THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



1846.] 



Bot in an undivided beam owin? to the connection between its parts the 

 portion a could not assume the form here represented. Its lower surface 



205 



must coincide with and be of the same length as the upper side of A, and 

 be much more extended than when separated from it. Consequently no 

 argument taken from the consideration of the slice a, in its separated atalei 

 will apply to it when considered as an integral portion of the beam. 



We proceed now lo the direct arguments establishing the actual existence 

 of the neutral boundary. 



Whatever may be the degree of deflection of a beam, and however great 

 the load which it bears, it is clear that while the system is in a state of 

 equilibrium, the internal forces of this beam and the external pressures 

 upon it are subject to the ordinary principles of statics. Let A B C D be 



-<C.-M 



^>II 



one-half of a uniform beam resting on an abutment at A. Then if we sup 

 pose for simplicity that the beam is loaded uniforndy throughout, the centre 

 of gravity of the system is half way between the two abutments; that is, 

 the pressures on each abutment are equal. Hence the pressure on A is 

 the weight of A B C D, and the load on it. Or if R be the reaction at A, 

 and W the weight of the beam and load together, R=4W. 



Now if we suppose the beam actually cut in half at CD, it is certain 

 that ihe equilibrium of A B G D will not be altered if we consider the molecu. 

 iar forces which the supplied by the connection with the other half of the beam 

 to be external forces acting at C U. And whatever these forces may be, 

 they are capable of being resolved parallel to three co-ordinate axes, and 

 the resultants maybe equated with the remaining forces of the system. 

 Let the axis of x be parallel to the length of the beam and horizo>.tal, let 

 ,he axis ofy be vertical, and that of s perpendicular to the two former. 

 Tlienif 2 (X), 2 (Y), 2(Z) be the sums of the corresponding molecular 

 forces we have equating all the forces of the system, 



2(X)=:0, 2(Y)+R-iW = 0, 2(Z) = 0. 

 With respect to the second of ihese equations we have since R — JW _ 0, 

 2(Y) = 0, and since these forces represented by 2 (Y) are all parallel 

 and lie in one plane, they have a single resultant, and this resultant is 

 zero, consequently there is no vertical force at CD. Similar reasoning 

 applies to the forces represented by 2 (Z). 



With respect however to the longitudinal forces 2 (X) the case is diffe- 

 rent • for these though parallel to each other do not lie all in one plane. 

 Consequently they do not necessarily have a single resultant. The equa- 

 tion 2 (X) = may be interpreted by supposing it to represent a couple 

 M -M ■ and if we take moments about A we shall see tliat this is the 

 correct interpretation, for the moment of JW about A must be balanced by 

 some equal and opposite moment, and this can only be supplied by the 



"Thl final conclusion is therefore that the resultants of all the molecular 

 forces at C D are two equal and opposite forces M,-M, such that if the 

 distance of the centre of gravity of A B C D from A be called a, and the 

 distance between the points of application of M and - M at C D be called b, 

 Uh- iW a. 

 It is certain also from the direction in which M acts that it arises from 

 the compression or tendency to compress the upper part of C D, and in the 

 same way - M arises from the extension or tendency to extend the lower 



^*We have therefore the upper and lower parts of C D in opposite states 

 pf action and these two portions must be perfectly distinct from each other. 

 It IS quite impossible to conceive that the compressed and extended parts 



can alternate, that as we examine successive portion of C D, we shall have 

 first an exteuded part, then a compressed part, then again an extended 

 part, and so on. The portion of the beam exerting thrust cannot contain 

 any part in a slate of tension, and the portion exerting tension cannot con- 

 tain any part exerting a thrust. Neither can these two portions overlie 

 each other, for then vie must have some part of the beam exerting both ten- 

 sion and ihiust, which is as absurd as to suppose a man can pull and push 

 a thing at one and the same time. 



It is clear then that the two parts of the beam in opposite states of action 

 are perfectly distinct from each other, and there must therefore be some 

 boundary which marks tlie transition from the one state to the other, some 

 place where the one kind of force ceases and the other begins. This place 

 is called the neutral boundary. 



It will be observed that this conclusion is deduced from foundamental 

 principles of statics and not from hypotheses respecting the molecular 

 structures of the beam which, however ingenious, are seldom trustworthy. 

 It is a necessary consequence of the existence of a neutral line that the 

 compression of the upper side and the tension of the lower should vary in 

 decree, and gradually diminish towards the neutral boundary, so that there 

 should be no abrupt transition from one state of action to another. It is 

 impossible to imagine that the molecular action can suiideiily char.ge in any 

 part; from the connection between each two successive Iamina3, it is clear 

 that the extension or compression of the one must be affected by the exten- 

 sion or compression of the other. We are therefore perfectly safe in sup- 

 posing that there is some general law by which these variations of action 

 may be represented, thisis,thattheamountof molecular force at any point of 

 either side of the beam is a continuous function of the distance from some 

 fixed point. Now this function changes its sign in passing through the 

 neutral boundary, for on the upper side of the beam it represents forces 

 contrary in direction to those on the lower side, and therefore, since no 

 continuous function can change its sign without passing through zero or 

 infinity, and since iu the present case it is obvious that it cannot pass 

 through infinity, it must pass through zero, and consequently there is no 

 longitudinal action whatever at the neutral boundary. 



It is usual to represent the molecular forces as a function of their dis- 

 tances from the neutral boundary. This is perfectly arbitrary and merely 

 corresponds to a convenient choice of co-ordinates, for by properly altering 

 the form of the fuuction we might make the molecular forces depend on 

 their distances from any other point of refereuie. 



The determination of the actual position of the neutral boundary may 

 frequently be dilficulf, but the fact of its existence does not depend on the 

 labour of calculating its place. It fortunately happens however that for 

 the cases which most commonly occur in practice — those in which the de- 

 flection of the beam is small — the place of the neutral boundary can be as- 

 certained with quite sufficient accuracy to answer all useful purposes. 



It may be as well to point out to those not familiar with the subject that 

 this neutral boundary is not a mere matter of curious speculation, but one 

 of direct practical importance. Every thing depends upon it. Without 

 we know its position, we cannot tell how much of the beam is in a state of 

 compression, and how much in a state of tension— in other words, we can 

 know nothing about the strength of the beam. 



We have wished to confine attention here to those things only which may 

 be strictly deduced from the fundamental principles of mechanics, an 

 which cannot be controverted without attacking those principles. We 

 therefore do not enter into the consideration of the /or/« of the function 

 above alluded to. The assumption usually made accords perfectly well 

 with the results of practice, and there does not seem the least reason for 

 disputing its accuracy. At the same time we must remember that its cor- 

 rectness depends on experiment, and not on independent reasons. When, 

 indeed the beam is perfectly homogeneous, there are good theoretical reasons 

 in favour of the usual assumption, and of this at least we are certain, from 

 both theory and experiment, that the true law must be so near the assumed 

 one, that any error introduced into the calculation by this assumption will 

 be inappreciable iu the general result when the deflection oi the beam is 

 small. 



We have slated our objections fully and freely, because if any mistaken 

 conclusion be here set forth, Mr. Byrne will be able to correct them. If 

 in error, we shall be sincerely obliged to him for setting us right, for we 

 have no other object but the advancement of the trulh ; but till he does 

 this, we shall confide little on a theory of the strength of beams founded oo 

 a denial of the existence of a neutral boundary. 



