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Nov. 29, 2005 As I mentioned in some previous articles months ago, there are basically two kinds of stress that figure into the stress tensor, which is used in physics and engineering to model the state of stress at any given point in a mechanical or structural element, or in a natural formation, such as the massif of a mountain. The first kind of stress is called axial stress, and the second kind is called polar or shear stress. This article will be about axial stress and some related ideas. An axial stress is one acting along the length of a member such as a column, a chain supporting a weight, or the connecting rod that transmits motion from wheel to wheel in a locomotive. An axial stress may be either compressive or tensile. An example of a compressive stress could be found in one of the columns that support the entablature in a Greek temple or in the modern counterpart in a high-rise steel structure. An example of a tensile stress could be found in a chain from a crane boom that is carrying a load, or in a structural hanger or suspension element, such as one of the many suspension elements in the Golden Gate Bridge hanging from the main cable down to the roadway. Some elements in machines and buildings may be subject to compression and tension alternately, as a train wheel connecting rod, which first pushes and then pulls as it completes one whole revolution, or bracing in a building which varies in stress with the direction of the wind. Suppose we have a steel element with a cross-sectional area of 10 square inches that may be subjected to either compression or tension. In American usage, a stress would be expressed in pounds or kilopounds, also called kips. Thus if a compressive stress of 50,000 pounds or 50 kips were introduced into the element, we would call that the total stress. Usually, however, stresses are reckoned on a unit basis, so we would say 5000 pounds per square inch or 5 kips per square inch. These are abbreviated to ‘psi’ and ‘ksi’. In the metric system one would use Newtons per square meter, which are also called Pascals. 200 psi equals about 1,379,000 Pascals, or 1.4 megaPascals. Note that stress has the same units as pressure. Really, they’re practically the same thing. In order to determine whether the element is adequate, we would compare the actual stress with the allowable stress, which is dictated by code. Thus for ordinary structural steel, the allowable tensile stress is 22 ksi, so if the actual stress is 5 ksi, we’ve satisfied code easily. The allowable stress includes a safety factor. The yield stress of ordinary steel is 36 ksi, and this is multiplied by .6 to provide a margin of safety, with the resultant 21.6 being rounded to 22. There are grades of steel with yield stresses of 50 and 100 ksi and even more, but the multiplier of .6 remains the same. In the case of compression, the allowable stress varies from 0 to 22 ksi depending on the length, because a member in compression is subject to buckling, and the tendency to buckle increases with the length, whereas a member in tension is not subject to buckling. So a column 20 feet tall might have an allowable compressive stress of only 10 ksi, whereas a very short column, say 2 or 3 feet, would have very nearly 22 ksi. Calculating allowable compressive stresses involves a complicated quartic equation. Therefore, given a length and a load, one may not proceed directly to the required size, but rather must guess a size and see if it is adequate by working the formula, that is, by trial and error, though in the age of computers, this is done instantaneously. A related topic is that of elasticity, strain and deformation. For steel, the modulus of elasticity, also called Young’s modulus, is 29,000 ksi. Of course, steel cannot be stressed nearly that greatly. What the modulus tells us is that if steel is stressed to 29 ksi, for example, which is over the allowable stress, but well under the yield stress, we will have a strain of .001, which is a dimensionless number. In other words: stress / elasticity = strain; 29 / 29,000 = .001. What this means is that the increase in the length of the piece subjected to a stress of 29 ksi will be one-one thousandth of its own length. For example, suppose we have a hanger with a length of 83’-4”, which is 1000 inches, and subject to a stress of 29 ksi, the increase in length, which is called the deformation, will be one inch. The hanger will increase in length to 1001 inches. So we have: strain x length = deformation Although the amount may seem relatively small, it is important. For instance, in a 100-story building, like the John Hancock Tower in Chicago, the major columns are those that are centrally located in the building. They are called core columns, rising the whole height of the building, and are stressed significantly more greatly than the peripheral columns which merely go from floor to floor, almost like wall studs. The result is that unless strain is accurately calculated, when construction reaches the upper floors, the beams will all be sloping downward towards the center of the building a couple of inches, the 1000-foot-long columns all having contracted. In the foregoing discussion of stress and strain, it has been assumed that the stress-strain relationship is proportional, but in actuality it is proportional only in the so-called elastic range. In this range, it is understood that the steel is not stressed so greatly that it will not resume its original shape when released from the stress. Then the deformation is called elastic, but when the steel undergoes a permanent deformation, it is called plastic. A simple example would be a wire coat-hanger. If you suspend maybe one pound in the middle of the horizontal piece, it will deflect a little. When you remove the weight, it will spring back. That is an elastic deformation. But if you hang 20 pounds, it will bend permanently. That is a plastic deformation. Note, however, that in the area of the bend, the steel actually becomes a little harder than it was originally. That phenomenon is called strain-hardening. The explanations given above all concern steel or other material acting in the elastic range. Most design is so-called elastic design, but there are structures that have been engineered on the basis of plastic design, a somewhat more complicated procedure that takes advantage of strain-hardening to get more strength out of the same steel. ------------ About the author Thomas Keyes: I have written two books: A SOJOURN IN ASIA (non-fiction) and A TALE OF UNG (fiction), neither published so far. I have studied languages for years and traveled extensively on five continents. Email: udikeyes@yahoo.com Tell a friend about this site! ------------ All articles are EXCLUSIVE to Useless-Knowledge.com and are not allowed to be posted on other websites. ARTICLE THIEVES WILL BE PROSECUTED! |
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