THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



rjuNE, 



164 



„,anv equal an.l parallel forces and may be considered as acting together 

 ZZ centre of ,Liy of .he section, so ,hat denoting "- -.stance a 

 single 6bre by /, and supposing the section to be a -^""^'fj^p;;^, 

 and height ft. then will/i ft express the sum of .he '^'^^'1°'''' ^"'\~^ 

 acts at the centre of gravity of the section, which is at the d.s.ance of Jft 



A _ /*" 

 fromtheline AB.itsmoment to turnabout AB will be/ftftXg - 2 



Leibnitz gave another hypo.hesis, which agreed with that of Galileo with 

 respect to the position of .he axis about which the segments would turn 

 But Leibnitz supposed .he filaments or fibres to exert forces proportional 

 to their dis.ances from the axis ; so that the middle fibre, according to .he 

 theory of Leibnitz, exerted but half the force of the extreme fibre. LM- 

 ing the force of the extreme fibre /, the sum of the forces would be-^- i 

 and since the centre of such a system of parallel forces is at the distance of 

 " from the axis about which the whole is supposed to turn, hence the 



and I will give you something like the result of experiment from a line so 

 determined Let j Y Z be a portion of a beam in the locality of fracture, 

 caused by the forces F F acting in the direction! of the arrows. The same 

 process of reasoningwhich points out a neutral axis in the whole A H D C, 

 will point out a neutral axis in any portion of the body ^ a6x <?, no 

 matter where it be situated ; in fact, every fibre may be sa.d to be cora- 

 pressed on one side and extended at the other, while the whole or each is 

 bent round a common centre, as S, entirely outside the body. Then S Y 

 is the radius of curvature of the arc C I^ D at the point p. 



Fig. 3. 



It is easily seen. 



fbh 2h_,fbl,- 

 moraent to turn will be expressed by-jp x j - 3 



that, as far as regards the comparative strength of rectangular beams of 

 the same material, or of similar beams whose transverse sections are rect- 

 angular, it is no matter which of these hypotheses be adopted, for both 

 pofnt out the law of resistance to be as the breadth multiplied by the 

 square of the height; we shall in future use the term height or depth for 

 the dimension in the direction of the pressure. 



Fig. 2. 



Galileo and Leibnitz supposed that the segments of a beam X Y, frac- 

 tured by a weight W, turned about the line A B where the fracture termi- 

 nates But James Bernoulli, Mariotte, and others, were of opinion that 

 the segments had a tendency to turn about a line, as nm, entirely within 

 the section ; the fibres on that side of the line where the fracture begins 

 are extended, and those on the other side compressed. If the beam X Y, 

 resting on two props at X and Y, be fractured by the weight W and mn, 

 the line inside of the section A B a 6, about which the segments of the 

 beam have a tendency to turn, then the fibres or filaments in the space 

 „ m 6 a are supposed to be extended, and those in the space B A m n com- 

 pressed. The imaginary line in n, which divides the section A B a 6 into 

 two parU-Ihe area of compression and the area of tension-is called the 

 neutral axis Mr. P. Barlow laboured much to find out the true position 

 of this neutral axis in different sorts of timber. The result of his labours 

 and experiments may be summed up in the following words, the truth of 

 which is very questionable :-" The centre of tension and the centre of 

 compression each coincide with the centre of gravity of its respective 

 area(') and the neutral line which divides the two is so situated, that the 

 area of tension into the distance of its centre of gravity from the neutral 

 axis is to the area of compression into the distance of its centre of gravity 

 from the same line, in a constant ratio for each distinct species of wood, 

 but approximating in all towards the ratio of 3 to 1." (?) It would take 

 up too much space to dwell on the absurd conclusions of Mr. Barlow; 

 there are one or two things which require but little consideration to detect, 

 first the neutral fibres do not arrange themselves in a right line in all forms 

 of beams, indeed, if such aline did exist, it would be a curve governed by 

 the external form of the beam and the force applied. In the second place, 

 the centre of gravity cannot agree in all cases with the centres of the forces 

 of the filaments of the compressed and extended areas, and any man at 

 first sight might suppose that the areas of compression and tension would 

 bear a constant ratio to each other in each distinct species of wood. In 

 the third place, I defy experiment either to confirm or contradict these 

 conclusions of Mr. Barlow, for they have nothing whatever to do with the 

 Strength of beams. He merely says, find where MY line is by experiment, 



Now let us take ff c d J- f y 9 Z, any portion of the beam , it is evident that 

 the filaments in the upper part near to di, are expanded, and those near 

 toxYZ are compressed; according to this reasoning there is a set of 

 fibres between d (9 and xYZ which are neither compressed nor expanded ; 

 hence each portion of the beam is entitled to a neutral axis, which is rela- 

 tively correct, but each neutral axis is itself bent round a centre in r S. 



It is stated by Tredgold (" Practical Essay on the Strength of Cast 

 Iron " page 53),—" When a rectangular beam is supported at the ends, 

 and loaded in a'ny manner between the supports, it may be observed that 

 the side against which the force acts is always compressed, and that the 

 opposite side is always extended ; while at the middle of the depth there 

 is a part which is neither extended nor compressed ; or, in other words, it 



is not strained at all. ,f.i,..i,- 



"Any one who chooses to make experiments may satisfy himself that this 

 is a correct statement of the fact, in any material whatever, whether it be 

 hard and brittle, as cast iron, zinc, or glass ; or tough and ductile, as wrought 

 iron and soft steel ; or flexible, as wood and caoutchouc ; or soft and ductile, 

 as lead and tin. In very flexible bodies it may be observed by drawing fine 

 parallel lines across the side of the bar before the force is applied ; when 

 the niece is strained, the lines become inclined, retaining their original 

 distance apart only at the neutral axis." Now this fallacious statement 

 ffirst made by M. Mariotte, which Mr. Barlow aud a host of others down 

 to Mr Mosely have endeavoured to support by experiments, conjectures, 

 and assertions,) may be exposed in the following descriptive manner and 

 afterwards by a mathematical investigation. Suppose X \ , V W , and Z T to 



