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= 4t13: this being substituted in the expression for w, actp the latter becomes W Id (Ibid., No 314.) 3p From Mr. Ilodgkinaon'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 weight 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 lbs. on the square inch (it is called by some engineers 67,200 lbs.) 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 1'745; whilst that of the beat Dantzic oak will be and that of red deal 00785. 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 of the crushing weight, the following table, translated from Clauders Formules h 1' usage des Ing6nieurs; may be considered to represent the loads it is advisable to bring upon solid wrought iron columns.
Ratio r > 5 10 20 30 40 50 00 70 80 Load in ibm. per In. 11,352 7095 0484 6670 4824 4044 3301 2838 2384 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 Messrs. Chevaudier and Wertheim, ' Sur lee Propri6t6s M6caniques des Bois' 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 npon will materially affect its resistance to compression, for according to Navier and Dulcau, with the majority of woods the resistance is in the ratio of , when the section is triangular ; it is L2 when the section is di square, and 73 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 props, and strained transversely by a weight acting perpendicularly to its length 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 points 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 ease, the beam did not slide on 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 n (fig. 1) be the face of a wall, and let at 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 as P; then, if these fibres were supposed to be rigid and incompressible, the effect of a weight at r would be to bring the beam to an inclined position, as in n, producing a fracture on the line as 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 as bears to the whole depth M @ the ratio of 1 tali- V3, or nearly that of 4 to 11. Therefore, d representing the depth as Q, let 4 OK be represented by 11 11 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 f represent the force of cohesion in the fibre at M P, we shall have 11 and the last term will express the force of 4 11 cohesion between two particles at a distance above o equal to x. Consequently, dx expressing the indefinitely small depth of a fibre, 11 we have f 4d x dx for the cohesive power of a fibre at the same place. But this power acting at a distance from o equal to a., we have 11 f 4d e dx for the momentum of that force; and its integral will express the strength of all the fibres in the vertical section represented by K N. The transverse section of the beam being supposed to be rectangular, the breadth will be constant; let it be represented by b: llb 4 then the integral of f x' dx (between x=0, x= d), that is 1Gb 1 f or nearly f, will express the strength by which all 363d the fibres above the axis of o resist the strain.