Strain Transformation Essays (897 words) - Solid Mechanics

Strain Transformation The following report will be on Strain Transformation. Strain transformation is similar to stress transformation, so that many of the techniques and derivations used for stress can be used for strain. We will also discuss methods of measuring strain and material-property relationships. The general state of strain at a point can be represented by the three components of normal strain, ?x, ?y, ?z, and three components of shear strain, gxy, gxz, gyz. For the purpose of this report, we confine our study to plane strain. That is, we will only concentrate on strain in the x-y plane so that the normal strain is represented by ?x and ?y and the shear strain by gxy . The deformation on an element caused by each of the elements is shown graphically below. Before equations for strain-transformation can be developed, a sign convention must be established. As seen below, ?x and ?y are positive if they cause elongation in the the x and y axes and the shear strain is positive if the interior angle becomes smaller than 90?. For relative axes, the angle between the x and x' axes, q, will be counterclockwise positive. If the normal strains ?x and ?y and the shear strain gxy are known, we can find the normal strain and shear strain at any rotated axes x' and y' where the angle between the x axis and x' axis is q. Using geometry and trigonometric identities the following equations can be derived for finding the strain at a rotated axes: ?x' = (?x + ?y)/2 + (?x - ?y)cos 2q + gxy sin 2q (1) gx'y' = [(?x - ?y)/2] sin 2q + (gxy /2) cos 2q (2) The normal strain in the y' direction by substituting (q + 90?) for q in Eq.1. The orientation of an element can be determined such that the element's deformation at a point can be represented by normal strain with no shear strain. These normal strain are referred to as the principal strains, ?1 and ?2 . The angle between the x and y axes and the principal axes at which these strains occur is represented as qp. The equations for these values can be derived from Eq.1 and are as followed: tan 2qp = gxy /(?x - ?y) (3) ?1,2 = (?x -?y)/2 ? {[(?x -?y)/2]2+ (gxy/2)2 }1/2 (4) The axes along which maximum in-plane shear strain occurs are 45? away from those that define the principal strains and is represented as qs and can be found using the following equation: tan 2qs = -(?x - ?y) / 2 (5) When the shear strain is maximum, the normal strains are equal to the average normal strain. We can also solve strain transformation problem using Mohr's circle. The coordinate system used has the abscissa represent the normal strain ?, with positive to the right and the ordinate represents half of the shear strain g/2 with positive downward. Determine the center of the circle C, which is on the ? axis at a distance of ?avg from the origin. Please note that it is important to follow the sign convention established previously. Plot a reference point A having coordinates (?x , gxy / 2). The line AC is the reference for q = 0. Draw a circle with C as the center and the line AC as the radius. The principal strains ?1 and ?2 are the values where the circle intersects the ? axis and are shown as points B and D on the figure below. The principal angles can be determined from the graph by measuring 2qp1 and 2qp2 from the reference line AC to the ? axis. The element will be elongated in the x' and y' directions as shown below. The average normal strain and the maximum shear strain are shown as points E and F on the figure below. The element will be elongated as shown. To measure the normal strain in a tension-test specimen, an electrical-resistance strain gauge can be used. An electrical-resistance strain gauge works by measuring the change in resistance in a wire or piece of foil and relates that to change in length of the gauge. Since these gauges only work in one direction, normal strains at a point are often determined using a cluster of gauges arranged in a specific pattern, referred to as a strain rosette. Using the readings on the three gauges, the data can be used to determine the state of strain, at that point using geometry and trigonometric identities. It

An Introduction to World War I Turning Point and Periodic Table of Nations

An Introduction to World War I Turning Point and Periodic Table of NationsThe essays on the World War I Turning Point and Periodic Table of Nations from this course are well worth reading. These essays will help you develop a clear understanding of history and the impacts of events, especially those affecting World War I. By the end of the course, you will have an overall feel for World War I and what took place at that time.The Turning Point in World History refers to the event from which we all began. It is defined as the point at which the various elements of the modern world began to grow and change into what they are today. There were many factors which influenced the development of the world as we know it today.World War I saw the birth of the nation-state. This is when individual nations began to take shape and form and one of the first is the modern nation-state. The periodic table of nations is used to chart the different groups and nations that existed at that time. These a re groups which include nations like Japan, Germany, France, Russia, China, Italy, Great Britain, the U.S., Japan, Germany, Poland, and many others.World War I Turning Point and Periodic Table of Nations will help you understand the impact of events that occurred during this time period and how they could impact the future. You will also learn about the same concepts that have been used since then. For example, you will learn about common names and suffixes and the names of places and regions.The lesson plan for this class uses historical data to illustrate several points. It also uses graphs, maps, charts, and more. In addition, you will find numerous supporting materials such as lesson plans, discussions, demonstrations, a glossary, and other aids.What you need to know about World War I Turning Point and Periodic Table of Nations depends on the information that you want to know. Some of the major topics include: the events and activities that happened during World War I, the histo ry of countries, the United States and its allies, and the events which led to the eventual end of the war. Additionally, you will learn about important and influential figures such as Franklin D. Roosevelt, Winston Churchill, Joseph Stalin, and others. Finally, you will learn about geography as it relates to WWI.You will get an overview of the many factors that led to the world as we know it today and how the events of World War I influenced this period. By the end of the class, you will have a firm grasp of the major factors that lead to world events today. If you want to better understand the events of world history, World War I Turning Point and Periodic Table of Nations is the right course for you.The World War I Turning Point and Periodic Table of Nations lesson plan will help you understand the events of World War I and how the world has evolved in the ensuing years. This course will help you better understand how World War I affects our world today. This type of information is necessary for anyone that is studying world history. World War I Turning Point and Periodic Table of Nations is designed to teach you about the world's history and to help you understand the events that took place in this period of history.