CHAPTER XXXVII.

BEAMS AND GIRDERS.

WR now proceed to investigate the second main division of constructional steel work known as beams and girders. We have already dealt with the theory of beams in a previous chapter, as we did with stanchions: it remains in our present chapter to place before the reader some examples whereby the foregoing theory may be successfully put into practice.

It will, however, be necessary to investigate some further rules for determining safe loads, skin stresses, etc., in addition to those already given.

First of all we have the very familiar equation—

=

where M the greatest bending moment on the beam in tons-inches. This will be found by the various means suggested in Chapter VIII. and which will not be here repeated.

I

=

greatest moment of inertia of any section about the neutral axis passing through the centre of gravity of that section.

y = distance in inches of the outermost fibre from the neutral axis (as fully explained in the chapter referred to).

Р

=

safe skin stress in tons per square inch, which may be taken as 61, because this figure is recognised by the Board of Trade.

Now let Z

=

section modulus.

Every beam has a section modulus. They are usually given with accuracy in tables of standard beams. When a beam section, however, is not symmetrical there will be two values of Z. For practical use, however, only one value is generally given.

The sectional area of the section is usually denoted by A square inches. Any other moment of inertia parallel to I is called I, and distance between them V.

The radius of gyration = p.

Then from the fundamental rule (461) which applies to any beam of any section we have

and so on, the above being the most useful and commonly occurring modifications.

Let us take a case. A beam has a span of 20 ft. and is to support a wall of a building whose total weight = 10 tons. The weight of roof say 4 tons and the floor with its load

.. total load

= 20 tons.

=

6 tons.

=

The load being evenly distributed, the maximum bending moment will WL 20 × 20 × 12

be equal to

beam. We find the value of I=629, and P=6, and y

600 tons-ins. M. Let us try a 15 in. x 6 in. I

(using Equation 466) P=

600 x 7.5 4500.0

629

This

sq. in. for extreme fibre stress. Now this is over 61, but as the particular load is quite steady and is uniformly distributed the beam will be quite safe. is a very simple case, given for sake of example only. No doubt the reader is aware that in section books which contain tables of the properties of rolled-steel joists each beam has appended to it a certain maximum safe load for a certain span. To use such tables, no doubt, affords a great saving of time, but the necessary competition amongst steelmakers causes the values to be the absolute maximum admissible, and hence great care should be exercised in using them, and for this reason we have given in detail means whereby the proper sections can be arrived at. The particular value of Formula 461 is that it is universally applicable to any shape of beam whatsoever, provided the moment of inertia and y and P are known. The values of I for all manner of shapes are easily got from tables, or we have also shown how to work them out if required. For convenience, however, we can state the more common values as follows:

1. For a rectangle with dimensions b and h in inches,

(471)

And for an I beam, as shown in fig. 902,

bh3 + b1h23
12

3

which also applies to the channel section, fig. 903.

(472)

Now in large cases of beams, the actual weight of the built-up girder will be in itself a considerable part of the total load, and must be included in our calculations.

Y

Y.

-X.

b=Thickness of Web) (b1 = Thickness of Web)

FIG. 902.

FIG. 903.

=

W12
CxSxd-12

Or if we have any girder and know its sectional area in square inches the probable weight of that girder in lbs. per yard

= 10A

(475)

Another way of finding the probable weight of any girder per foot-run

(476)

We have already discussed the effect of fixing a beam, that is, it causes principally a higher safe load and three possible points of rupture. In no case whatsoever should any cast-iron beam be fixed. Cast-iron beams have rapidly fallen into disuse, but when designing it is a good idea to make the bottom flange equal in area to the distributed load in tons (square inches), the top flange being anything greater than of this value.

We shall return to the question.

Now in some cases we shall require web stiffeners in a built-up box or plate girder to take up the extra amount of shear which usually comes on

the web towards the supports. By any usual method find the reaction on the R. Let D depth of web, then we have supports of the proposed beam = R.

a value

=

x

Then =required thickness of web in inches. If the proposed web fails У

to comply with this condition, the extra thickness required must be supplied by web stiffners.

Now if we know the bending moment of a beam at any point, and divide by the effective depth of that beam (viz. depth between centres of flanges), we shall have the stress set up in the flanges at that point. Hence for a distributed load we shall have a gradual diminution of this value towards the ends of the beam in parabolic form. To the beginner this will naturally suggest diminishing the flanges towards the ends for the sake of economy. Practical engineers have found, however, that it is the best practice not to do this, but in place of so doing the depth of the web is decreased instead uniformly towards the supports, which gives us the commonly-called fishbellied girder used for a variety of purposes, especially overhead travelling cranes which are designed for a maximum centrally applied load.

Our previous remarks on riveting also apply to girders. Of course zigzag riveting is in most cases preferable to chain riveting, because the effective depth will undergo less reduction. In cases such as figs. 868 to 875 special care must be taken over those rivets which are under shear, allowing only 4 tons per sq. in. for those in single shear and 7 for those in double. The practice of notching, practised by some engineers, tends, however, to take much of the load off these rivets. We also know that towards the end of built-up girders we find a larger amount of rivets, for obvious reasons of shear; but the tendency of reducing this pitch towards the centre must not be carried too far; 16 times the diameter is quite the maximum permissible. This is not because we shall have insufficient strength, but for the reason of buckling occurring, and causing the two or more riveted plates to separate and allow of the admission of air, which will tend to hasten any corrosion. Now it was stated that to find the flange stress in any beam we divided the maximum value of the BM by the depth, which gave us the maximum flange stress. This will very often be too much for a single joist, and yet to use a larger size would not be economy. What do we do then? will be asked. Flange plates will be riveted on to provide for this extra stress in a manner which we are going to describe. It is, however, entirely a matter for the engineer to decide whether it will be cheaper to install the extra heavy joist, or rivet two or more plates on. With small loads the former will no doubt be the case; but when we come to large built-up girders, plates will invariably have to be used. The plates should, however, not have much side overlap, or else we sacrifice stiffness in the beam, which is just as important as mere strength, which is also a reason for the box girder being in most cases of structural work, preferable to the plate girder. We must always be careful. to deduct the area of the rivet-holes. As we pointed out in our example, a girder is very often required to carry a wall of a building to allow of an entrance, shop window, or other case. The beam is called a bressummer.

Architects usually take the whole wall as solid into consideration when

calculating the total load. This is a practice to be advocated as it errs on the side of safety, but it is not done in contractors' offices, where the all-important consideration is profit. Hence great care is necessary when examining tenders and the usual way to calculate this value is shown in fig. 904, the windows, of course, being deducted. The shaded area alone is calculated.

If no windows occur, upon the span as base, draw an equilateral triangle and calculate upon the enclosed portion.

In the same way as girders and beams, we can calculate for cantilevers and semibeams. Knowing the method of loading, find the maximum bending moment, which usually occurs at the support. Thence we find the flange stress by dividing by the proposed depth, and hence the sectional area.

For this purpose a tee section is usually selected, or two angles back to back, sometimes with a web plate. Owing to the rapid decrease of stress towards the ends of cantilevers, except in very small cases they are rarely

Clear Span

FIG. 904.

made of one section or an uniform depth. When, however, a single joist is used, we can select one from a section book capable of standing four times the proposed load of the cantilever when used as a beam uniformly loaded. We can then easily see what a lavish waste of material we shall have. To reduce this and yet allow of a single member being used, cast-iron brackets are sometimes used, but they cannot be advised. They do not really add much strength.

Now the practice of constructing floors of rolled-steel joists filled in with monolithic concrete still survives. It is a practice about which much good may be said, although advocates of concrete steel construction readily condemn it, and adhere to such principles as have been set forth in Chapter XX. A particular case, however, of economy of such construction is when old steel railway metals are obtainable for the purpose, as they are usually cheap and readily purchased. The rust usually thereon has a decidedly beneficial effect on the mutual adhesion of concrete and steel. A very strong and cheap floor can be made in this way; in fact, many an economical structure can be made of old railway and tram rails. Many examples will present themselves to the intelligent engineer. The writer recently designed a boat-slip which had piles mainly of old steel railway metals (80 lb. flange rails), and tramway rails as stringers. It has withstood the severest gales, and was very cheaply executed. For ease of erection in country places, structures of concrete and old rails are hard to beat. So important, in fact, is this question that we will take the trouble to investigate a special case.

Calculate the weight an old railway metal will safely carry, loaded in the centre for a 10 ft. span.

The conditions of stability of any beam are given by the equation M=R, or bending moment moment of resistance. In the case of a beam carrying

=