where to
More informationComputer Graphics. Geometric Transformations
Contents coordinate sstems scalar values, points, vectors, matrices right-handed and left-handed coordinate sstems mathematical foundations transformations mathematical descriptions of geometric changes,
More informationSection 9.3: Functions and their Graphs
Section 9.: Functions and their Graphs Graphs provide a wa of displaing, interpreting, and analzing data in a visual format. In man problems, we will consider two variables. Therefore, we will need to
More informationTo graph the point (r, θ), simply go out r units along the initial ray, then rotate through the angle θ. The point (1, 5π 6
Polar Coordinates Any point in the plane can be described by the Cartesian coordinates (x, y), where x and y are measured along the corresponding axes. However, this is not the only way to represent points
More informationComputer Graphics. Geometric Transformations
Computer Graphics Geometric Transformations Contents coordinate sstems scalar values, points, vectors, matrices right-handed and left-handed coordinate sstems mathematical foundations transformations mathematical
More informationAUTOPIPE NOZZLE. LOCAL STRESS ANALYSIS V8i Release Tutorial with Examples
AUTOPIPE NOZZLE LOCAL STRESS ANALYSIS V8i Release 8.11 Tutorial with Examples 1 Table of Contents List of Examples... 2 Example 1: The basic operating procedures of AutoPIPE Nozzle... 3 Starting AutoPIPE
More informationTheoretical and experimental study on the hydroforming of bifurcation tube
Journal of Materials Processing Technology 142 (2003) 367 373 Theoretical and experimental study on the hydroforming of bifurcation tube Quang-Cherng Hsu Department of Mechanical Engineering, National
More informationMath 265 Exam 3 Solutions
C Roettger, Fall 16 Math 265 Exam 3 Solutions Problem 1 Let D be the region inside the circle r 5 sin θ but outside the cardioid r 2 + sin θ. Find the area of D. Note that r and θ denote polar coordinates.
More informationShallow Spherical Shell Rectangular Finite Element for Analysis of Cross Shaped Shell Roof
Electronic Journal of Structural Engineering, 7(7) Shallow Spherical Shell Rectangular Finite Element for Analsis of Cross Shaped Shell Roof A.I. Mousa Associated Professor of Civil Engineering, The Universit
More informationPre-Calc Unit 14: Polar Assignment Sheet April 27 th to May 7 th 2015
Pre-Calc Unit 14: Polar Assignment Sheet April 27 th to May 7 th 2015 Date Objective/ Topic Assignment Did it Monday Polar Discovery Activity pp. 4-5 April 27 th Tuesday April 28 th Converting between
More informationAttendance Problems. 1. Sketch a right angle and its angle bisector.
Page 1 of 10 ttendance Problems. 1. Sketch a right angle and its angle bisector. 2. Draw three different squares with (3, 2) as one verte. 3. Find the values of and y if (3, 2) = ( + 1, y 3) Vocabulary
More informationROTATIONAL DEPENDENCE OF THE SUPERCONVERGENT PATCH RECOVERY AND ITS REMEDY FOR 4-NODE ISOPARAMETRIC QUADRILATERAL ELEMENTS
COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng, 15, 493±499 (1999) ROTATIONAL DEPENDENCE OF THE SUPERCONVERGENT PATCH RECOVERY AND ITS REMEDY FOR 4-NODE ISOPARAMETRIC QUADRILATERAL
More informationTutorial 1: Welded Frame - Problem Description
Tutorial 1: Welded Frame - Problem Description Introduction In this first tutorial, we will analyse a simple frame: firstly as a welded frame, and secondly as a pin jointed truss. In each case, we will
More informationASME Verification and Validation Symposium May 13-15, 2015 Las Vegas, Nevada. Phillip E. Prueter, P.E.
VVS2015-8015: Comparing Closed-Form Solutions to Computational Methods for Predicting and Validating Stresses at Nozzle-to-Shell Junctions on Pressure Vessels Subjected to Piping Loads ASME Verification
More informationINSTRUCTIONAL PLAN L( 3 ) T ( ) P ( ) Instruction Plan Details: DELHI COLLEGE OF TECHNOLOGY & MANAGEMENT(DCTM), PALWAL
DELHI COLLEGE OF TECHNOLOGY & MANAGEMENT(DCTM), PALWAL INSTRUCTIONAL PLAN RECORD NO.: QF/ACD/009 Revision No.: 00 Name of Faculty: Course Title: Theory of elasticity L( 3 ) T ( ) P ( ) Department: Mechanical
More information17. SEISMIC ANALYSIS MODELING TO SATISFY BUILDING CODES
17. SEISMIC ANALYSIS MODELING TO SATISFY BUILDING CODES The Current Building Codes Use the Terminology: Principal Direction without a Unique Definition 17.1 INTRODUCTION { XE "Building Codes" }Currently
More informationRobotics - Projective Geometry and Camera model. Marcello Restelli
Robotics - Projective Geometr and Camera model Marcello Restelli marcello.restelli@polimi.it Dipartimento di Elettronica, Informazione e Bioingegneria Politecnico di Milano Ma 2013 Inspired from Matteo
More informationEngineering Effects of Boundary Conditions (Fixtures and Temperatures) J.E. Akin, Rice University, Mechanical Engineering
Engineering Effects of Boundary Conditions (Fixtures and Temperatures) J.E. Akin, Rice University, Mechanical Engineering Here SolidWorks stress simulation tutorials will be re-visited to show how they
More informationOn the Kinematics of Undulator Girder Motion
LCLS-TN-09-02 On the Kinematics of Undulator Girder Motion J. Welch March 23, 2009 The theory of rigid body kinematics is used to derive equations that govern the control and measurement of the position
More informationDigging deeper using GeoGebra: An exploration of quadratics and more.
Digging deeper using GeoGebra: An exploration of quadratics and more. Abstract Using GeoGebra students can explore in far more depth topics that have until recently been given a standard treatment. One
More informationFinite Element Analysis of Ellipsoidal Head Pressure Vessel
Finite Element Analysis of Ellipsoidal Head Pressure Vessel Vikram V. Mane*, Vinayak H.Khatawate.**Ashok Patole*** * (Faculty; Mechanical Engineering Department, Vidyavardhini s college of Engineering.
More information1 Points and Distances
Ale Zorn 1 Points and Distances 1. Draw a number line, and plot and label these numbers: 0, 1, 6, 2 2. Plot the following points: (A) (3,1) (B) (2,5) (C) (-1,1) (D) (2,-4) (E) (-3,-3) (F) (0,4) (G) (-2,0)
More informationThe Three Dimensional Coordinate System
The Three-Dimensional Coordinate System The Three Dimensional Coordinate System You can construct a three-dimensional coordinate system by passing a z-axis perpendicular to both the x- and y-axes at the
More information6.7. POLAR COORDINATES
6.7. POLAR COORDINATES What You Should Learn Plot points on the polar coordinate system. Convert points from rectangular to polar form and vice versa. Convert equations from rectangular to polar form and
More informationThermal Deformation Analysis Using Modelica
Thermal Deformation Analsis Using Modelica Eunkeong Kim 1 Tatsurou Yashiki 1 Fumiuki Suzuki 2 Yukinori Katagiri 1 Takua Yoshida 1 1 Hitachi, Ltd., Research & Development Group, Japan, {eunkeong.kim.mn,
More informationApplication of Shell elements to buckling-analyses of thin-walled composite laminates
Application of Shell elements to buckling-analyses of thin-walled composite laminates B.A. Gӧttgens MT 12.02 Internship report Coach: Dr. R. E. Erkmen University of Technology Sydney Department of Civil
More informationA rotation is a transformation that turns a figure around a point, called the.
Name: # Geometr: Period Ms. Pierre Date: Rotations Toda s Objective KWBAT represent a rotation as a function of coordinate pairs and rotate a figure in the plane following a rule described in words or
More informationturn counterclockwise from the positive x-axis. However, we could equally well get to this point by a 3 4 turn clockwise, giving (r, θ) = (1, 3π 2
Math 133 Polar Coordinates Stewart 10.3/I,II Points in polar coordinates. The first and greatest achievement of modern mathematics was Descartes description of geometric objects b numbers, using a sstem
More informationAppendix D Trigonometry
Math 151 c Lynch 1 of 8 Appendix D Trigonometry Definition. Angles can be measure in either degree or radians with one complete revolution 360 or 2 rad. Then Example 1. rad = 180 (a) Convert 3 4 into degrees.
More information3D Finite Element Software for Cracks. Version 3.2. Benchmarks and Validation
3D Finite Element Software for Cracks Version 3.2 Benchmarks and Validation October 217 1965 57 th Court North, Suite 1 Boulder, CO 831 Main: (33) 415-1475 www.questintegrity.com http://www.questintegrity.com/software-products/feacrack
More informationI can identify reflections, rotations, and translations. I can graph transformations in the coordinate plane.
Page! 1 of! 14 Attendance Problems. 1. Sketch a right angle and its angle bisector. 2. Draw three different squares with (3, 2) as one vertex. 3. Find the values of x and y if (3, 2) = (x + 1, y 3) Vocabulary
More informationPosition and Displacement Analysis
Position and Displacement Analysis Introduction: In this chapter we introduce the tools to identifying the position of the different points and links in a given mechanism. Recall that for linkages with
More informationHFAN Rev.1; 04/08
pplication Note: HFN-0.0. Rev.; 04/08 Laser Diode to Single-Mode Fiber Coupling Efficienc: Part - Butt Coupling VILBLE Laser Diode to Single-Mode Fiber Coupling Efficienc: Part - Butt Coupling Introduction
More information3.4 Warm Up. Substitute the given values of m, x, and y into the equation y = mx + b and solve for b. 2. m = 2, x = 3, and y = 0
3.4 Warm Up 1. Find the values of x and y. Substitute the given values of m, x, and y into the equation y = mx + b and solve for b. 2. m = 2, x = 3, and y = 0 3. m = -1, x = 5, and y = -4 3.3 Proofs with
More informationPTC Newsletter January 14th, 2002
PTC Email Newsletter January 14th, 2002 PTC Product Focus: Pro/MECHANICA (Structure) Tip of the Week: Creating and using Rigid Connections Upcoming Events and Training Class Schedules PTC Product Focus:
More informationFinite Element Simulation Models for Mechanics of Materials
Paper ID #18627 Finite Element Simulation Models for Mechanics of Materials Dr. Shahnam Navaee, Georgia Southern University Dr. Navaee is currently a Full Professor in the Civil Engineering and Construction
More informationTo sketch the graph we need to evaluate the parameter t within the given interval to create our x and y values.
Module 10 lesson 6 Parametric Equations. When modeling the path of an object, it is useful to use equations called Parametric equations. Instead of using one equation with two variables, we will use two
More informationMixed Mode Fracture of Through Cracks In Nuclear Reactor Steam Generator Helical Coil Tube
Journal of Materials Science & Surface Engineering Vol. 3 (4), 2015, pp 298-302 Contents lists available at http://www.jmsse.org/ Journal of Materials Science & Surface Engineering Mixed Mode Fracture
More informationLagrange multipliers 14.8
Lagrange multipliers 14.8 14 October 2013 Example: Optimization with constraint. Example: Find the extreme values of f (x, y) = x + 2y on the ellipse 3x 2 + 4y 2 = 3. 3/2 Maximum? 1 1 Minimum? 3/2 Idea:
More informationLagrange multipliers October 2013
Lagrange multipliers 14.8 14 October 2013 Example: Optimization with constraint. Example: Find the extreme values of f (x, y) = x + 2y on the ellipse 3x 2 + 4y 2 = 3. 3/2 1 1 3/2 Example: Optimization
More informationMath : Differentiation
EP-Program - Strisuksa School - Roi-et Math : Differentiation Dr.Wattana Toutip - Department of Mathematics Khon Kaen University 00 :Wattana Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou. Differentiation
More informationMATHEMATICS FOR ENGINEERING TRIGONOMETRY
MATHEMATICS FOR ENGINEERING TRIGONOMETRY TUTORIAL SOME MORE RULES OF TRIGONOMETRY This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves
More informationf xx (x, y) = 6 + 6x f xy (x, y) = 0 f yy (x, y) = y In general, the quantity that we re interested in is
1. Let f(x, y) = 5 + 3x 2 + 3y 2 + 2y 3 + x 3. (a) Final all critical points of f. (b) Use the second derivatives test to classify the critical points you found in (a) as a local maximum, local minimum,
More information9 CARTESIAN SYSTEM OF COORDINATES You must have searched for our seat in a cinema hall, a stadium, or a train. For eample, seat H-4 means the fourth seat in the H th row. In other words, H and 4 are the
More informationIsometry: When the preimage and image are congruent. It is a motion that preserves the size and shape of the image as it is transformed.
Chapter Notes Notes #36: Translations and Smmetr (Sections.1,.) Transformation: A transformation of a geometric figure is a change in its position, shape or size. Preimage: The original figure. Image:
More informationCHAPTER 4 INCREASING SPUR GEAR TOOTH STRENGTH BY PROFILE MODIFICATION
68 CHAPTER 4 INCREASING SPUR GEAR TOOTH STRENGTH BY PROFILE MODIFICATION 4.1 INTRODUCTION There is a demand for the gears with higher load carrying capacity and increased fatigue life. Researchers in the
More information16.6. Parametric Surfaces. Parametric Surfaces. Parametric Surfaces. Vector Calculus. Parametric Surfaces and Their Areas
16 Vector Calculus 16.6 and Their Areas Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. and Their Areas Here we use vector functions to describe more general
More informationNEW WAVE OF CAD SYSTEMS AND ITS APPLICATION IN DESIGN
Vol 4 No 3 NEW WAVE OF CAD SYSTEMS AND ITS APPLICATION IN DESIGN Ass Lecturer Mahmoud A Hassan Al-Qadisiyah University College of Engineering hasaaneng@yahoocom ABSTRACT This paper provides some lighting
More information0618geo. Geometry CCSS Regents Exam
0618geo 1 After a counterclockwise rotation about point X, scalene triangle ABC maps onto RST, as shown in the diagram below. 3 In the diagram below, line m is parallel to line n. Figure 2 is the image
More informationUnderstanding Rotations
Lesson 19 Understanding Rotations 8.G.1.a, 8.G.1.b, 8.G.1.c, 8.G., 8.G.3 1 Getting the idea A rotation is a tpe of transformation in which ou turn a figure about a fied point. The image formed b a rotation
More information11 cm. A rectangular container is 12 cm long, 11 cm wide and 10 cm high. The container is filled with water to a depth of 8 cm.
Diagram NOT accurately drawn 10 cm 11 cm 12 cm 3.5 cm A rectangular container is 12 cm long, 11 cm wide and 10 cm high. The container is filled with water to a depth of 8 cm. A metal sphere of radius 3.5
More informationGoals: Course Unit: Describing Moving Objects Different Ways of Representing Functions Vector-valued Functions, or Parametric Curves
Block #1: Vector-Valued Functions Goals: Course Unit: Describing Moving Objects Different Ways of Representing Functions Vector-valued Functions, or Parametric Curves 1 The Calculus of Moving Objects Problem.
More informationMATH-G Transversal l cuts lines a, b, c, and d. Which two lines are parallel? A a and c B a and d C b and c D b and d
MATH-G 2007 [Exam ID:PZRMS1] Scan Number:3697 1 Transversal l cuts lines a, b, c, and d. Which two lines are parallel? A a and c B a and d C b and c D b and d 2 In the figure above, 2 and 6 are a pair
More informationPolar Functions Polar coordinates
548 Chapter 1 Parametric, Vector, and Polar Functions 1. What ou ll learn about Polar Coordinates Polar Curves Slopes of Polar Curves Areas Enclosed b Polar Curves A Small Polar Galler... and wh Polar
More informationSimulation of Flow Development in a Pipe
Tutorial 4. Simulation of Flow Development in a Pipe Introduction The purpose of this tutorial is to illustrate the setup and solution of a 3D turbulent fluid flow in a pipe. The pipe networks are common
More informationContents Metal Forming and Machining Processes Review of Stress, Linear Strain and Elastic Stress-Strain Relations 3 Classical Theory of Plasticity
Contents 1 Metal Forming and Machining Processes... 1 1.1 Introduction.. 1 1.2 Metal Forming...... 2 1.2.1 Bulk Metal Forming.... 2 1.2.2 Sheet Metal Forming Processes... 17 1.3 Machining.. 23 1.3.1 Turning......
More informationBell Crank. Problem: Joseph Shigley and Charles Mischke. Mechanical Engineering Design 5th ed (New York: McGraw Hill, May 2002) page 87.
Problem: A cast-iron bell-crank lever, depicted in the figure below is acted upon by forces F 1 of 250 lb and F 2 of 333 lb. The section A-A at the central pivot has a curved inner surface with a radius
More informationMathematical Analysis of Spherical Rectangle by H.C. Rajpoot
From the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot Winter February 9, 2015 Mathematical Analysis of Spherical Rectangle by H.C. Rajpoot Harish Chandra Rajpoot Rajpoot, HCR Available at: https://works.bepress.com/harishchandrarajpoot_hcrajpoot/30/
More informationMath (Spring 2009): Lecture 5 Planes. Parametric equations of curves and lines
Math 18.02 (Spring 2009): Lecture 5 Planes. Parametric equations of curves and lines February 12 Reading Material: From Simmons: 17.1 and 17.2. Last time: Square Systems. Word problem. How many solutions?
More informationApplication nr. 2 (Global Analysis) Effects of deformed geometry of the structures. Structural stability of frames. Sway frames and non-sway frames.
Application nr. 2 (Global Analysis) Effects of deformed geometry of the structures. Structural stability of frames. Sway frames and non-sway frames. Object of study: multistorey structure (SAP 2000 Nonlinear)
More informationFind the volume of a solid with regular cross sections whose base is the region between two functions
Area Volume Big Ideas Find the intersection point(s) of the graphs of two functions Find the area between the graph of a function and the x-axis Find the area between the graphs of two functions Find the
More informationThink About. Unit 5 Lesson 3. Investigation. This Situation. Name: a Where do you think the origin of a coordinate system was placed in creating this
Think About This Situation Unit 5 Lesson 3 Investigation 1 Name: Eamine how the sequence of images changes from frame to frame. a Where do ou think the origin of a coordinate sstem was placed in creating
More informationPLAXIS 2D - SUBMERGED CONSTRUCTION OF AN EXCAVATION
PLAXIS 2D - SUBMERGED CONSTRUCTION OF AN EXCAVATION 3 SUBMERGED CONSTRUCTION OF AN EXCAVATION This tutorial illustrates the use of PLAXIS for the analysis of submerged construction of an excavation. Most
More informationUSE OF PMAC S CIRCULAR INTERPOLATION FOR ELLIPSES, SPIRALS AND HELICES:
USE OF PMAC S CIRCULAR INTERPOLATION FOR ELLIPSES, SPIRALS AND HELICES: PMAC allows circular interpolation on the X, Y, and Z aes in a coordinate sstem. As with linear blended moves, TA and TS control
More informationHonors Pre-Calculus. 6.1: Vector Word Problems
Honors Pre-Calculus 6.1: Vector Word Problems 1. A sled on an inclined plane weighs 00 lb, and the plane makes an angle of 0 degrees with the horizontal. What force, perpendicular to the plane, is exerted
More informationPARAMETERIZATIONS OF PLANE CURVES
PARAMETERIZATIONS OF PLANE CURVES Suppose we want to plot the path of a particle moving in a plane. This path looks like a curve, but we cannot plot it like we would plot any other type of curve in the
More informationLearning Module 8 Shape Optimization
Learning Module 8 Shape Optimization What is a Learning Module? Title Page Guide A Learning Module (LM) is a structured, concise, and self-sufficient learning resource. An LM provides the learner with
More informationMATHEMATICS FOR ENGINEERING TUTORIAL 5 COORDINATE SYSTEMS
MATHEMATICS FOR ENGINEERING TUTORIAL 5 COORDINATE SYSTEMS This tutorial is essential pre-requisite material for anyone studying mechanical engineering. This tutorial uses the principle of learning by example.
More informationSeptember 23,
1. In many ruler and compass constructions it is important to know that the perpendicular bisector of a secant to a circle is a diameter of that circle. In particular, as a limiting case, this obtains
More information