The total thickness of the cover-plates, K inch, is too thick to be punched. In such cases as this, the section is made up of two or more plates whose total thickness is equal to that required. If plates of more than one thickness are decided upon, then their thickness should decrease from the flange angles- outward. For the case in hand, one plate inch thick and one plate i inch thick will be decided upon. The flange section at the center as finally designed is: The above is the section required at the center of the girder; for any other point it will be less, decreasing toward the end, where it will be zero. Evidently, then, the cover-plates will not be required to extend the entire length. The following analysis will determine where they should be stopped. If the load were uniform, the moment curve would be a parabola. Although under wheel loading the curve of moments is not a parabola, yet it is sufficient for practical purposes to consider it as such. The curve of flange areas, like that of moments, is to be considered a parabola (see Fig. 141).
Let a, = Net area, in square inches, of the outer cover-plate; a, = Net area, in square inches, of the next cover-plate; a„ etc. = Net areas of the other cover-plates; A = Net area of all the cover-plates and the flange angle. Then, from the properties of the parabola, where L = Length of cover-plate in question; = Length of span, center to center; a = Net area of that cover-plate and all above it; and A = Total net area of the flange, l< of the gross area of the web not being considered in this quantity.
The lengths of the cover-plates for the section above designed (see Fig. 141) are: Lc--- 61.75 = 26.45 feet.• 24.38 4.50 + 6.00 La = 61.75 24.38 — 0.00 feet.
One foot is usually added on each end of the cover-plate as theoretically determined above. The results are also usually rounded off to the nearest half-foot. This is done in order to allow a safe margin because of the fact that the curve of flange areas is not a true parabola. The final measurements of the cover-plates are: 14 in. by c in. by 28 ft. 6 in. long.
14 in. by . in. by 42 ft. 6 in. long.
In most cases the cover-plate next to the angle on the top flange only is made to extend the entire length of the girder. Although this is not required for it is done in order to provide additional stiffness to the flange angles toward the ends of the span, and to prevent the action of the elements from deteriorating the angles and the web by attacking the joint at the top (see broken lines, Fig. 141, for length of first cover-plate extended).
1. The dead-load moment equals 469 000 pound-inches; and the live load moment, 4 522 000 pound-inches. Design a flange section of angles, if the distance back to back of angles is 451 inches.
2. The dead-load moment is 3 340 000 pound-inches, and the live load moment, 21 235 000 pound-inches. Design a flange section using 6 by G-inch angles and three 14-inch cover-plates, the distance back to back of flange angles being 781 inches.
3. In each of the above cases, design the flange section considering that r of the web area is taken as effective flange area. (For demonstra tion of the methods to be employed in the solution of this problem, see the succeeding text.) While the section of a plate-girder is composite—that is, it con sists of certain shapes joined together, and is not one solid piece— nevertheless these shapes are joined so securely that the section may be considered as a solid one and its moment of resistance computed accordingly. Let Fig. 142 be considered.
The moment of resistance of the section is: ( d X + A in the derivation of which the moment of inertia of the flange its own neutral axis is considered as zero, and A equals the net area one flange. Now, as the values of and It seldom differ by more than one inch, for all practical purposes they may be considered* as equal. The above expression then reduces to: = S X h (net area of flange + one-sixth gross area of web) Since the rivet-holes decrease the moment of resistance of the web, one-sixth of the gross area cannot be considered, as is theoreti cally indicated in the above formu la. It is common practice to take one-eighth, instead of one-sixth, of the gross web area. Substituting this value in the above equation, and transposing, there results: Area of flange + x gross web area = The flange section will now be designed for the moments previously given, considering s of the gross web area as efficient in withstanding the moment.
The gross area of the web is 74 X a = 27.75 square inches; and of this is 3.47 square-inches. The total approximate amount of flange area required is, as in the first case, 24.48 square inches.
According to the above formula, of the web area, or 3.47 square inches, may be considered as flange area, and therefore 24.48 — 3.47 21.01 square inches, is the approxi= mate area of the angles and cover-plates of the flange. The ap proximate area of one angle is then 21.01 - (2 x 2) = 5.25 square inches. A 6 by 6 by angle gives the gross area of 6.43 square inches and, two rivet-holes being deducted, a net area of 5.305 square inches (see "Steel Construction," Part I, Table VIII, or Carnegie Handbook, p. 117). As this is qtiite close to the approxi mate area determined above, this angle will he taken. The ap proximate area of the cover-plates is 21.01 — 2 X 5.305 = 10.40 square inches. As before, the gross width of the cover-plate will be taken as 14 inches. The thickness is then — 10.40 14-2(1+s) = 0.867 inch—say a inch.