LECTURE XXXI.-QUESTIONS.

1. Define "bending moment" and "shearing force." A uniform beam weighing 15 cwts. rests on supports at its ends 20 feet apart. The beam is loaded with three weights of 4, 6, and 10 cwts. at distances of 2, 7, and 12 feet respectively from one of the supports. Find the B.M. and S.F. at a point 8 feet from the same support. Ans. B. M. = 98 ft.-cwts.;

S.F. 3 cwts.

=

2. A bar of pine 44 inches long rests on props at its extremities, and just supports 10 weights, of 14 lbs. each, hung at equal intervals of 4 inches along the rod. Find the amount of a single weight, which, if hung at the centre of the bar, would strain it to the same extent. Ans. 43 27 lbs.

3. A batten of fir, 6 feet in length and supported at its extremities, will just sustain a load of 520 lbs. when hung at the centre. If this weight be removed, and two weights, each equal to P lbs., be hung at distances of 2 and 4 feet along the bar, what is the greatest value which may be assigned to P? Ans. 390 lbs.

4. A beam, 20 feet long, whose weight is neglected, is supported at both ends and loaded with 1 ton evenly distributed along its length. Find the bending moment at a distance of 7 feet from one end. Ans. 5,096 ft.-lbs.

5. A beam, whose weight may be neglected, rests on supports at its ends 15 fect apart. Weights of 10, 6, 5, and 12 cwts. rest on the beam at intervals of 3 feet apart, the weight of 10 cwts. being 3 feet from one support. Find the points where the maximum bending moment and shearing force occur, and obtain their values. Construct the diagrams of bending moments and shearing force for the whole beam. Ans. The тах. В.М. 66 ft.-cwts., and occurs at all points between the weights 6 and 5 cwts.; the max. S. F. = 17 cwts., and occurs at the point where the weight of 12 cwts. rests.

=

6. A uniform cantilever, or beam fixed at one end and free at the other, 10 feet long, weighs 6 cwts., and carries two loads, one of 2 cwts. at the free end, and the other 4 cwts. at its middle point. Construct the shearing force diagram for the whole cantilever, and find the shearing forces at points 2 feet and 6 feet from fixed end. Ans. 105 cwts.;

6.4 cwts.

7. A block of wood weighing 800 lbs., 20 feet long and 12 inches square, floats in water, and is loaded

(1) By a weight of 200 lbs., placed at each extremity;

(2) By a weight of 400 lbs. at the centre.

Show what forces act on the beam, and draw the curves of shearing force and bending moment for each case. (S. & A. Exam., 1892.)

8. A girder is supported at both ends, and has a clear span of 30 feet. Show by means of a curve the position and magnitude of the greatest bending moment produced by a load of 20 tons as it rolls from one end to the other of the girder. Obtain the numerical results for distances respectively of 10 and 15 feet from one end. (S. & A. Hons. Exam., 1895.)

9. Prove an algebraic formula to show that, with a continuous load of uniform intensity passing over a beam A B such as when a long train passes over a bridge A to B, the maximum shearing stress to any point K of the beam occurs when the part AK is fully loaded while the part

KB is entirely unloaded, and that the magnitude of the stress is propor tional to the square of the distance of K from the point A. A train of 1 ton per foot run, and upwards of 100 feet in length, passes over a bridge of 100 feet span; what would be the maximum shearing stresses at distances of 25 and 50 feet respectively from one end of the bridge? Show how to determine graphically the shearing stresses in the beam. (S. & A. Hons. Exam., 1896.)

10. What occurs at the cross-section of a horizontal beam, carrying vertical loads? Where is the neutral line? What is the value of the stress at any place? What is meant by bending moment? Describe any model which illustrates, however roughly, what occurs at a section of the beam. (Adv. S. & A. Exam., 1898.)

294

LECTURE XXXII.

CONTENTS.-Resistance of Beams to Flexure-Examples I., II., III., and IV. Thin Wrought-Iron Girders-Example V.-Curvature and Deflection of Beams-Example VI.—Uniform Beam on Three Supports -Uniform Beam fixed at one end and supported at the other-Beams fixed at both ends and loaded at centre-Beams fixed at both ends and loaded uniformly-Tables-Questions.

Resistance of Beams to Flexure.-In the previous Lecture we saw that the effect of loading a beam was to give rise to both shearing and bending.

From the theory of couples set forth in Vol. I. we know that nothing but a couple can balance a couple. The resistance which a beam offers to bending must be of this nature, and therefore a couple of equal magnitude to that of the applied load, but of opposite tendency. The tendency of the applied couple is to bend or curve the beam, whilst the tendency of the induced couple is to oppose this curving action.

When a beam is curved the longitudinal fibres on the convex side of it are stretched beyond their normal length, and consequently they are in tension. On the concave side the fibres

Neutral

Surface

Neutral Axis

2

ILLUSTRATING FLEXURE OF BEAMS.

are shortened, and, therefore, they are in compression. Somewhere within the beam there must be a layer of fibres that are neither lengthened nor shortened, and are therefore unstressed.

This layer is termed the neutral surface of the beam, and the intersection of this surface with any cross-section of the beam is termed the neutral axis of that section. The neutral axis is of fundamental importance in the theory of beams, because it is the fulcrum about which both the bending and resisting couples act. We shall now find the position of the neutral axis of any given section of a beam.

Let be the length of a small portion of the neutral surface; that of a parallel layer of fibres on the stretched side of the beam, and at a distance y, from the neutral surface. If l' = l when the beam is straight, it is evident that the amount of stretch in the fibres at distance, y, from the neutral surface will

bell, and the strain "Let p denote the radius of cur

vature of the neutral surface at the cross-section bisecting 7. Then the radius of curvature corresponding to l' will be = p + y.

If ƒ be the tensile stress at distance y, from the neutral axis, and E the modulus of elasticity of the material, we already know that:

If we had considered in the same way a layer of fibres at a distance y', to the concave side of the neutral surface, and denoted the stress there as -f' (the minus sign indicating compressive stress), we should have arrived at the equation: