Suppose, for example, this constant is formed for the case in which the beam is fixed at one end and loaded at the other, and the weight W were required that would break any given beam, under any of the circumstances stated above, then we should have I. Beam fixed at one end, and loaded at the other S — 1 2. Beam fixed at one end and loaded uniformly throughout w= 2 S b d' 1 3. Beam supported at each end and loaded in the 4 S — 1 4. Beam supported as above and loaded uniformly 8 S W 5. Beam fixed at each end and loaded in the middle W= S bd.
1 6. Beam fixed as above and loaded uniformly W 1 2 S / And in the third and fourth cases, if the load be ap plied in any other point than the middle, then calling nt and n the distance from the ends. The above results are to be respectively divided by 4 inn /' for the resistance in these cases, which thus become 3 S lb W= S lb d' and M 72 2 m n It has been explained under CARPENTRY that we ought to proceed correctly to introduce into these formula the cosine of the beam's deflections, but this leads to considerable intricacy, and is of little value, because we do not generally require to know the actual but rela• tine resisting power, and in all practical cases the de flection is too inconsiderable to be regarded. Indeed it has been asserted that our inquiry ought not to be directed to the ultimate strength ; we think, however, T that it is desirable in all cases to know what this is, and we may then keep as much within the limits as we please, or as the ease seems to require.
The following table, containing the value of S, deduced from a variety of well conducted experiments, is taken from Tredgold's " Elementary Principles of Carpentry." We have only multiplied that author's constant by 3, to make it correspond with our con stant denoted above by S, and added a few other results from Barlow's Essay on the Strength of Timber, Ece.
Comparing the relative strength of oak and cast iron to resist a transverse strain, it appears that the latter is about five times that of the former, while its direct cohesive power is not more than one and a half times, and its weight is about eight or nine times greater than an equal bar of oak.
The application of the numbers given in the above tables to the formula preceding them, is so obvious as to require very little further illustration, we shall therefore confine ourselves to a single example.
To find the weight which, (applied at the centre of a rectangular beam supported at both ends, and of given dimensions,) is necessary to produce fracture.
The rule, in words, is here obviously as follows : Multiply the breadth by the square of the depth, and again by four times the constant value S of the par ticular material given in the table. Then divide the
product by the length, in inches, for the weight re quired.
Of the Stiffizess of Beams in Resisting a Transverse Strain.
In the preceding section our inquiry has been di rected to the ultimate strength of materials, viz. the greatest weight they will support, or rather, perhaps, the least weight that will destroy the beam ; but this in general is not the information a practical man most requires, although, it is obvious, if he knows the ultimate strength, he may keep as much within those limits as his case may seem to require, and thus far the tables and rules laid down in the preceding pages will be found highly useful, but commonly it is not the strength but the stiffness of his beams and rafters, that is of greatest consequence to an archi tect or engineer for a certain deflection, in many cases, is nearly as dangerous or injurious as an actual fracture, we should therefore leave this subject incom plete, if we did not also furnish such experimental data connected with it, as have been obtained from the sour ces already referred to since our article CARPENTRY was published.
When a beam is supported at its two ends, and loaded either at its middle point or uniformly through out, the centre of the beam will sink below the hori zontal line, and this sinking measured where it is greatest, is called the deflection.
The stiffness of a beam is the proportion between its length and deflection, the weight and dll other things being the same, therefore, when the length is different, and the stiffness the the deflection must be proportional to the length. Now, it has been found from numerous experiments, as well as from theoretical investigation, (see CARPENTRY and Bar low's Essay,) that the deflection of beams of the same material similarly loaded, varies as the weight and cube of the length directly, and as the breadth and cube of the depth inversely, or d varies as /3XW but the stiffness is as the length, divided by the de P flection, this therefore varies as . Let now d be an experimental deflection found to be due to a given length 1, depth d, breadth b, and weight W and d, any proposed deflection with the same beam, to find the corresponding weight w that will produce it ; this is found by the simple proportion d: rh : Then this value of w being substituted in the formula LP t (instead of '\V,) ought to produce a constant ' quantity = a, d, being always taken proportional to the length, and from this constant the proper dimen sions in any other case may be computed.