*Ex. 464. If ABC is a section of a rectangular wall, p the pressure applied to every foot of its length at a, the inner edge of its summit; determine the equation to the line of resistance.

FIG. 129.

P

A

[Take any horizontal section of the wall MN; let AN=x, BC=a, then the weight w of ANM=axw, where w is the weight of a cubic foot of the wall; now, if the direction of the resultant cuts MN in R, this will be a point in the line of resistance, and if RN=y we are to determine a relation between x and y. The relation in question can easily be shown to be

2 P sin a it may be carried to any height whatever with safety.

W

*Ex. 466. If the wall in Ex. 464 has to support the pressure of earth or water reaching to the top of the wall, show that the line of resistance is a parabola with its axis horizontal, and show that in the latter case its focus is in the summit of the wall at a distance from the intrados equal to

a

2

(1+3), where w is the weight of a cubic foot of masonry and w1 of

water.

*Ex. 467. If ABCD is the section of a reservoir wall the vertical face of which (BC) is towards the water; the width of the top of the wall (AB) is a; the inclination of AD to vertical is 0, and s is the specific gravity of the wall; show that when the water reaches to the top of the wall the equation to the line of resistance is x and y being measured as in Ex. 464

x2 (+tan20) -3xy tan 0+3ax tan 0-6ay + 3 a2 = 0

S

*Ex. 468. Show that if the wall in the last Example stand, whatever be the depth of the water whose pressure it sustains, then tan e must be >

1 √2s

*Ex. 469. Determine the equation to the line of resistance in a river wall of Aberdeen granite the thickness of which is 4 ft., and which sustains the pressure of water whose surface is on the level of the top of the wall.

Ans. x2=63 (y-2).

CHAPTER IX.

ON THE DEFLECTION AND RUPTURE OF BEAMS BY
TRANSVERSE STRAIN.*

96. Notation. The cases of deflection that we shall in the first place consider will be those of beams with a rectangular section. The following is the notation employed: a denotes the natural length of the beam, b its depth, and c its breadth, i. e. in a direction perpendicular to the plane of the paper; in the numerical examples the measurements are to be taken in inches, since the values of the modulus of elasticity E, given in Table III. p. 11, are referred to a square inch of section.†

97. Neutral Surface and Neutral Line of a Beam.If we consider a long beam of wood AD Supported at its

CD will suffer extension, and the upper surface A B com

* This chapter cannot be read with advantage by any student who has not some acquaintance with the Integral Calculus.

† The term modulus of elasticity was introduced by Dr. Young, to whom is due the first correct investigation of the flexure of beams; his views are to be found in his Lectures, vol. ii. p. 46, &c.: he denotes the modulus by the letter m, which is equivalent to Ebc of the text. The reader will find the question fully discussed in Mr. Moseley's Mechanics of Engineering, Part V., which has been frequently referred to in drawing up the present chapter.

pression: so that there will be some section PQ which will be intermediate to the compressed and extended parts, having undergone neither compression nor extension; this surface is called the neutral surface. It sometimes happens that the whole of the substance is either compressed or extended; in such a case the neutral surface will not have a real existence, but there will exist without the body an imaginary surface bearing the same relation to the compressions or extensions as that borne by the actual neutral surface in other cases.

If we were to divide the beam into any number of thin parts by vertical planes parallel to P1BD, the forms of the surfaces would be unaffected, consequently any part of the neutral surface is like any other; we may therefore confine our attention to the section of that surface made by a vertical plane passing lengthwise through the centre of gravity of the beam: this section is called the neutral line of the beam; by the term axis of the beam is intended the geometrical axis of the beam considered as a prism. In the following examples it is assumed that the forces act in a plane passing through the axis and parallel to the face of the beam. It is also assumed that the deflection of the beam is small, so that the moments of the forces that bend it are not changed by the deflection of the beam.

Ex. 471. If a line AB is subjected to a continuous pressure throughout its length of such a nature that the pressure at any point p is at the rate of k. AP per inch, then the resultant pressure equals k. A B2, and its moment round a equals k.AB3.

[The solution is similar to that already given (Art. 82) of the friction on a pivot.]

Proposition 21.

If a rectangular beam be held firmly by one end, and be acted on at the other by a force P, in a direction perpendicular to its length, the neutral line will coincide with the axis of the beam, and at any point distant p

inches from the end at which P acts the radius of curvature

Let ABCD represent the beam brought into its present position by the action of the force P: let LL1 be the neutral line; consider any small portion of the beam HKMN, which in its original state had the thickness EF,

but owing to the action of P the ends MK and NH converge too; we are to determine the position of the point F, and the distance Fo; the former will give the position of the neutral line, the latter the radius of curvature at the point F.

We may suppose HM to be made up of thin laminæ parallel to EF, of which mn represents one; all those within MF are in a state of extension, while those within FK are in a state of compression. Now, since each part of the beam is in equilibrium we may confine our attention to the portion MH, and may regard NH as a fixed surface; then the expansive forces within FK and the contractile forces within FM must be in equilibrium with P. But it is plain that the contractile force of any lamina such as nm acts in a direction perpendicular to that of P, and similarly of the expansive forces of any lamina. Hence (Prop. 15) the sum of the contractile forces of MF= the sum of the expansive forces of KF. Let EF and oF be denoted by land p, NF and HF by b1 and b2, and nr by z, the width of the lamina being dz; now the natural length of mn is l, therefore mn-lis the extent by which it is stretched; therefore the force T necessary to produce this extension is given by the proportion (see Art. 6)

But by smr triangles mn: l::z+p:p

...mn-l:l:: z: p

EC

ρ

Now the force necessary to produce the extension equals that with which the lamina tends to contract, therefore T gives the contractile force of the lamina mn, and the same being true of all the others, their sum (by Ex. 471) will equal

and in like manner the sum of the expansive forces will equal

And these being equal we have b1=b2; also since the same will be true of any other section, the neutral line