CHAPTER XIV.

GENERAL PRINCIPLES OF THE

STRENGTH OF

BEAMS, AND STRENGTH OF IRON BEAMS.

By the term "beam" is meant any piece of material which supports a load whose tendency is to break the piece across, or at right angles to, the fibres, and which also causes the piece to bend before breaking. When a load of any kind is applied to any beam, it will cause it to bend by a certain amount; and as it is impossible to bend a piece of any material without stretching the fibres on the outer side, and compressing the fibres on the inner side, the bending of the beam will produce tension in its lower fibres, and compression in its upper ones. This tension and compression are also greatest in those fibres which are the farthest from the neutral axis of the beam. The neutral axis is the line along which the fibres of the beam are neither lengthened nor shortened by the bending of the beam. For beams of wrought-iron and wood the neutral axis practically passes through the centre of gravity of the crosssection of the beam.

To determine the strength of any beam to resist the effects of any load, or series of loads, we must determine two things: first, the destructive force tending to bend and break the beam, which is called the "bending-moment;" and, second, the combined resistance of all the fibres of the beam to being broken, which is called the "moment of resistance."

The methods for finding the bending-moments for any load, or series of loads, have been given in Chap. XII.; and rules for finding the moment of resistance, which is equal to the moment of inertia divided by the distance of the most extended or compressed fibres from the neutral axis, and the quotient multiplied by the strength of the material, have been given in Chap. XIII., together with tables of the moment of inertia for rolled iron sections of the usual patterns.

Now, that a beam shall just be able to resist the load, and not break, we must have a condition where the bending-moment in the beam is equal to the moment of resistance multiplied by the strength of the material. That the beam may be abundantly safe to resist the given load, the moment of resistance multiplied by

strength of material must be several times as great as the bendingmoment; and the ratio in which this product exceeds the bending-moment, or in which the breaking-load exceeds the safe load, is known as the "factor" of safety.

By the strength of the material" is meant a certain constant quantity, which is determined by experiment, and which is known as the "Modulus of Rupture." Of course this value is different for each different material. The following table contains the values of this constant divided by the factor of safety, for most of the materials used in building-construction. The moment of resistance multiplied by these values will give the safe resisting-power of the beam.

MODULUS OF RUPTURE FOR SAFE STRENGTH.

The above values of R for wrought-iron and steel are one-fourth that for the breaking-loads; for cast-iron, one-sixth; for wood, onethird; and for stone, one-sixth. The constants for wood are based upon the recent tests male at the Massachusetts Institute of Technology upon full-size timbers of the usual quality found in buildings. The figures given in the above table are believed to be amply safe for beams in floors of dwellings, public halls, roofs, etc.; but, for floors in mills and warehouse-floors, the author recommends that not more than two-thirds of the above values be used. The safe loads for the Trenton, Phoenix, and Carnegie sections, used as beams, are all computed with 12,000 pounds for the safe value of R, or with 12,000 pounds fibre strain, as it is generally called, for iron, and 16,000 pounds for steel.

There are certain cases of beams which most frequently occur in building construction, for which formulas can be given by which the safe loads for the beams may be determined directly; but it often happens that we may have either a regularly shaped beam

irregularly loaded, or a beam of irregular section, but with a common method of loading, or both; and in such cases it is necessary to determine the bending-moment, or moment of resistance, and find the beam whose moment of resistance multiplied by Ris equal to this bending-moment, or what load will give a bendingmoment equal to the moment of resistance of a beam multiplied by R.

For example, suppose we have a rectangular beam of yellow pine loaded at irregular points with irregular loads: what dimensions shall the beam be to carry these loads? We will suppose that we have found the bending-moment caused by these loads to be 480,000 inch pounds.

Then, as bending-moment equals moment of resistance multiplied by R,

the beam should be 11 inches by 12 inches.

If, instead of a hard-pine beam, we should wish to use an iron beam to carry our loads in the above example, we must find a beam whose moment of resistance multiplied by 12,000 equals 480,000 inch pounds. We can only do this by trial, and for the first trial we will take the Trenton 124-inch 125-pound beam. The moment of inertia of this beam is given as 288; and its moment of resistance is one-sixth of this, or 48. Multiplying this by 12,000, we have 576,000 pounds as the resisting-force of this beam, or 96,000 pounds over the bending-moment. Hence we should probably use this beam, as the next lightest beam would probably not be strong enough. In this way we can find the strength of a beam of any cross-section to carry any load, however irregularly disposed it may be.

Strength of Wrought-Iron Beams, Channels, Angle and T Bars.

It is very seldom that one needs to compute the strength of wrought-iron beams, channels, etc.; because, if he uses one of the regular sections to be found in the market, the computations have already been made by the manufacturers, and are given in their handbook. There might, however, be cases where it would be necessary to make the calculations for any particular beam; and t meet such cases we give the following formulas.

Beams fixed at one end, and loaded at the other (Fig. 1).

Safe load in pounds =

1000 x moment of inertia

length in feet x y

(1)

W

Fig. 1.

Beams fixed at one end, loaded with uniformly distributed load

Beams supported at both ends, loaded at middle (Fig. 3).

Beams supported at both ends, loaded with concentrated load not at centre (Fig. 5).

Beams supported at both ends, loaded with W pounds, at a distance m from each end (Fig. 6).

Safe load W, in pounds at each point =

1000 x moment of inertia

m in feet x y

(6)

The letter y in the above formulas is used to denote the distance of the farthest fibre from the neutral axis; and, in beams of symmetrical section, y would be one-half the height of the beam in inches. These formulas apply to iron beams of any form of crosssection, from an I-beam to an angle or T bar. For steel beams, increase the value of W one-third.

Weight of Beam to be subtracted from its Safe

Load.

As the weight of iron beams often amounts to a considerable proportion of the load which they can carry, the weight should always be subtracted from the maximum safe load: for beams with concentrated loads, and for beams with distributed loads, one-half the weight of the beam should be subtracted.

EXAMPLE 1. - What is the safe load for a Trenton 124-inch light I-beam, 125 pounds per yard, having a clear span of 20 feet, the load being concentrated at a point 5 feet from one end? 1000 x I x span

Ans. Safe load (For. 5) = 12,500 pounds.

m x n x y

1000 × 288 X 20

5 X 15 X 6

=