Weights of Bridges 20

31. Stresses in Chord Members. The stresses in chord may be obtained by either the method of moments or the method of resolu tion of forces, this latter being usually the resolution of horizontal forces.

In accordance with the text of Article 29, the following rule may be stated with regard to the solution of stresses in chord members by the method of moments: Pass a plane section cutting the member whose stress is to be computed, and as few others as possible; then take the center of moments at such a point that the lines of action of as many forces as possible, the unknown one excepted, pass through that point; write an equation of the moments about this point of the known loads and forces to the left of the section, assuming the unknown force to act away from the section, and taking the known forces to act as given, the tensile stresses to act away from the section, and the compressive stresses to act towards the section; place the equation equal to zero, and solve.

The stress will solve out with its correct characteristic sign.

In the majority of cases a section can be made to cut three mem bers only, one of the three being the one whose unknown stress is desired. In such cases, take the center of moments at the inter section of the other two, and proceed as before. As examples of this latter case, note the centers of moments at Fig. 26, and 0, Fig. 27, and also the equations resulting therefrom.

When the method of resolution of forces is used, it is usually designated as the tangent method or the chord increment method. The simplest application of this method is to trusses with horizontal chords and vertical posts in the web members. Then the stress in any chord member is equal to the product of the sum of the shears in the panels up to that section, and the tangent of the angle which the diagonals make with the vertical.

This can readily be proved by reference to Fig. 29. Let it be re quired to determine the stress in the chord member S,. Pass the section a— a. The stresses S„ S" and S, are now computed, and are —V, sec ’; S,= + V, sec 0; S, = + V, sec ’; and S,— + V, sec II. Now noting the directions of the known stresses and assuming S. to act away from the section, the equation of the hori zontal component is: +S, sin ’+ S. sin +&sin =0.

Now, substituting the values of S„ S„ etc., and remembering that sec ’ the equation becomes: sin sin ’ sin 0 sin ’ • V + i + V. + t S. = 0, . ' cos ’ ' cos ’ ' cos ’ ' cos 0 from which, .

8, = tan 0; • S. — —BV tan 0.

From inspection of Fig. 29, it will be noticed that the stress in any section of the chord is equal to that in the section to the left of it, plus the increment (horizontal component) of the diagonal; hence the name chord increment method.

32. Notation. The practice hitherto used in designating stresses by S„ S,, etc., will now be discontinued, as it is inconvenient in the extreme; moreover, it is not the method used in practical work. The notation to be used is that given in Figs. 30 and 31, the former

being for a through and the latter for a deck truss.

The practical advantages of this system are very great. When U, U, is noted, it is at once known to be the top chord of the second panel; U, L is known to be the second vertical; while U, L, is at once recognized as the diagonal in the third panel. A stress in a member, as well as the member itself, is designated by the subscript letters at its ends. Thus U, may mean the member itself or the stress in the member. The text will clear this up. In analysis, the stress would be implied, while in design the member itself would be intended.

33. Warren Truss under Dead Loads. The Warren truss has its web members so built of angles and plates or of channels, that they can take either tension or compression. The top chord is of structural shapes, while the lower chord may be of built-up shapes or simply of bars.

Let it be required to determine all of the stresses in the six panel truss of a through Warren highway 120-foot span for country traffic. The height is to be 20 feet. The outline is given in Fig. 32. According to Fig. 16, the total weight of the span, including wooden floor, is 76 000 pounds. Each truss carries one-half of this, or 76 000 2 = 38 000 pounds. As there are six panels,, each panel load is 38 000 - 6 = 6 333 pounds. This means that we must compute the stresses in the abovd truss by considering that a load of 6 333 pounds is at points L„ L,, L„ and Of course there is some weight at and but this does not stress the bridge, as it is directed over the abutments or supports. The reactions at and are each equal to (5 X 6 333) _ 2 = 15 833 pounds.

The shears are next computed, and are: It is unnecessary to go past the center of the bridge, as it is symmetri cal. The V, represents the shear on any section between and represents the shear on any section between L, and and represents the shear on any section between and The secant of the angle 4 The stresses in the web members are computed as follows: For L„ Pass section a— a. Assume stress acting away from the section, as shown. Then, which shows that has a compressive stress of 17 700 pounds. For L,. Pass section b—b. Assume stress acting away from the section, as shown. Then, which shows that U,L, has a tensile stress of 17 700 pounds. For Pass section c — c. Then, For Pass section e e. Then, For Pass section / — f . Then, +3 167 — cos ’ = 0; = +3 167 X 1.12 = +3 540.

The computation of the stresses in the chords is made by the method of moments, and is as follows: For Section b — b cuts and besides the mem ber whose stress is desired, and therefore the center of moments will be taken at their intersection U,. The equation is: For Either section c — c or d — d may be used, and each shows the center of moments to be at The equation is: