25 



MATERIALS, STRENGTH OF. 



MATERIALS, STRENGTH OF. 



620 



particles in some of the transverse sections are forced outwards by 

 lateral pressures arising from the particles above and below their 

 intervals being thrust between them, and then the pillar swells on its 

 whole periphery. The consequence in either case is, that the cohesion 

 of the longitudinal fibres is impaired or destroyed, and the pillar is at 

 length broken or crushed. 



The strength of a pillar when so compressed must evidently depend 

 upon the number of particles in a transverse section that is, upon the 

 area of such section; but since, besides the displacement of those 

 particles from the longitudinal pressure, their lateral cohesion must 

 be overcome before they can be thrust outwards, it is evident that the 

 strength is not proportional to the area, simply, but to some function 

 of that area. No law on which any dependence can be placed has yet 

 been discovered for the strength of a pillar in such circumstances. 

 Euler, from analytical considerations, concluded that it varies as the 

 square of the area ; but late engineers have supposed that the square 

 root of the third power of the area more correctly represents the law 

 of the strength. 



If a bar or pillar, resting on one end in a vertical position, and con- 

 sidered as a perfectly elastic body, be compressed by a weight acting 

 vertically above it, the purely mathematical theory gives the following 

 equation for the value of the compressing weight when the pillar 

 begins to bend : 

 w= *L^LS (Poisson, ' Mecanique,' torn, i., No. 313); where w= the 



compressing weight; Z=the length of the pillar; o=the area of the 

 urge section; <Z=the thickness perpendicularly to the bending 

 surface; 8=the element of deflection ; and ir = 3'1416. It followsthat, 

 when in two bars of like material a and d are respectively equal, the 

 weights which those bars will sustain without bending are inversely 

 proportional to the squares of the lengths. 



It is also found, if w be a weight applied as above, and producing a 

 flexure p, measured at the middle of the bar perpendicularly to its 



length, that 8 * : this being substituted in the expression for w, 

 a&p 



the latter becomes w = *. (Itid., No 314.) 



Sp 



From Mr. Hodgkinson's experiments on the resistance of columns 

 it appears that the shape of their bearing-ends has an important 

 influence upon their power of supporting an insistent weight; for 

 columns with flat and firmly bedded ends carried about three times the 

 'it which broke other columns with rounded ends. It is customary 

 at the present day to consider that the resistance of wrought iron to 

 an effort of compression may be considered to be, at the minimum, 

 66,760 Ibs. on the square inch (it is called by some engineers 67,200 

 Ibs.) ; and according to Mr. Hodgkinson, if the resistance of cast iron 

 be taken as unity, that of cast steel will be 2'518 ; that of wrought 

 iron 1745; whilst that of the best Dantzie oak will be 0-1088 ; and 

 that of red deal 0'0785. In the cases of cast iron or of other columns 

 the resistance is, however, affected] by the ratios of the diameters to 

 the length ; and if the value of the resistance be taken at 56,760, and 

 the safety load be taken at J of the crushing weight, the following table, 

 translated from Claudel's ' Formules a 1' usage des Ingemeurs,' may be 

 it -red to represent the loads it ia advisable to bring upon solid 

 wrought iron columns. 



Load in Ibs. per in. 



This table is taken in preference to the one given in the same work for 

 the strength of cast iron pillars ; because the latter table is based upon 

 a coefficient of resistance which seems to be rather too high, and it 

 would therefore be preferable to affect the results of the table given by 

 the ratios of the various materials above quoted. The various essays 

 by Meagre. Chevaudier and Wertheim, ' Sur les Propri<5tes M<5caniques 

 .is ' should be consulted by all engineers and architects who may 

 be called upon to use those materials under heavy loads. It must also 

 be observed that the shape of the column or body operated upon will 

 materially affect its resistance to compression, for according to Navier 

 and Duleau, with the majority of woods the resistance is in the ratio 



of _, when the section is triangular ; it is when the section is 



ii 4 

 square, and -p when the section is circular. 



The most important inquiry concerning the strength of materials is 

 that which relates to a beam or bar supported at its extremities on two 

 . and strained transversely by a weight acting perpendicularly to 

 ,'th at a given point between the props. 



In order to simplify the investigation, it is usual to imagine that the 

 beam, its breadth and depth being supposed uniform, is made to rest 

 on one prop at the place where the weight may have been applied in 

 the former case, suppose in the middle of its length, and that from the 

 point* where the two props were situated weights are suspended equal 

 to the reactions of those props in consequence of the first weight ; 



that is, to half the whole weight in the middle. Then, supposing the 

 deflection of the beam to be very small, so that, in the former case, 

 the beam did not slide ou its point of support, the effect of the two 

 weights to break the beam on its single prop will be the same as that 

 of the one weight applied as at first supposed. Again, if a beam of 

 equal dimensions with respect to breadth and depth were fixed at one 

 end horizontally in a wall, the part projecting from the face of the 

 wall being equal in length to half that of the former beam ; and if a 

 weight were applied at the opposite end equal to each of the two 

 weights applied to the beam on one prop, the effect of this weight to 

 break the beam at the face of the wall will be equal to that of the two 

 weights to break the beam on the one prop, or of the double weight to 

 break the same beam on two props. The investigation for the case at 

 first supposed is therefore reduced to that of finding the strength of a 

 beam attached at one end to a wall, and strained by a weight at the 

 opposite extremity. 



Let A B (fig. 1) be the face of a wall, and let M N represent a vertical 

 section of the beam in the direction of its length. Let it be supposed 

 that the beam consists of an infinite number of fibres parallel to M p ; 

 then, if these fibres were supposed to be rigid and incompressible, the 

 effect of a weight at p would be to bring the beam to an inclined 

 position, as m n, producing a fracture on the line M Q by drawing the 

 particles on that line away from those which were at first nearly in 

 contact with them. But from experiment it is found that, when a 

 beam is so strained, while the upper fibres are in a state of tension, the 

 lower ones are in a state of compression ; and consequently that there 

 is a certain point o in the depth of the beam at which neither of these 

 effects takes place. A line passing through this point perpendicularly 

 to the plane M N is therefore called the neutral axis of the beam, and 

 the termination of the fracture may be supposed to be at o instead of 

 Q ; the fibres below the former point having no effect in resisting the 

 tendency of those above to be broken, yet constituting part of the 

 strength of the beam by the power with which they resist compression, 

 and thus oppose the tendency of the beam to turn about the neutral 

 axis. The position of this neutral axis is uncertain ; but Mr. Barlow, 

 from experiment, has found that in rectangular beams of wood (the 

 faces being in vertical and horizontal positions) its distance from the 

 upper surface at M bears to the whole depth M Q the ratio of 1 to 1 + -\/3, 

 or nearly that of 4 to 11. Therefore, d representing the depth M Q, let 



o M be represented by d. 



Now adopting the hypothesis of Leibnitz, which is founded on the 

 elasticity of the fibres, that the force of cohesion in any one fibre is 

 proportional to the tension to which it is subject, or to the distance of 

 that fibre from the axis about which the beam turns in consequence 

 of the strain; that is, from the neutral axis just mentioned: if 

 x be the distance of any fibre above o from the latter place, and / 

 represent the force of cohesion in the fibre at MP, we shall have 



;pr d : f : : x : f TjZ- ar 'd the last term will express the force of 



cohesion between two particles at a distance above o equal to x. 

 Consequently, dx expressing the indefinitely small depth of a fibre, 



we have / -r-, x dx for the cohesive power of a fibre at the same place. 

 But this power acting at a distance from o equal to x, we have 

 / -r-j .1? dx for the momentum of that force ; and its integral will 



express the strength of all the fibres in the vertical section represented 

 by MN. The transverse section of the beam being supposed to be 

 rectangular, the breadth will be constant ; let it be represented by b : 



116 4 



then the integral oif-^x'dx (between x=0, x= jy d), that is 



jjgjr^ f> or near 'y 23 bd-f, will express the strength by which all 



the fibres above the axis of o resist the strain. 



A corresponding expression for the strength arising from the resist- 

 ance of the fibres below the neutral axis to the force of compression 



116 7 



would be the integral of / -=-j- 3? dx (between x=0, x= jj d), that 



2 

 is, nearly r; id 2 /' (/' being the force by which a fibre at Q N would 



resist compression), and the sum of the two integrals will bo the 

 whole strength of the beam to resist a transverse strain. Now the 



