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July 29, 2005 Anyone who has driven down an interstate or other major highway in the US cannot have failed to notice and appreciate the smoothness of the pavement, the regularity of transitions, the presence of banks in just the right places along curves, and the neat merges and ramps. There are no angular turns, no bumps, no crooked roads or awkward stretches. Everything is beautiful and orderly. One of the most important factors in bringing this about is the method of highway geometry used in the surveying and layout of roads being built, which ordinarily include bridges, and that is what I worked on. Let us suppose we have a section of road several miles long on hilly land and crossing three or four streams or rivers. This could conceivably be a straight line, but more often than not will involve one or more horizontal curves, that is, the kind of curves where you turn left or right. These are usually arcs of circles, but occasionally an arc of a spiral will be used. A spiral is a plane curve of increasing radius. What sometimes is known popularly as a spiral, as, for example, in a spiral notebook or spiral staircase, is in reality a helix. Circles are preferred to spirals in most cases, because they are easier. The first thing that is done is to set up the ‘profile grade line’ (PGL). This is a single line, straight or curved, along the length of the highway, sometimes but not always on the center of the highway. At every hundred feet from some starting point, like a state line, a benchmark is placed. If a given point is 161,300 feet from the starting point, the benchmark is labelled ‘Sta. 1613+00’. The next bench mark will be ‘Sta. 1614+00’, etc. Locally, within a given 100 foot section, one may just say ‘Station 80’, which will mean ‘Sta. 1613+80.’ If the road is curved, then the stations are measured along the arc of the curve rather than along chords. Usually, two-decimal-place accuracy is what is sought, so a particular point may be referred to as ‘Sta. 1613+81.55’, and this signifies that the point is 161,381.55 feet from the starting point. A hundredth of a foot is about an eighth of an inch. Any point not on the PGL, then, is either right or left of it, and will be noted thus, ‘Sta.+1613+81.55, R 19.17’, with ‘R’ for ‘right’ and ‘L’ for ‘left’, as you look ahead, that is, towards the next higher station. A highway may also have a vertical curve. This is usually a series of straight lines and parabolic arcs tangent to them. Parabolas are used instead of circles, because, in this case, they are easier to calculate, but the amount of curvature is usually so slight that for all practical purposes a circular arc and a parabolic arc are identical. At every point along the PGL, we have an elevation, usually from sea level. Thus we may say that at ‘Sta.+1613+81.55, R 19.17’, the surface of the road is at ‘El.+654.19’, which means 654.19 feet above sea level. At each point there is also a grade, like +5% or -3%. Thus the road rises 5 feet or falls 3 feet in 100 feet as you go upstation. On a parabolic curve, the grade changes instantaneously. In we have two sections of road with constant but different grades, as +5% and -2%, and terminating, at points at a given station-to-station distance from each other, at fixed elevations, as El.+671.19 and El.+685.19, we have too many known quantities to fit an ordinary parabola tangent to both grades at its ends, so a number of transitions of vertical curves and grades may be necessary to maintain a continuous, smooth drivable surface. Once the vertical profile has been worked out, we have produced a station equation. We have a mathematical formula that will give us the elevation at the top of the pavement along the profile grade line, at any point in the length of road under consideration. The surface of a highway is never horizontal. It has either a camber (curvature) or two straight lines (like a flat chevron) in cross section. The transverse pitch may be only 1%, mostly for the sake of drainage. However, a road with a horizontal radius, which may vary from 15,000 feet on a barely perceptible curve in the country down to 300 feet on one of the ramps in a cloverleaf, may have to be banked, depending on the shortness of the radius and vehicle speed. Looking in cross-section through a banked road, we see that the surface describes a straight but sloping line. The bank is called the superelevation and is also specified as a percentage, as for example SE+4%, the maximum being 6%. Here the SE is positive if the road banks upward to the right as you look in the direction of increasing stations, negative if downward. So for any point right or left of the PGL, we can determine the elevation at the top of the pavement, by adding or subtracting the superelevation to or from the elevation at the PGL, as determined by the vertical curve. The superelevation will remain constant for a distance, but as the curved section of the road comes to a point of tangency with a straight section, there will be a distance of instantaneously increasing or decreasing superelevations. All the particulars of the horizontal curve, the vertical curve and the superelevation are shown on a plot plan accompanied by a table as methodical as mathematics itself. It’s beautiful to contemplate, and a real pleasure to work with! There’s yet another curve in bridges, due to deflection, and this is much more complicated. For a single simple span, like a plank on two sawhorses, with a continuous load, a fourth-order differential equation is required to calculate deflection. For two or three spans, that is like one continuous plank over three or four equidistant sawhorses and carrying a continuous load, several differential equations must be figured into each other by transport matrices, which becomes complicated. Beyond that, it becomes impossible. Twenty years or more ago, engineering firms involved in major bridge projects, like Golden Gate, would actually have precisely manufactured models made, and measure deflections by simulating loading conditions in miniature. Today, I suppose, with the finite-element method and sophisticated software, the modelling can probably be done in virtual space. The idea is that the bridge should assume the correct configuration AFTER deflection. Simply, if a bridge is going to deflect 3 feet, you camber it 3 feet upwards first, so that when it sags, it will be where it is supposed to be. But getting the correct numbers all along the length of the bridge is a very complicated procedure. ------------ 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 Comment on this article here! ------------ 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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