- 1. AHSANULLAH UNIVERSITY OF SCIENCE & TECHNOLOGY CE-416 PRE-STRESSED CONCRETE SESSIONAL PRESENTATION ON “AXIS SYSTEM” Submitted by Faria Haider 10.01.03.057
- 2. AXIS SYSTEM An axis is a straight line, sometimes seen, sometimes not, that is important in mathematics, art, science and our survival. The most famous axis is the one the earth spins around, giving us the 24hour day. This task explains how to define a new three-axis system locally. There are two ways of defining it: either by selecting geometry or by entering coordinates. The definition of an axis is a real or imaginary line on which something rotates, or a straight line around which things are evenly arranged. An example of axis is an imaginary line running through the earth on which the earth rotates. An example of axis is the line running through the body from head to feet determining left and right sides.
- 3. CARTESIAN COORDINATE SYSTEM A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, The coordinates 0). can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.
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- 5. Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering, and many more. They are the most common coordinate system used in computer graphics, computeraided geometric design, and other geometryrelated data processing.
- 6. CARTESIAN COORDINATES IN TWO DIMENSIONS The modern Cartesian coordinate system in two dimensions (also called a rectangular coordinate system) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis. (Early systems allowed "oblique" axes, that is, axes that did not meet at right angles.) The lines are commonly referred to as the x and y-axes where the x-axis is taken to be horizontal and the y-axis is taken to be vertical. The point where the axes meet is taken as the origin for both.
- 8. CARTESIAN COORDINATES IN THREE DIMENSIONS Choosing a Cartesian coordinate system for a three-dimensional space means choosing an ordered triplet of lines (axes), any two of them being perpendicular; a single unit of length for all three axes; and an orientation for each axis. As in the two-dimensional case, each axis becomes a number line. The coordinates of a point p are obtained by drawing a line through p perpendicular to each coordinate axis, and reading the points where these lines meet the axes as three numbers of these number lines.
- 10. QUADRANTS AND OCTANTS The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the two coordinates are I (+,+), II (−,+), III (−,−), and IV (+,−). When the axes are drawn according to the mathematical custom, the numbering goes counterclockwise starting from the upper right ("northeast") quadrant. Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or octants, according to the signs of the coordinates of the points. The convention used for naming a specific octant is to list its signs, e.g. (+ + +) or ( - + -). The n-dimensional generalization of the quadrant and octant is the orthant, and the same naming system applies.
- 12. CARTESIAN FORMULAS FOR THE PLANE Distance between two points The Euclidean distance between two points of the plane with Cartesian coordinates and is This is the Cartesian version of Pythagoras' theorem. In threedimensional space, the distance between points and is Which can be obtained by two consecutive applications of Pythagoras’ theorem.
- 13. POLAR COORDINATE SYSTEM In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. The fixed point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the fixed direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the angular coordinate, polar angle.
- 15. CONVERTING BETWEEN POLAR AND CARTESIAN COORDINATES The polar coordinates r and φ can be converted to the Cartesian coordinates x and y by using the trigonometric functions sine and cosine: The Cartesian coordinates x and y can be converted to polar coordinates r and φ with r ≥ 0 and φ in the interval (−π, π] by:
- 17. SPHERICAL COORDINATE SYSTEM In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. The radial distance is also called the radius or radial coordinate. The polar angle may be called colatitude, zenith angle, normal angle, or inclination angle.
- 18. The use of symbols and the order of the coordinates differs between sources. In one system frequently encountered in physics (r, θ, φ) gives the radial distance, polar angle, and azimuthal angle, whereas in another system used in many mathematics books (r, θ, φ) gives the radial distance, azimuthal angle, and polar angle. In both systems ρ is often used instead of r. Other conventions are also used, so great care needs to be taken to check which one is being used.
- 20. CYLINDRICAL COORDINATE SYSTEM A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point. The origin of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis. The axis is variously called the cylindrical or longitudinal axis, to differentiate it from the polar axis, which is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction.
- 22. BIPOLAR CYLINDRICAL COORDINATES The term "bipolar" is often used to describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term bipolar coordinates is never used to describe coordinates associated with those curves, e.g., elliptic coordinates. The classic applications of bipolar coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which bipolar coordinates allow a separation of variables. A typical example would be the electric field surrounding two parallel cylindrical conductors.
- 24. AXIS SYSTEM IN CIVIL ENGINEERING Axis system exists in almost every sphere of universe. It has its massive usage in the field of cosmic system, medical fraternity , music world, data processing & communication system, mathematics & as well as in every engineering sector. Axis system has its wings stretched in every little aspect of civil engineering chapters. As for example, in engineering mechanics, hydraulics, centroid, force system, moment of inertia and so on. The importance of axis system in civil engineering can be demonstrated briefly in following slides.
- 25. IMPORTANCE OF AXIS SYSTEM IN CIVIL ENGINEERING: In a sense, axis system is stitched with the base of civil engineering mechanics. Civil engineers deal with frequent analysis & design which involves every kind of forces, stresses and their projections, which includes the application if axis system. Some basic but very important phases of application of axis system are described in the following slides-
- 26. GLOBAL & LOCAL AXIS SYSTEM: Generally we work with the axis system which has its one axis horizontal to the ground or parallel to building beams which are horizontal to the ground. This simple axis system is referred as the global axis system.
- 27. But when we deal with any inclined member, then difficulty arises in analyzing that inclined member with respect to the global system. In order to decrease the complicacy of analysis, the local axis system has been introduced where one of its axes is parallel to the length of that inclined member.
- 28. MAJOR & MINOR AXIS: A structural member (like beams, columns etc.) has three axes in 3D Cartesian axis system. On the basis of the dimension of the member, there is one major axis and two minor axes. As for example, consider a beam whose horizontal length is the largest dimension, then its major axis (also called longitudinal axis) will be along its horizontal length and other two axes will be referred as minor axes for that particular beam.
- 29. CROSS-SECTION WITH RESPECT TO AXIS: Again, a structural member is sometimes analyzed through its cross-section. It is essential to note that not just any cut-section is called cross-section. In case of beam, we cut the beam vertically and found its cross-section, whereas we will not get the crosssection of a column if we cut it vertically, instead we have to cut it horizontally to get the cross-section of a column.
- 30. So we see, even in getting crosssections, we face the influence of axis system. In order to get a cross-section of a member, we need to cut the member perpendicular to its major axis.
- 31. RELEVANCE OF FORCE WITH AXIS SYSTEM: In analysis of structural member, we must assess the type, magnitude and direction of every kind of forces that act on the member. The line of acting force changes along with the change of major axis. E.g. - For a normal rectangular beam of a building, shear force acts perpendicular to the major axis of the beam i.e. perpendicular to ground. But in case of a column of the same building the shear force does not act perpendicular to ground, instead it acts parallel to ground.
- 32. Because, in case of column, the major axis aligns with its longest dimension (i.e. the height of column). Hence, the shear force acts perpendicular to the major axis of the column which is parallel to the ground. Therefore, the acting line of shear force changes along with the major axis of the structure. Same rule applies when the direction of axial force is to be determined for any structural member. Only in this case, the axial force acts parallel to the major axis of a structural member.
- 33. APPLICATION OF AXIS SYSTEM IN MOMENT OF INERTIA: It is practically impossible to analysis a structural member without estimating its moment of inertia. Moment of inertia is an indicator of the stability of the member. But it is very important to keep in mind that along which axis we are calculating the moment of inertia. Because if we analyze this 2D rectangular member, we find along X-axis it has maximum moment of inertia.
- 34. That means if same amount of moment is produced along both axes, the member shows more resistance along X-axis compared to Yaxis. Thus the member will be difficult to bend along X-axis than Y-axis. Therefore, we can estimate along which axis, moment can be produced to design safe structures.
- 35. MOHR’S CIRCLE Mohr’s circle is one of the important contents which is also defined by axis system. Mohr’s circle is a geometric representation of the 2-D transformation of stresses and is very useful to perform quick and efficient estimations, checks of more extensive work, and other such uses.
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