THE

MECHANICAL PRINCIPLES

OF

CIVIL ENGINEERING.

PART I.

STATICS.

1. FORCE is that which tends to cause or to destroy motion, or which actually causes or destroys it.

The direction of a force is that straight line in which it tends to cause motion in the point to which it is applied, or in which it tends to destroy the motion in it.*

When more forces than one are applied to a body, and their respective tendencies to communicate motion to it counteract one another, so that the body remains at rest, these forces are said to be in EQUILIBRIUM, and are called

PRESSURES.

It is found by experiment + that the effect of a pressure, when applied to a solid body, is the same at whatever point in the line of its direction it is applied; so that the conditions of the equilibrium of that pressure, in respect to other pressures applied to the same body, are not altered, if, with out altering the direction of the pressure, we remove its point of application, provided only the point to which we remove it be in the straight line in the direction of which it

.acts.

The science of STATICS is that which treats of the equili brium of pressures. When two pressures only are applied to † Note () Ed. Appendix.

*Note (a) Ed. Appendix.

a body, and hold it at rest, it is found by experiment that these pressures act in opposite directions, and have their directions always in the same straight line. Two such pressures are said to be equal.

If, instead of applying two pressures which are thus equal in opposite directions, we apply them both in the same direction, the single pressure which must be applied in a direction opposite to the two to sustain them, is said to be double of either of them. If we take a third pressure, equal to either of the two first, and apply the three in the same direction, the single pressure, which must be applied in a direction opposite to the three to sustain them, is said to be triple of either of them: and so of any number of pressures. Thus, fixing upon any one pressure, and ascertaining how many pressures equal to this are necessary, when applied in an opposite direction, to sustain any other greater pressure, we arrive at a true conception of the amount of that greater pressure in terms of the first.

That single pressure, in terms of which the amount of any other greater pressure is thus ascertained, is called an UNIT of pressure.

Pressures, the amount of which are determined in terms of some known unit of pressure, are said to be measured. Different pressures, the amounts of which can be determined in terins of the same unit, are said to be commensurable.

The units of pressure which it is found most convenient to use, are the weights of certain portions of matter, or the pressures with which they tend towards the centre of the earth. The units of pressure are different in different countries. With us, the unit of pressure from which all the rest are derived is the weight of 22-815 cubic inches of distilled water. This weight is one pound troy; being divided into 5760 equal parts, the weight of each is a grain troy, and 7000 such grains constitute the pound avoirdupois.

If straight lines be taken in the directions of any number of pressures, and have their lengths proportional to the numbers of units in those pressures respectively, then these lines having to one another the same proportion in length that the pressures have in magnitude, and being moreover drawn in the directions in which those pressures respectively. act, are said to represent them in magnitude and direction."

*This standard was fixed by Act of Parliament, in 1824. The temperature of the water is supposed to be 62 Fahrenheit, the weight to be taken in air, and the barometer to stand at 30 inches.

A system of pressures being in equilibrium, let any number of them be imagined to be taken away and replaced by a single pressure, and let this single pressure be such that the equilibrium which before existed may remain, then this single pressure, producing the same effect in respect to the equilibrium that the pressures which it replaces produced, is said to be the RESULTANT.

The pressures which it replaces are said to be the COMPONENTS of this single pressure; and the act of replacing them by such a single pressure, is called the COMPOSITION Of

pressures.

If, a single pressure being removed from a system in equilibrium, it be replaced by any number of other pressures, such, that whatever effect was produced by that which they replace singly, the same effect (in respect to the conditions of the equilibrium) may be produced by those pressures conjointly, then is that single pressure said to have been RESOLVED into these, and the act of making this substitution of two or more pressures for one, is called the RESOLUTION of pressures.

THE PARALLELOGRAM OF PRESSURES.

2. The resultant of any two pressures applied to a point, is represented in direction by the diagonal of a paral lelogram, whose adjacent sides represent those pressures in magnitude and direction.*

(Duchayla's Method.†)

To the demonstration of this proposition, after the excellent method of Duchayla, it is necessary in the first place to show, that if there be any two pressures P, and P, whose directions are in the same straight line, and a third pressure P, in any other direction, and if the proposition be true in respect to P, and P,, and also in respect to P, and P1, then it will be true in respect to P, and P2+P,.

Let P1, P,, and P, form part of any system of pressures in equilibrium, and let them be applied to the point A; take AB and AC to represent, in magnitude and direction, the pressures P, and P,, and CD the pressure P, and complete the parallelograms CB and DF. Suppose the proposition to be true with regard

*This proposition constitutes the foundation of the entire science of Statics. Note (e) Ed. App.

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to P, and P, the resultant of P, and P, will then be in the direction of the diagonal AF of the parallelogram BC, whose adjacent sides AC and AB represent P, and P, in magnitude and direction. Let P, and P, be replaced by this resultant. It matters not to the equilibrium where in the line AF it is applied; let it then be applied at F. But thus applied at Fit may, without affecting the conditions of the equilibrium, be in its turn replaced by (or resolved into) two other pressures acting in CF and BF, and these will manifestly be equal to P, and P, of which P, may be transferred without altering the conditions to C, and P, to E. Let this be done, and let P, be transferred from A to C, we shall then have P, and P, acting in the directions CF and CD at C and P,, in the direction FE at E, and the conditions of the equilibrium will not have been affected by the transfer of them to these points. Now suppose that the proposition is also true in respect to P, and P, as well as P, and P,. Then since CF and CD represent P, and P, in magnitude and direction, therefore their resultant is in the direction of the diagonal CE. Let them be replaced by this resultant, and let it be transferred to E, and let it then be resolved into two other pressures acting in the directions DE and FE; these will evidently be P, and P,. We have now then transferred all the three pressures P., P., P,, from A to E, and they act at E in directions parallel to the directions in which they acted at A, and this has been done without affecting the conditions of the equilibrium; or, in other words, it has been shown that the pressures P1, P2, P., produce the same effect as it respects the conditions of the equilibrium, whether they be applied at A or E. The resultant of P1, P, P1, must therefore produce the same effect as it regards the conditions of the equilibrium, whether it be applied at A or E. But in order that this resultant may thus produce the same effect when acting at A or E, it must act in the straight line AE, because a pressure produces the same effect when applied at two different points only when both those points are in the line of its direction. On the supposition made, therefore, the resultant of P,, P, and P, or of P, and P, + P, acts in the direction of the diagonal AE of the parallelogram BD, whese adjacent sides AD and AB represent I,+P, and P, in magnitude and direction; and it has been shown, that if the proposition be true in respect to P, and P, and also in respect to P, and P,, then it is true in respect to P, and P, + P. Now this being the case for all values of P., P., P. it is the case when P, P., and P,, are equal

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