This document describes methods of assessing the potential for wedge sliding instability, using stereonet kinematic analysis.


 Grace Blankenship
 2 years ago
 Views:
Transcription
1 Wedge Failure Kinematic analysis on a stereonet This document describes methods of assessing the potential for wedge sliding instability, using stereonet kinematic analysis. 1. The traditional approach has been to consider the orientation of the intersection line of two planes. 2. An alternative approach considers the normal to the intersection line (wedge pole method). Wedge versus Planar sliding Kinematic wedge sliding and planar sliding analyses using a stereonet are fundamentally very similar. Planar sliding is sometimes considered a special case of wedge sliding, or vice versa. One difference between planar and wedge sliding analyses, is that for wedge sliding analysis, it is not necessary to consider lateral kinematic limits on the sliding direction (i.e. the plus/minus 20 degree limits), as is done with pure planar sliding analysis. In general, as long as the intersection line of a wedge satisfies the daylighting condition, it is not necessary to impose additional lateral restrictions on the sliding direction. Because the joint planes act as release surfaces, it is always possible to remove a wedge with a daylighting intersection line. (In practice, wedges may not be easily removable, e.g. thin wedges can wedge themselves in place, but we do not consider those issues here). Wedges do not always slide along the line of intersection. Depending on the wedge geometry, they may slide on only one of the joint planes. For example, a wedge line of intersection can be nearly parallel to the strike of the slope, and the wedge can still slide out of the slope on one joint plane). This might be called planar wedge sliding, when a wedge slides along one joint plane, rather than the line of intersection. For the purpose of stereonet analysis of wedge sliding, it is assumed: both sliding planes of a wedge have the same friction angle. cohesion of the joint planes equals zero (same as the planar analysis assumption) This friction only analysis assumption allows a rough safety factor to be calculated (tan theta / tan phi).
2 Intersection Line Method The intersection of two arbitrary planes forms a line in 3dimensional space. Kinematic analysis of wedge stability is simply based on comparing the orientation of the line of intersection of two planes with respect to the slope plane orientation. In order for a wedge to slide, the line of intersection must daylight in the slope face. The most common check for wedge sliding is illustrated in the following figure. If a wedge intersection line falls within the crescent shaped zone defined by the slope plane (great circle) and the friction circle, then wedge sliding can occur. Figure 1: critical wedge sliding zone (slope plane = 65/180, friction angle = 30) Wedge intersection lines which fall within the region indicated in Figure 1, represent daylighting wedges which can slide (i.e. the wedge intersection line dips out of the slope, and more steeply than the friction angle). Because we are analyzing actual sliding directions (rather than poles), the friction circle is defined with respect to the perimeter of the stereonet. This is defined by a cone (small circle) with a vertical axis, and angular radius equal to 90 minus the friction angle. In the above example, the friction angle = 30, so the cone angle = = 60 degrees. The orientation of the slope plane in the above example is 65/180.
3 Wedge Sliding Modes The wedge intersection line must always daylight in order for wedge sliding to be possible, however sliding does not necessarily occur along the line of intersection. For the critical zone shown in Figure 1, there are two possible wedge sliding modes: Sliding along the line of intersection (i.e. sliding on both planes simultaneously) Sliding along one wedge plane The wedge sliding mode can be determined on a stereonet according to Hocking (1976). Assuming an intersection line which daylights in the critical zone shown in Figure 1: If the dip vector of one of the joint planes lies between the slope dip direction, and the line of intersection trend, then the wedge will slide on one plane If the dip vectors of both planes lie outside of this region, then sliding will be along the line of intersection. This is illustrated in the following figures.
4 This could be offered as an optional analysis feature (i.e. show the wedge sliding mode on the stereonet). Another case should be considered: it is possible to have wedges where the line of intersection dips LESS THAN the friction angle, but sliding can still take place on one plane, if the dip vector of the plane falls within the critical zone of Figure 1. This is shown in the following figure. If the wedge intersection line falls in either of the yellow regions, AND The dip vector of one plane falls in the red region. Then wedge sliding can occur on the one plane. Basically, these are wedges where the intersection line has a shallow dip, and strikes subparallel to the slope. Figure 2: critical zones for wedge sliding on one plane This wedge failure mode is not usually considered for stereonet analysis (probably due to the difficulties involved in graphically evaluating these wedges). However, it is a valid case, and was noticed while viewing the output from Swedge. This is shown in the following figure, a Combination Analysis with output displayed on the stereonet, and failed wedge intersections highlighted in red. Notice that the extent of the failed wedge intersections corresponds (approximately) to the sum of the yellow and red areas in the above figure, and is larger than the traditional crescent shape considered for wedge sliding on a stereonet. (The output does not correspond exactly, because the actual distribution of poles in the original dataset is not uniformly distributed on the stereonet. If
5 a larger number of uniformly distributed poles were analyzed, the regions would correspond closely.) Figure 3: Swedge combination analysis, stereonet output, failed wedge intersections occupy crescent shaped region (note: failed poles are also highlighted in red). Notice that the crescent shaped region of failed intersections in Figure 3, corresponds to the total crescent shaped region of Figure 2 (red + yellow zones). Failed intersections corresponding to the red zone of Figure 2 may represent sliding on one plane or two planes. Failed intersections corresponding to the yellow regions of Figure 2 can only represent sliding on one plane, because the line of intersection dips less than the friction angle. Note the following special cases: If the dip direction of the intersection line = slope dip direction, then sliding will occur along the line of intersection.
6 Conversely, if the dip vector of one plane has the same dip direction as the slope, then sliding will take place on this plane. In the special case you could have the same dip direction for the intersection line, one plane and the slope, and sliding would be in the common direction of all three. The intersection point could be coincident with the dip vector of one plane, in which case the wedge would slide along the line of intersection and the dip vector of the plane simultaneously.
7 Wedge Pole method of stereonet wedge stability An alternative method of evaluating wedge sliding using stereonets, has been proposed. Rather than plotting the wedge intersection lines, normals to the wedge intersection lines are plotted. This has been called the wedge pole method. The procedure is described in detail in a discussion paper by Laurie Richards (ref). One of the main goals of this alternative method, is to allow the assessment of planar and wedge stability, simultaneously, using the same constructions on the stereonet (i.e. use the same daylight envelope and pole friction cone, and plot poles to planes and wedge poles). Figure : Wedge Pole construction for wedge stability (Note: in Laurie Richards paper, he does not mention the plus/minus strike limits for planar sliding, and implies that planar and wedge sliding both use the full daylight envelope (this is also implied in the original Dips manual). However, the planar strike limits are important, and constitute a practical difference between wedge and planar sliding, which lessens the value of trying to use the same templates for planar and wedge sliding.)
8 Figure : Contouring of wedge poles However, the advantages of the wedge pole method are not clear, when compared to the dip vector method of planar sliding and the intersection line method of wedge sliding. These are also complementary methods, which use the same constructions (slope plane great circle, and plane friction cone measured from the stereonet perimeter). The wedge pole method requires that normals to the wedge intersection lines are generated, a procedure which is not very intuitive and requires an additional step of work. Furthermore, many practitioners and students find the dip vector method of stereonet kinematic analysis, much more intuitive and easier to understand than the pole plot methods. In view of these considerations, the value of the wedge pole method is not clear, although it certainly is a valid alternative.
9 Methods of determining wedge intersection lines We should note that there are various ways of generating the wedge intersection lines. 1. One common method is to first determine the distinct joint sets in your data (usually determined from a contour plot). Then determine the mean orientation plane for each joint set. Then determine all of the possible intersections of the mean planes, and check to see if any of these intersections fall in the critical zone. This is the method presented in the original Dips manual example of wedge sliding analysis. The obvious drawback of this method is that only mean joint planes are considered. This will highlight any obvious wedge stability problem, but does not consider the many individual wedges which can be formed by all possible joint combinations. Figure x: Intersections determined from mean (or major) planes only. 2. Other methods attempt to bracket the range of orientations represented by each joint set, by estimating an upper and lower bound for each set. The intersections of these bounding planes then form curved rectangular regions, which can be checked against the critical wedge sliding zone. A rather inefficient and clumsy attempt to improve upon the mean plane method. This method is suitable as a graphical manual method.
10 Figure x: joint sets bracketed by bounding planes, defines a region of possible intersections. 3. The most comprehensive check, is to generate all possible combinations of two joint planes, and plot all resulting intersection lines on the stereonet. With current computing power, this is now easily done, and allows every possible wedge orientation to be checked. This is available as the Combination Analysis option in Swedge. Other Notes Plotting of upper face plane on stereonet (reduces the number of valid daylighting intersection lines if the upper face has a nonzero dip angle).
Swedge. Verification Manual. Probabilistic analysis of the geometry and stability of surface wedges. 19912013 Rocscience Inc.
Swedge Probabilistic analysis of the geometry and stability of surface wedges Verification Manual 1991213 Rocscience Inc. Table of Contents Introduction... 3 1 Swedge Verification Problem #1... 4 2 Swedge
More informationChapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis
Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >
More informationGroundwater Analysis Overview
Finite Element Groundwater Seepage (Overview) 71 Groundwater Analysis Overview Introduction Within the Slide program, Slide has the capability to carry out a finite element groundwater seepage analysis
More informationArrangements And Duality
Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,
More informationStereographic projections
Stereographic projections 1. Introduction The stereographic projection is a projection of points from the surface of a sphere on to its equatorial plane. The projection is defined as shown in Fig. 1. If
More informationLab 3: Stereonets. Fall 2005
Lab 3: Stereonets Fall 2005 1 Introduction In structural geology it is important to determine the orientations of planes and lines and their intersections. Working out these relationships as we have in
More informationUnderstanding Poles and Zeros
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function
More informationReflection and Refraction
Equipment Reflection and Refraction Acrylic block set, planeconcaveconvex universal mirror, cork board, cork board stand, pins, flashlight, protractor, ruler, mirror worksheet, rectangular block worksheet,
More informationFoundation Engineering Prof. Mahendra Singh Department of Civil Engineering Indian Institute of Technology, Roorkee
Foundation Engineering Prof. Mahendra Singh Department of Civil Engineering Indian Institute of Technology, Roorkee Module  03 Lecture  09 Stability of Slopes Welcome back to the classes of on this Stability
More informationArea in Polar Coordinates
Area in Polar Coordinates If we have a circle of radius r, and select a sector of angle θ, then the area of that sector can be shown to be 1. r θ Area = (1/)r θ As a check, we see that if θ =, then the
More informationBack Analysis of Material Properties
Back Analysis of Material Properties 231 Back Analysis of Material Properties This tutorial will demonstrate how to perform back analysis of material properties using sensitivity analysis or probabilistic
More informationMap Patterns and Finding the Strike and Dip from a Mapped Outcrop of a Planar Surface
Map Patterns and Finding the Strike and Dip from a Mapped Outcrop of a Planar Surface Topographic maps represent the complex curves of earth s surface with contour lines that represent the intersection
More informationStructural Geology. Practical 1. Introduction to Stereographic Projection
Structural Geology Practical 1 Introduction to Stereographic Projection Lecture Practical Course Homepage Contact Staff 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 71 8 9 10 STEREONETS 1 INTRODUCTION TO STEREOGRAPHIC
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationChapter 12. The Straight Line
302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,
More informationChapter 14: Production Possibility Frontiers
Chapter 14: Production Possibility Frontiers 14.1: Introduction In chapter 8 we considered the allocation of a given amount of goods in society. We saw that the final allocation depends upon the initial
More informationProbabilistic Analysis
Probabilistic Analysis Tutorial 81 Probabilistic Analysis This tutorial will familiarize the user with the basic probabilistic analysis capabilities of Slide. It will demonstrate how quickly and easily
More informationTWODIMENSIONAL TRANSFORMATION
CHAPTER 2 TWODIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization
More informationIntroduction to the Smith Chart for the MSA Sam Wetterlin 10/12/09 Z +
Introduction to the Smith Chart for the MSA Sam Wetterlin 10/12/09 Quick Review of Reflection Coefficient The Smith chart is a method of graphing reflection coefficients and impedance, and is often useful
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationSurface Area of Rectangular & Right Prisms Surface Area of Pyramids. Geometry
Surface Area of Rectangular & Right Prisms Surface Area of Pyramids Geometry Finding the surface area of a prism A prism is a rectangular solid with two congruent faces, called bases, that lie in parallel
More informationAlgebra II: Strand 7. Conic Sections; Topic 1. Intersection of a Plane and a Cone; Task 7.1.2
1 TASK 7.1.2: THE CONE AND THE INTERSECTING PLANE Solutions 1. What is the equation of a cone in the 3dimensional coordinate system? x 2 + y 2 = z 2 2. Describe the different ways that a plane could intersect
More informationLecture L222D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L  D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L3 for
More informationThis function is symmetric with respect to the yaxis, so I will let  /2 /2 and multiply the area by 2.
INTEGRATION IN POLAR COORDINATES One of the main reasons why we study polar coordinates is to help us to find the area of a region that cannot easily be integrated in terms of x. In this set of notes,
More informationMathematics 1. Lecture 5. Pattarawit Polpinit
Mathematics 1 Lecture 5 Pattarawit Polpinit Lecture Objective At the end of the lesson, the student is expected to be able to: familiarize with the use of Cartesian Coordinate System. determine the distance
More informationStraight Line motion with rigid sets
Straight ine motion with rigid sets arxiv:40.4743v [math.mg] 9 Jan 04 Robert Connelly and uis Montejano January, 04 Abstract If one is given a rigid triangle in the plane or space, we show that the only
More informationInteractive Math Glossary Terms and Definitions
Terms and Definitions Absolute Value the magnitude of a number, or the distance from 0 on a real number line Additive Property of Area the process of finding an the area of a shape by totaling the areas
More informationSolidWorks Implementation Guides. Sketching Concepts
SolidWorks Implementation Guides Sketching Concepts Sketching in SolidWorks is the basis for creating features. Features are the basis for creating parts, which can be put together into assemblies. Sketch
More informationThe NotFormula Book for C1
Not The NotFormula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationCHAPTER 35 GRAPHICAL SOLUTION OF EQUATIONS
CHAPTER 35 GRAPHICAL SOLUTION OF EQUATIONS EXERCISE 143 Page 369 1. Solve the simultaneous equations graphically: y = 3x 2 y = x + 6 Since both equations represent straightline graphs, only two coordinates
More informationFall 12 PHY 122 Homework Solutions #8
Fall 12 PHY 122 Homework Solutions #8 Chapter 27 Problem 22 An electron moves with velocity v= (7.0i  6.0j)10 4 m/s in a magnetic field B= (0.80i + 0.60j)T. Determine the magnitude and direction of the
More informationThe calibration problem was discussed in details during lecture 3.
1 2 The calibration problem was discussed in details during lecture 3. 3 Once the camera is calibrated (intrinsics are known) and the transformation from the world reference system to the camera reference
More informationARE211, Fall2012. Contents. 2. Linear Algebra Preliminary: Level Sets, upper and lower contour sets and Gradient vectors 1
ARE11, Fall1 LINALGEBRA1: THU, SEP 13, 1 PRINTED: SEPTEMBER 19, 1 (LEC# 7) Contents. Linear Algebra 1.1. Preliminary: Level Sets, upper and lower contour sets and Gradient vectors 1.. Vectors as arrows.
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More information12.510 Introduction to Seismology Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 04/30/2008 Today s
More informationPure Math 30: Explained!
www.puremath30.com 45 Conic Sections: There are 4 main conic sections: circle, ellipse, parabola, and hyperbola. It is possible to create each of these shapes by passing a plane through a three dimensional
More informationPore pressure. Ordinary space
Fault Mechanics Laboratory Pore pressure scale Lowers normal stress, moves stress circle to left Doesn Doesn t change shear Deviatoric stress not affected This example: failure will be by tensile cracks
More informationTHE TRANSITION FROM OPEN PIT TO UNDERGROUND MINING: AN UNUSUAL SLOPE FAILURE MECHANISM AT PALABORA
THE TRANSITION FROM OPEN PIT TO UNDERGROUND MINING: AN UNUSUAL SLOPE FAILURE MECHANISM AT PALABORA Richard K. Brummer*, Hao Li* & Allan Moss *Itasca Consulting Canada Inc., Rio Tinto Limited ABSTRACT At
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationCopyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass
Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of
More information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which the
More informationINSTANTANEOUS CENTER (IC) OF ZERO VELOCITY (Section 16.6) Today s Objectives: Students will be able to: a) Locate the instantaneous center (IC) of
INSTANTANEOUS CENTER (IC) OF ZERO VELOCITY (Section 16.6) Today s Objectives: Students will be able to: a) Locate the instantaneous center (IC) of zero velocity. b) Use the IC to determine the velocity
More informationComputer Numerical Control
Training Objective After watching the video and reviewing this printed material, the viewer will gain knowledge and understanding of the basic theory and use of computer numerical control, or CNC, in manufacturing.
More informationDigital Image Processing. Prof. P.K. Biswas. Department of Electronics & Electrical Communication Engineering
Digital Image Processing Prof. P.K. Biswas Department of Electronics & Electrical Communication Engineering Indian Institute of Technology, Kharagpur Lecture  27 Colour Image Processing II Hello, welcome
More informationUNIVERSITY OF CALIFORNIA AT BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences. EE105 Lab Experiments
UNIVERSITY OF CALIFORNIA AT BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences EE15 Lab Experiments Bode Plot Tutorial Contents 1 Introduction 1 2 Bode Plots Basics
More informationComputer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2D
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2D Welcome everybody. We continue the discussion on 2D
More informationangle Definition and illustration (if applicable): a figure formed by two rays called sides having a common endpoint called the vertex
angle a figure formed by two rays called sides having a common endpoint called the vertex area the number of square units needed to cover a surface array a set of objects or numbers arranged in rows and
More informationMaths for Computer Graphics
Analytic Geometry Review of geometry Euclid laid the foundations of geometry that have been taught in schools for centuries. In the last century, mathematicians such as Bernhard Riemann (1809 1900) and
More information5. Möbius Transformations
5. Möbius Transformations 5.1. The linear transformation and the inversion. In this section we investigate the Möbius transformation which provides very convenient methods of finding a onetoone mapping
More informationSection 2.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates
Section.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O,the rotating ray or half line from O with unit tick. A point P in
More informationaxis axis. axis. at point.
Chapter 5 Tangent Lines Sometimes, a concept can make a lot of sense to us visually, but when we try to do some explicit calculations we are quickly humbled We are going to illustrate this sort of thing
More information8 th grade mathematics Team: T. Kisker, S. Miller, B. Ricks, B. Byland April 2012
Compare and order all rational numbers including percents and find their approximate location on a number line. N.1.A.8 Number and Operations Order positive rational numbers on a number line 1, 2, 5,
More informationLecture L6  Intrinsic Coordinates
S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6  Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed
More informationChapter 23. The Reflection of Light: Mirrors
Chapter 23 The Reflection of Light: Mirrors Wave Fronts and Rays Defining wave fronts and rays. Consider a sound wave since it is easier to visualize. Shown is a hemispherical view of a sound wave emitted
More informationElementary triangle geometry
Elementary triangle geometry Dennis Westra March 26, 2010 bstract In this short note we discuss some fundamental properties of triangles up to the construction of the Euler line. ontents ngle bisectors
More informationChapter 5: Working with contours
Introduction Contoured topographic maps contain a vast amount of information about the threedimensional geometry of the land surface and the purpose of this chapter is to consider some of the ways in
More informationAwellknown lecture demonstration1
Acceleration of a Pulled Spool Carl E. Mungan, Physics Department, U.S. Naval Academy, Annapolis, MD 40506; mungan@usna.edu Awellknown lecture demonstration consists of pulling a spool by the free end
More informationSection 2.1 Rectangular Coordinate Systems
P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is
More informationGraphical Representation of Multivariate Data
Graphical Representation of Multivariate Data One difficulty with multivariate data is their visualization, in particular when p > 3. At the very least, we can construct pairwise scatter plots of variables.
More informationMaterials & Loading Tutorial
Materials & Loading Tutorial 21 Materials & Loading Tutorial This tutorial will demonstrate how to model a more complex multimaterial slope, with both pore water pressure and an external load. MODEL FEATURES:
More informationData Envelopment Analysis: A Primer for Novice Users and Students at all Levels
Data Envelopment Analysis: A Primer for Novice Users and Students at all Levels R. Samuel Sale Lamar University Martha Lair Sale Florida Institute of Technology In the three decades since the publication
More information1051232 Imaging Systems Laboratory II. Laboratory 4: Basic Lens Design in OSLO April 2 & 4, 2002
05232 Imaging Systems Laboratory II Laboratory 4: Basic Lens Design in OSLO April 2 & 4, 2002 Abstract: For designing the optics of an imaging system, one of the main types of tools used today is optical
More informationCalculating Areas Section 6.1
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Calculating Areas Section 6.1 Dr. John Ehrke Department of Mathematics Fall 2012 Measuring Area By Slicing We first defined
More informationChapter 6 Work and Energy
Chapter 6 WORK AND ENERGY PREVIEW Work is the scalar product of the force acting on an object and the displacement through which it acts. When work is done on or by a system, the energy of that system
More informationMagnetic Field of a Circular Coil Lab 12
HB 112607 Magnetic Field of a Circular Coil Lab 12 1 Magnetic Field of a Circular Coil Lab 12 Equipment coil apparatus, BK Precision 2120B oscilloscope, Fluke multimeter, Wavetek FG3C function generator,
More informationEngineering Geometry
Engineering Geometry Objectives Describe the importance of engineering geometry in design process. Describe coordinate geometry and coordinate systems and apply them to CAD. Review the righthand rule.
More informationMath 215 HW #1 Solutions
Math 25 HW # Solutions. Problem.2.3. Describe the intersection of the three planes u+v+w+z = 6 and u+w+z = 4 and u + w = 2 (all in fourdimensional space). Is it a line or a point or an empty set? What
More information3 Drawing 2D shapes. Launch form Z.
3 Drawing 2D shapes Launch form Z. If you have followed our instructions to this point, five icons will be displayed in the upper left corner of your screen. You can tear the three shown below off, to
More informationArea and Arc Length in Polar Coordinates
Area and Arc Length in Polar Coordinates The Cartesian Coordinate System (rectangular coordinates) is not always the most convenient way to describe points, or relations in the plane. There are certainly
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More information10.5. Click here for answers. Click here for solutions. EQUATIONS OF LINES AND PLANES. 3x 4y 6z 9 4, 2, 5. x y z. z 2. x 2. y 1.
SECTION EQUATIONS OF LINES AND PLANES 1 EQUATIONS OF LINES AND PLANES A Click here for answers. S Click here for solutions. 1 Find a vector equation and parametric equations for the line passing through
More informationAxisymmetry Tutorial. A few representations of simple axisymmetric models are shown below. Axisymmetry Tutorial 61 0, 18 28, MPa 0, 12.
Axisymmetry Tutorial 61 Axisymmetry Tutorial 0, 18 28, 18 0, 12 10 MPa 10 MPa 0, 6 0, 0 4, 0 x = 0 (axis of symmetry) userdefined external boundary 4, 24 12, 24 20, 24 28, 24 This tutorial will illustrate
More informationWe call this set an ndimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.
Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to
More informationCalypso Construction Features
Calypso The Construction dropdown menu contains several useful construction features that can be used to compare two other features or perform special calculations. Construction features will show up
More informationCBE 6333, R. Levicky 1. Potential Flow
CBE 6333, R. Levicky Part I. Theoretical Background. Potential Flow Potential Flow. Potential flow is irrotational flow. Irrotational flows are often characterized by negligible viscosity effects. Viscous
More informationPHY121 #8 Midterm I 3.06.2013
PHY11 #8 Midterm I 3.06.013 AP Physics Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension
More informationUsing The TINspire Calculator in AP Calculus
Using The TINspire Calculator in AP Calculus (Version 3.0) You must be able to perform the following procedures on your calculator: 1. Plot the graph of a function within an arbitrary viewing window,
More informationCabri Geometry Application User Guide
Cabri Geometry Application User Guide Preview of Geometry... 2 Learning the Basics... 3 Managing File Operations... 12 Setting Application Preferences... 14 Selecting and Moving Objects... 17 Deleting
More informationCHAPTER 1 Linear Equations
CHAPTER 1 Linear Equations 1.1. Lines The rectangular coordinate system is also called the Cartesian plane. It is formed by two real number lines, the horizontal axis or xaxis, and the vertical axis or
More informationApplied Geomorphology. Lecture 4: Total Station & GPS Survey Methods
Applied Geomorphology Lecture 4: Total Station & GPS Survey Methods Total Station Electronic version of Alidade Accurate to ±3 ppm horizontal & vertical 3x106 (5000 feet) = 0.2 inches Total Station Advantages
More informationAjit Kumar Patra (Autor) Crystal structure, anisotropy and spin reorientation transition of highly coercive, epitaxial PrCo films
Ajit Kumar Patra (Autor) Crystal structure, anisotropy and spin reorientation transition of highly coercive, epitaxial PrCo films https://cuvillier.de/de/shop/publications/1306 Copyright: Cuvillier Verlag,
More informationStability Of Structures: Basic Concepts
23 Stability Of Structures: Basic Concepts ASEN 3112 Lecture 23 Slide 1 Objective This Lecture (1) presents basic concepts & terminology on structural stability (2) describes conceptual procedures for
More informationarxiv: v1 [math.mg] 6 May 2014
DEFINING RELATIONS FOR REFLECTIONS. I OLEG VIRO arxiv:1405.1460v1 [math.mg] 6 May 2014 Stony Brook University, NY, USA; PDMI, St. Petersburg, Russia Abstract. An idea to present a classical Lie group of
More informationChapter 4: Rock Slope Stability Analysis: Limit Equilibrium Method
Chapter 4: Rock Slope Stability Analysis: Limit Equilibrium Method 1. Plane failure analysis 2. Wedge failure analysis 3. Toppling failure analysis 4.1 Planar Failure Analysis Planar failure of rock slope
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Firms that survive in the long run are usually those that A) remain small. B) strive for the largest
More informationCHAPTER 22 COSTVOLUMEPROFIT ANALYSIS
CHAPTER 22 COSTVOLUMEPROFIT ANALYSIS Related Assignment Materials Student Learning Objectives Conceptual objectives: C1. Describe different types of cost behavior in relation to production and sales
More informationPhysics: Principles and Applications, 6e Giancoli Chapter 2 Describing Motion: Kinematics in One Dimension
Physics: Principles and Applications, 6e Giancoli Chapter 2 Describing Motion: Kinematics in One Dimension Conceptual Questions 1) Suppose that an object travels from one point in space to another. Make
More informationDMU Space Analysis. Preface What's New Getting Started Basic Tasks Workbench Description Customizing Glossary Index
DMU Space Analysis Preface What's New Getting Started Basic Tasks Workbench Description Customizing Glossary Index Dassault Systèmes 199499. All rights reserved. Preface DMU Space Analysis is a CADindependent
More informationPOLAR COORDINATES DEFINITION OF POLAR COORDINATES
POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand
More informationExploring Spherical Geometry
Exploring Spherical Geometry Introduction The study of plane Euclidean geometry usually begins with segments and lines. In this investigation, you will explore analogous objects on the surface of a sphere,
More informationCentroid: The point of intersection of the three medians of a triangle. Centroid
Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:
More informationGauss's Law. Gauss's Law in 3, 2, and 1 Dimension
[ Assignment View ] [ Eðlisfræði 2, vor 2007 22. Gauss' Law Assignment is due at 2:00am on Wednesday, January 31, 2007 Credit for problems submitted late will decrease to 0% after the deadline has passed.
More informationREVISED GCSE Scheme of Work Mathematics Higher Unit 6. For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012
REVISED GCSE Scheme of Work Mathematics Higher Unit 6 For First Teaching September 2010 For First Examination Summer 2011 This Unit Summer 2012 Version 1: 28 April 10 Version 1: 28 April 10 Unit T6 Unit
More informationEQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS
EQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS Today s Objectives: Students will be able to: 1. Analyze the planar kinetics of a rigid body undergoing rotational motion. InClass Activities: Applications
More informationGraphical Presentation of Data
Graphical Presentation of Data Guidelines for Making Graphs Titles should tell the reader exactly what is graphed Remove stray lines, legends, points, and any other unintended additions by the computer
More informationLecture 6 : Aircraft orientation in 3 dimensions
Lecture 6 : Aircraft orientation in 3 dimensions Or describing simultaneous roll, pitch and yaw 1.0 Flight Dynamics Model For flight dynamics & control, the reference frame is aligned with the aircraft
More information( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those
1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make
More informationLecture L5  Other Coordinate Systems
S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5  Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates
More informationConstructing Möbius Transformations with Spheres
Rose Hulman Undergraduate Mathematics Journal Constructing Möbius Transformations with pheres Rob iliciano a Volume 13, No. 2, Fall 2012 ponsored by RoseHulman Institute of Technology Department of Mathematics
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More information