A BEAM FREE AT THE ENDS AND LOADED
IN THE MIDDLE.
377. In the preceding lecture we have examined some general circumstances in connection with the condition of a beam acted on by a transverse force; we proceed in the present to inquire more particularly into the strength under these conditions. We shall, as before, use for our experiments rods of pine only, as we wish rather to illustrate the general laws than to determine the strength of different materials. The strength of a beam depends upon its length, breadth, and thickness; we must endeavour to distinguish the effects of each of these elements on the capacity of the beam to sustain its load.
We shall only employ beams of rectangular section; this being generally the form in which beams of wood are used. Beams of iron, when large, are usually not rectangular, as the material can be more effectively disposed in sections of a different form. It is important to distinguish between the stiffness of a beam in its capacity to resist flexure, and the strength of a beam in its capacity to resist fracture. Thus the stiffest beam which can be made from the cylindrical trunk of a tree 1' in diameter is 6" broad and 10"·5 deep, while the strongest beam is 7" broad and 9"·75 deep. We are now discussing the strength (not the stiffness) of beams.
378. We shall commence the inquiry by making a number of experiments: these we shall record in a table, and then we shall endeavour to see what we can learn from an examination of this table. I have here ten pieces of pine, of lengths varying from 1' to 4', and of three different sections, viz. 1" × 1", 1" × 0"·5, and 0"·5 × 0"·5. I have arranged four different stands, on which we can break these pieces: on the first stand the distance between the points of support is 40", and on the other stands the distances are 30," 20", and 10" respectively; the pieces being 4', 3', 2', and 1' long, will just be conveniently held on the supports.
379. The mode of breaking is as follows:—The beam being laid upon the supports, an S hook is placed at its middle point, and from this S hook the tray is suspended. Weights are then carefully added to the tray until the beam breaks; the load in the tray, together with the weight of the tray, is recorded in the table as the breaking load.
380. In order to guard as much as possible against error, I have here another set of ten pieces of pine, duplicates of the former. I shall also break these; and whenever I find any difference between the breaking loads of two similar beams, I shall record in the table the mean between the two loads. The results are shown in Table XXIV.
Table XXIV.—Strength of a Beam.
Slips of pine (cut from the same piece) supported freely at each end; the length recorded is the distance between the points of support; the load is suspended from the centre of the beam, and gradually increased until the beam breaks;
| Formula, P = 6080 | area of section × depth |
| span |
| Number of Experiment. | Dimensions. | Mean of the observations of the breaking load in lbs. | P. Calculated breaking load in lbs. | Difference of the observed and calculated values. | ||
|---|---|---|---|---|---|---|
| Span. | Breadth. | Depth. | ||||
| 1 | 40"·0 | 1"·0 | 1"·0 | 152 | 152 | 0·0 |
| 2 | 40"·0 | 0"·5 | 1"·0 | 77 | 76 | -1·0 |
| 3 | 40"·0 | 1"·0 | 0"·5 | 38 | 38 | 0·0 |
| 4 | 40"·0 | 0"·5 | 0"·5 | 19 | 19 | 0·0 |
| 5 | 30"·0 | 1"·0 | 0"·5 | 59 | 51 | -8·0 |
| 6 | 30"·0 | 0"·5 | 0"·5 | 25 | 25 | 0·0 |
| 7 | 20"·0 | 1"·0 | 0"·5 | 74 | 76 | +2·0 |
| 8 | 20"·0 | 0"·5 | 0"·5 | 36 | 38 | +2·0 |
| 9 | 10"·0 | 1"·0 | 0"·5 | 154 | 152 | -2·0 |
| 10 | 10"·0 | 0"·5 | 0"·5 | 68 | 76 | +8·0 |