EXAMPLE.

- What is the greatest bending-moment in a beam of 20 feet span, loaded with a distributed load of 800 pounds and a concentrated load of 500 pounds 6 feet from one end, and a concentrated load of 600 pounds 7 feet from the other end?

Ans. 1st, The moment due to the distributed load is W ×

2000 pounds. We therefore lay off to a scale, say 4000 pounds to the inch, B1 = 2000 pounds, and draw a parabola between the

points A, B, and

Hence we draw E2 = 2100 pounds, to the same scale as B1, and then draw the lines AE and CE.

is

3d, The bending-moment for the concentrated load of 600 pounds 600 × 7 × 13

or 2730 pounds; and we draw D3 = 2730 pounds,

and connect D with A and C.

4th, Make EII = 2 — 4, and DG3-5, and connect G and H with C and A and with each other.

The greatest bending-moment will be represented by the longest vertical line which can be drawn between the parabola ABC and the broken line AHGC. In this example we find the longest vertical line which can be drawn is xy; and by scaling it we find the greatest bending-moment to be 5550 pounds, applied 10 feet 11 inches from the point A.

In this case, the position of the line Xy was determined by drawing the line TT, parallel to HG, and tangent to ABC. The line Xy is drawn through the point of tangency.

NOTE.

As the measurements used for determining the bending moment are in feet, we must multiply the moment by 12, to get it into inch pounds; otherwise, in working out the dimensions of the beam, they would be in feet instead of inches.

CHAPTER XIII.

MOMENTS OF INERTIA AND RESISTANCE, AND RADIUS OF GYRATION.

MOMENT OF INERTIA.

THE strength of sections to resist strains, either as girders or as posts, depends not only on the area, but also on the form of the cross-section. The property of the section which represents the effect of the form upon the strength of a beam or post is its moment of inertia, usually denoted by I. The moment of inertia for any cross-section is the sum of the products obtained by multiplying the area of each particle in the cross-section by the square of its distance from the neutral axis.

NOTE.-The neutral axis of a beam is the line on which there is neither tension nor compression; and, for wooden or wrought-iron beams or posts, it may, for all practical purposes, be considered as passing through the centre of gravity of the cross-section.

For most forms of cross-section the moment of inertia is best found by the aid of the calculus; though it may be obtained by dividing the figure into squares or triangles, and multiplying their areas by the squares of the distance of their centres of gravity from the neutral axis.

MOMENT OF RESISTANCE.

The resistance of a beam to bending and cross-breaking at any given cross-section is the moment of the two equal and opposite forces, consisting of the thrust along the longitudinally compressed layers, and the tension along the longitudinally stretched layers. This moment, called "the moment of resistance," is, for any given cross-section of a beam, equal to

moment of inertia

extreme distance from axis

In the general formula for strength of columns, given on p. 231, the effect of the form of the column is expressed by the square of the radius of gyration, which is the moment of inertia of the section divided by its area; or = r2 The moments of inertia of the principal elementary sections, and a few common

I

A

forms, are given below, which will enable the moment about any given neutral axis for any other section to be readily calculated by merely adding together the moments about the given axis of the elementary sections of which it is composed.

In the case of hollow or re-entering sections, the moment of the hollow portion is to be subtracted from that of the enclosing area.

Moments of Inertia and Resistance, and Radii of Gyration.

This is the formula generally used by the engineers for the iron. companies.

Moments of Inertia and Radii of Gyration of Merchant Shapes of Iron and Steel.

For the sections of rolled iron beams and bars to be found in the market, the moments of inertia are given in the "Book of Sections " published by the manufacturers. The following tables give the moments of inertia and radii of gyration for the principal sections manufactured by Carnegie, Phipps & Co., the New Jersey Steel and Iron Company, and the Phoenix Iron Company (revised to October 1, 1891). The Pencoyd Iron Works have recently made changes in a number of their sections, and some of the old sections of iron beams and channels have been abandoned, and they are not at present prepared to furnish the revised data.

The tables give the least weight for each section of iron beam, and the minimum and maximum weights for channels, deck beams, and angle irons. These shapes can be rolled for any weight between the two given, while the weight of the beams can also be greatly increased. With the quantities given in these tables, one can find all the data required in usual calculations.

The tables on pages 322-24 will be found very convenient in computing the strength of struts formed of two or four angie bars.