37 



RESISTANCE OF MATERIALS. 



RESISTANCE OF MATERIALS. 



found that the resistance to compression was much greater when the 

 effort was applied in a direction perpendicular to the bedding, than 

 when it was applied in a direction parallel to the beds ; and in fibrous 

 materials the resistance to efforts, either of extension or of compres- 

 sion, is the greatest when those efforts are applied in the direction of 

 the fibres. In this last-named class of materials it is especially 

 necessary to preserve them from flexure in their length ; and thence 

 also the necessity for observing the proportions before stated between 

 the various conditions of base, height, and load. As it has been 

 ascertained, theoretically and experimentally, that when the mass of a 

 body Is arranged in the form of a hollow body, the resistance is nearly 

 doubled, (when the thickness of the cylinder is made about .-5 of the 

 diameter,) it becomes an additional reason for using hollow columns of 

 metal to support heavy loads, because, in the first place, the powers of 

 resistance to compression are increased, and, in the second, there is 

 less danger of flexure, when the diameter of the body is thus made as 

 large as possible. 



In the works of Tredgold, Hodgkinson, Tate, Barlow, Moseley, 

 Willis, Whewell, Morin, Navier, Prony, Bresse, Claudel, Boordau, 

 Daguin, Jamin, &c., the various conditions of the resistance of solid 

 bodies, and of the forms of greatest resistance, are discussed in great 

 detail; and the reader is referred to them, should he require to 

 examine any complicated problem of this description. It may suffice 

 here to say that the condition most commonly occurring in practice is, 

 when a rectangular beam is exposed to a load acting either longitudinally, 

 or transversal!}-, to its axis. In the former case the whole action is 

 either of compression or of extension, as the case may be, and in 

 addition to what has been before stated, it is only necessary to observe 

 that it is essential, in order that the action should be uniform, that the 

 load should be brought to bear evenly over the whole area. When 

 beams are, however, exposed to efforts acting transversally to their axes, 

 the laws of their resistance become more complicated, for the deflections 

 produced cause some of the fibres to pass into a state of tension, whilst 

 some of the others are compressed, and some of them remain in a 

 neutral state, as long as the limit* of elasticity of the extended or 

 compressed fibres are not excee led. The modes of loading solid 

 bodies are usually considered to be classed under the following beads, 

 and the formula; for calculating their resistances have been deduced 

 from the known laws of mechanics. 1. When one extremity of the 

 beam is firmly embedded in masonry at one end and acted upon by a 

 force, p, applied at the other ; then, calling the lever of the beam L ; 

 the resistance to compression and extension, R ; the moment of inertia 

 of the beam at the point of its bedding, I ; and the distance of the 

 line of the neutral fibres from the most distant point of the section of 

 the part fixed, , 



*-!, 



As in a regular prism of a rectangular section n - _ and the 



4A* 



nft/i 



moment of inertia is I - _; this formula becomes PL = _ 1" in 



6 12 ' 



which It = the transverse section of the prism perpendicular to the 

 direction of the force p, and A = the section parallel to that direction. 



The deflection / would be represented by / = ; in which the new 



E'f/l-' 

 term E represents the modulus of elasticity. 



2. If the solid be of the form shown, then calling 6' the 

 internal dimension in the same direction as 4; and A' the 

 internal ilimcnqion in the same direction as A ; then, 



n(U J --///*) 

 p i. = - ^ ^ - ; and 



4PL 1 



f~ F(iA' VV>) retaining the preceding notation. 



3. In a beam of the section in the margin, calling V the 

 sum of the two deficiencies from the full section, and A their 



R(4A-4'A' S ) 4 PL 



h e ,ght,thenalBO,PL = 



4. The section being circular, and the radius = r ; then the formulio 



iwr 3 4 p i J 



become PL = __,nd/=_ - ; when the section is circular and 



4, SWEI-* 



hollow, calling r the external, and r 1 the internal radius, the formuUe 



become PL 



4r 



; and /= 



4l> '-' 



5. If now the beam under consideration be carried at a point in 

 its length, and be acted upon at its extremities by two forces which 

 balance one another upon this point of support, the formula for a pris- 

 matic beam of a rectangular section becomes, (calling m the leverage 

 of the force acting at OIK; end, and n the leverage of the force acting 

 on the other ; so that m + n = L the total length of the prism, and p -(- 

 JM + 'in _ Rl'li" L 



q = p the total load ;) z --- ^~- und if m=n = -. or if the point 



1*L It Wl* 



of support be in the middle of the length, then -j- = ~g~. 



6. If the load applied to a prism, fixed at oue of its extremities, 

 instead of being applied at the other extremity, be evenly distributed 

 over its length, then the load per unity of length being called p, the 



total load becomes pi. ; and t, the leverage of the total load pL ; the 

 formula; expressing the fundamental conditions of resistance become 

 L = /, retaining the preceding notation for the old 



terms. From these formulic, it appears that a beam loaded uniformly 

 over its whole length, can resist an effort which would be double the 

 one required to break it if applied only at the extremity the farthest 

 removed from the section of rupture ; and in order to produce equal 

 deflections at the extremity, the load, distributed evenly over the 

 whole length, should bear to the load applied solely at the extremity, 

 the proportion of 8 : 3. 



7. In the case of a beam resting upon two points of support at its 

 extremities, if we suppose that the weight of the beam itself can be 

 neglected, and that the weight, or load, P, be placed on the middle of 

 its length ; then as the effect upon the beam would be the same as if 

 it had been fixed in the middle, and loaded at each end, by a weight 



= -, the first get of formula; would apply, excepting that r would be re- 

 2 



placed by - and L by ; eo that for a rectangular prism the formula; 



become ^=?E; and /= FL 

 4 n 4Ei/i :l 



From this it appears that a beam 



supported at the extremities, and loaded in the middle, is able to 

 support a load four times as great as a similar beam fixed at one end 

 and loaded in the middle ; and that the deflection would be sixteen 

 times less than in the latter case. If the load, instead of being con- 

 centrated in the centre, were evenly distributed over the length, and 

 the load per unity of length be called p, the total load would be pL f 



and the formula; would become * = ; and / = ^il. 



8 n 334 EI 



It has been proved, both theoretically and practically, that a beam 

 faatened at both ends will bear a load applied in the middle of its clear 

 span, which would be double the one it would be able to support if it 

 merely rested upon two points of support, and that the deflection in 

 the former case would be four times less than it would be in the 

 latter. As it generally happens that in building operations, beams 

 and joists have bearings of only about one foot (which is insufficient 

 to constitute an effective fixing of their extremities), it is essential to 

 calculate their dimensions on the supposition that they are merely 

 beams resting on two points of support. 



Before closing these remarks on compression and extension, it may 

 be desirable to add a few practical observations on the resistance of 

 materials used in buildings. These are, 1st, that as the form of 

 greatest resistance is almost always one in which the mass of the 

 elements is concentrated at the top and bottom of a beam, leaving the 

 portion about the neutral axis as light as possible, it follows that with 

 plastic materials it is advisable to make the sections of the beam of a 

 girder shape, that is to say with top and bottom flanges : the material 

 must be distributed in those flanges according to its powers of resist- 

 ance to efforts either of compression or of extension. [GiRDER.] 2nd, 

 It is usually considered that a load acting with a shock, or able to 

 produce sudden vibrations, acts in a manner far more injurious than 

 if the same load were to act steadily ; and in practice engineers have 

 adopted the rule of never exposing a construction to a rolling weight 

 greater than j of the permanent breaking weight. 3rd, It is usually 

 considered that a deflection of ^ of the span, is the maximum which 

 should be tolerated ; but that the safe deflection would only 1 > 

 of the span. 4th, The resistance of cast iron to compression is, com- 

 pared to its resistance to extension, as 6.J (nearly) to 1 ; on the contrary 

 the resistance of wrought iron to compression is, compared to its 

 resistance to extension, as 4 to 5. 



Tariion. With respect to the resistance to torsion it may bu 

 observed, that in a prismatic body submitted to such an action, the 

 relation of the effort to the angle of torsion is constant for the same 

 material, so long as the limits of elasticity are not exceeded. Calling 

 this relation o ; the effort Q ; and the angle of torsion 6, for a rod of a 



given unity of length and of section ; we have = a, which may be 



called the co-efficient of torsion. If then, we call r the force tending 

 to twist a cylindrical or prismatic solid, in a plane normal to the axis; 

 n, the radius of the leverage with which P acts ; t, the angle of torsion ; 

 L, the length of the solid ; and j, the moment of polar inertia ; we have 



o t 

 the moment of the force p = p B = I ; and from this we derive t 



ui 



It is considered that the value of I becomes, 



when the section is circular, i = ; 



. 4 : W 

 when the section is rectangular, i= 0/^2 + / 



