This section is from the book "Cyclopedia Of Architecture, Carpentry, And Building", by James C. et al. Also available from Amazon: Cyclopedia Of Architecture, Carpentry And Building.
For Buildings: ft (tension) = 15,000 pounds per square inch, net area. fc (compression) = 12,000 pounds per square inch, gross area, reduced for ratio of unsupported length to width of flange. fs (shearing stress) = 12,000 pounds per square inch, net area.
For Highway Bridges: ft = 13,000 pounds per square inch, net area.
fc = 11,000 pounds per square inch, gross area, reduced for ratio of length to width of flange. fs = 10,000 pounds per square inch, net area.
For Railway Bridges: ft = 10,000 pounds per square inch, net area. fc = 8,000 pounds per square inch, gross area, reduced for ratio of length to width of flange. fs = 8,000 pounds per square inch net area.
The practice regarding the reduction of allowable compression stress varies somewhat; but the following formula is a conservative one for general use: f = fc/l2
1+/5,000W2 where f = Fiber stress to be used in compression; fc = Specified fiber stress unreduced; l = Length of unsupported flange (in inches); W = Width of flange (in inches). In ordinary construction, the fact that the two flanges are generally made of the same section makes it unnecessary in many instance.; to consider this reduced compression-fiber stress. If the unsupported length of top flange is long, however, so as to make the section determined for bottom flange insufficient, then both flanges should be made the same as that required by the compression value.
When the girder is short, and the web-plate is not spliced, allow-ance is sometimes made for the portion of the compression and tension which the web may carry. In doing this, the net area of the web - deducting rivet-holes - is considered concentrated at the centers of gravity of the flanges, and as reducing the required area of the flanges by an amount equal to 1/8 t h1 in which t = thickness of web, and ht = depth. When this assumption is made, therefore, the required area of each flange is F/f - 1/8th1 ,in which / is the compression value for the top flange and the tension value for the bottom flange.
There is a considerable saving in the templet and shop work if both flanges are made alike; the extra weight in one flange which may be added, will often be more than offset by the saving in shop work.
It is a very general practice, therefore, to make both flanges alike in section, determining this by whichever flange requires to be the larger.
Economical Depth of Web. It will be seen that the areas required for the flanges are dependent on the depth of the web. Where there are no conditions limiting this depth to certain values, it is desirable, therefore, to fix it so as to give the most economical section. For a uniformly distributed load, this depth is generally from 1/9 to 1/10 of the span. Sometimes several approximations of this depth can be made, and the corresponding areas determined; and then, by computing the weights of flanges and web-plates so determined, the most economical section can be chosen.
In a great many cases, especially in building construction, the economical depth cannot be used, because of conditions fixing this depth with relation to other portions of the construction. In other cases, certain sections of plates and angles must be used in order to obtain quick delivery; and accordingly, the depth must be fixed to harmonize with these sections.
Proportioning the Web. As before stated, the function of the web is to resist the shear.
In highway and railway girders, this loading is always different. In building construction it is very often different, because certain beams may frame into the girder over the support and these beams must be considered in determining the shear although they are not con-
STOCK EXCHANGE BUILDING, CHICAGO, ILL.
Louis H. Sullivan, Architect, Chicago, 111.
Walls of Terra-Cotta. Completed in 1903, Building Operations Covering a Period of One sidered in determining the bending moment. Again, a girder may carry a wall, and a portion of this wall may come directly over the end supports of the girder. This portion will materially increase the shear while perhaps not affecting the bending moment.
Year. Cost, about $1,250,000.
BORLAND BUILDING, CHICAGO, ILL.
Shepley, Rutan & Coolidge, Architects. E. C. & R. M. Shankland, Engineers.
Building completed in fall of 1906.
The general statement of loads to be considered in determining the shear where all loads are fixed in position, is to include all loads which directly or indirectly can come upon the girder, and to determine the maximum end reaction for these loads. (The determination of web shear for moving loads, will be treated under "Bridge Engineering)." Sometimes the shear at one end is greater than at the other, in which case the section is fixed by the requirements at the end having greatest shear.
Having determined therefore, the maximum shear, the required
S/fs = 3/4 t h area of web is in which S = Maximum shear; fs = Allowable shearing stress per square inch of net area of web; t = Thickness of web; and h = Depth of web. The net area is assumed as 3/4 the gross area.
Crippling of Web, and Use of Stiffeners. The value of fs to be used depends on the clear distance between the adjacent edges of the top and bottom flange angles, and upon whether or not stiffener angles are to be used.
The distribution of the shear over the web causes compression forces acting at angles of 45 degrees with the axis of the girder, in the manner indicated by Fig. 245. The web, therefore, under these compression stresses, is subject to failure laterally, just as a long column. The allowable shearing stress must therefore be reduced by a formula similar to the column formula, which may be taken as in which d c = distance between flanges; and t = thickness of web.
Either the web must be made thick enough not to exceed this allowable stress on a length 1,414 d c, which is the length on a 45-degree line between the adjacent edges of flange angles, or this unsupported length must be reduced by using stiffeners so spaced as to cut this 45-degree length down to limits which will conform to the allowable shearing stress given by the formula and to the thickness of web which it is desired to use.
Webs less than 5/16 inch thick are rarely used. For greater thicknesses, it is a matter of economy generally to use stiffeners. For very heavy loads, however, or for long spans, 3/8-inch or 2-inch webs would be used, with or without stiffeners, as might be required.
It will be seen from the above consideration, that, where the shear varies from the end towards the center, the required spacing of stiffeners will increase towards the center, since the area of the web is constant.
When the shear has reduced to the point where the area of web is sufficient to resist buckling on a length of 1.414 d c, then the stiffeners may be omitted. A convenient diagram for determining spacing of stiffeners is shown in Fig. 246; the use of this diagram will be illustrated by a problem.
Suppose the shear at the end of a girder is 100,000 pounds; and the clear distance between flange angles is 22 inches, and the web which it is desired to use is 30 inches by § inch. The gross area of web is then 11.25 square inches, and the shear per square inch of gross area is 8,900 pounds. Following up the vertical side of the diagram until the line corresponding to 8,900 is found, then following this line until it meets the line of a 3/8-inch web, and then looking under this intersection to the lower horizontal line, it is found that stiffeners must be spaced about 12 inches apart in order to conform to the above conditions.
If it was desired to find what thickness of web was necessary in order not to require stiffeners, the flange angles being 22 inches apart in the clear, this would be determined as follows:
Follow up the vertical line corresponding to 22 inches as given at the bottom of the diagram, until this line meets the line corresponding to such a thickness of web that the gross area is sufficient to bring the shearing stress within the limit by the horizontal line at this intersection of web-line and vertical through 22.
In this case the nearest intersection is found to be the 1/2-inch web. The area of a 30-inch by 1/2-inch web is 15 square inches, and this gives a shearing stress per square inch of 6,675 pounds. The allowable stress as given by the diagram is 7,400 pounds; but the 7/16-inch web found to give a shearing stress of 7,640 pounds, whereas the allowable shear for a 7/16-inch web with angles 22 inches apart is only 6,600 pounds.
It would be found more economical to use a 3/8-inch web with stiffeners, than a 1/2-inch web without stiffeners.
Another use of stiffeners is to stiffen the web at concentrated loads. The most important case under this head is the reaction at the bearings of the girder. Stiffeners are always used here, and they are generally placed so that the outstanding legs will come nearly over the edge of the bearing plate, as illustrated by Fig. 247. Sometimes the special nature of the bearing - as, for instance, the disposition of column members - makes it desirable to place these stiffeners close together, or in three lines instead of two. The fundamental idea is to place the stiffeners so as to distribute the reaction in the most direct way to the bearing. If this bearing is masonry, the stiffeners will be placed so as to give uniform bearing; if a column, they will be placed so as to correspond as closely as possible with the line members of the column. Wherever heavy concentrated loads from beams, other girders, masonry piers, etc., occur, stiffeners should be used to stiffen the web against this concentrated application of load. Stiffeners over bearings should be fitted to both the top and bottom flange angles. Stiffeners at loads on the top flange need be fitted only to the top flange angles.
Stiffeners used simply to prevent buckling from the shear, need not be fitted to either flange. Sometimes stiffeners used for this latter purpose are not carried over the flange angles, but stop clear so as to avoid the necessity of fillers, as indicated by Fig. 247. It is better practice, and more generally followed, to carry these angles over the flange angles, as shown by Fig. 248.