Imágenes de páginas
PDF
EPUB

(425.) The analytical method of investigation and calculation. adopted in the above may be laborious, but it has two great advantages: 1st, it covers and includes all possible cases, of any number of weights, and any method or order of distribution, for obviously it is only necessary to calculate the form and dimensions of beam for each weight independently, and then combine them in the manner we have illustrated; 2nd, this method has the great advantage of allowing every step in the process of calculation to be seen by the operator, and the rationale to be understood.

We have applied the method of finding the theoretical forms to timber beams for convenience of illustration: for that material the parallel form would almost always be adopted in practice from motives of economy of labour, &c.; the saving of material which would accrue from the adoption of the theoretical form would not compensate for the labour and trouble required to obtain it. With cast- and wrought-iron beams the case is different.

(426.) "Equally Distributed Load."-When a load is spread equally all over the length of a beam, it may be considered as divided into any given number of equal weights equidistant from each other, and then the effect of each of these imaginary weights may be separately calculated. Let Fig. 125 be a beam 16 feet long supported at each end, with its load divided into 16 parts-this is equivalent to a cantilever, Fig. 126, of half the length with its load divided into 8 parts, and we require 1st the depth at the points a, b, c, &c., the breadth being constant, and 2nd we require to find the breadth at those points when the depth is constant. For this purpose we must suppose each weight to rest on the beam at two points, each giving half the strain due to the whole weight. Thus the weight W being 1.0 will give a pressure of at a and being also 10 will give at b, and effect of W, and w at b is therefore (427.) The transverse strain at calculated; thus the strain at b is acting with a leverage of 1.0, or ratio of the breadth at that point when the depth is constant;

at b; then the weight at c, &c.; the combined

= 1.0.

every point may now be that due to the weight of x 1 = 0.5, which is the

but if the breadth is constant, we have √0.5.707, which is the ratio of the depth at that point. Again at c we have a weight of with leverage of 2, plus the weight B or 1.0 with a leverage of 1·0 or ( × 2) + (1 × 1) = 2, which is the ratio of the breadth at that point when the depth is constant; but if the breadth is constant, then √2 1.414 is the ratio of the depth at c: again at d, we have (1 × 3) + (1 × 2) + (1 ×1) = 4.5, the ratio of breadth when the depth is constant, and √4.5 2.12, the ratio of the depth when the breadth is constant, &c. Calculating in this way we find that at

=

[blocks in formation]

=

[blocks in formation]

where the depth is constant, the breadths should be in the ratios Fig. 129, or

2.0

4.5 8.0 12.5 18.0 24.5 32

0.0 0.5 but when the breadth of the beam is constant throughout its length, then the depths should be in the ratios Fig. 127, or

0.0 0.707 1.414 2.12 2.83 3.53 4.24 4.95 5.65

It will be observed that when the depth of the beam is constant, the breadths are in the ratio 1, 4, 9, 16, &c., or as the squares of the distances from the end of the beam, as in Fig. 128. But when the breadth is constant, the depths follow the simple arithmetical ratio 1, 2, 3, 4, &c., and the profile of the beam is then a triangle, as in Fig. 127. Applying this to a beam supported at each end and with the load equally distributed, then, when the depth is constant, the breadths are as given by Fig. 129, but when the breadth is constant, the profile is that of two triangles united at the base, as at x x in Fig. 127. The proportions of depths to breadths may be varied to any extent so long as d x b follow the ratio of the middle line above, or 0.5, 2.0, 4.5, &c.

(428.) "Form, as governed by Taste, &c."-The forms of beams which we have thus obtained, although theoretically correct for the transverse strains, are not such as to satisfy the requirements of taste, moreover, they do not provide for the shearing strain at the ends (403) nor for a fair area of bearings at the supports so as to spread over a large surface the insistent

R

weight which otherwise would crush the material,—stone,— brick, &c., on which the beam rests. To meet these requirements, the theoretical forms may be modified at pleasure, care being taken, however, that the sizes demanded by theory are not curtailed by the lines required by taste, convenience, or other considerations. Thus Fig. 117 might be modified to Fig. 130, in which the length is increased by the supplementary pieces m, n, the amount of which must be fixed by judgment so as to give a good bearing. The two semi-parabolas o, p, &c., are the same as in Fig. 117, and the curve r, s, t is an ellipsis which is perhaps the most beautiful of simple curves, and may be. easily described by taking a piece of paper with a perfectly straight edge P, making the distance a, b, equal to S, V, and b, c, equal to U, V; then passing it over the latter so that b and c are always in contact with the major and minor axes of the ellipse respectively, and making dots with a pencil or needle point at b, b, &c., a sufficient number of guide-dots is obtained, through which the perfect ellipse may be drawn by a French curve, &c.

(302

=

(429.) The same principles may be applied to find the section at different points in the length of I girders, whose profile has been determined by taste or convenience. Let Fig. 131 be a girder 16 feet between bearings, resting 18 inches on the wall at each end, 30 inches deep in the centre, whose section there is given at A, and the load being a central one, or by the Pillar E. The strength at the centre (350), or the reduced value of dxb, is for the top flange 2 × 8 = 32; for the vertical web (28° − 2*) × 1 1170: and for the bottom flange 282) × 18 = 2088-altogether 32+ 1170 + 2088 3290. Dividing the extreme half-length of the girder into four parts, we can now determine the strength, or d x bat each point B, C, D, that at A being 3290 on the principles explained in (423) and by Fig. 122. Thus, at D it will be 3290 ÷ 4 = 822; at C, 3290 ÷ 2 1645; and at B, 3290 × 3 ÷ 4 = 2467. The depth of the girder at those points having been predetermined by Fig. 131, we have now to find the breadths of the bottom flange necessary to give the required strength, that of the top flange being maintained at 8 inches throughout. Thus, at B, the depth being 25 inches, we have:

=

819 =

top flange, 22 × 8 = 32; vertical web (23 - 2) × 1 = 787; or 32 + 787 = 819 together; and as we require 2467 at that point, the bottom flange must yield 2467 1648, and as the depth there squared is 252 bottom flange must be 164896

=

232 = 96, the breadth of the 17·1 inches. Similarly,

=

at C we have: top flange, 22 × 8 = 32; vertical web (19a – 22) x 11 = 535; or 32+ 535 = 567 together; hence the bottom flange must yield 1645 567 1078, and the depth squared being 212 192 = 80, the width must be 1078 80 131 inches. Finally, at D we have for the top flange and vertical web (2a × 8) + (162 − 22] × 11⁄2) = 410, 822, the bottom flange must yield 822 depth squared being 182 162 68, 41268 6.06 inches, &c.

=

=

and as we require

410 = 412, and the the width must be

In many cases the form of the bottom flange as thus found would need modification to meet the requirements of taste, care of course being taken that the calculated sizes are not curtailed (428).

(430.) "Effect of Modes of Fixing and Loading."-There are three principal methods of fixing beams:-1st, when supported at the two ends; 2nd, when fixed, or built into walls at the two ends; and, 3rd, when fixed or built into a wall at one end only, the other end being free, and the beam then becomes a cantilever.

With each of these modes of fixing beams there are two principal methods of arranging the load, namely, 1st, a single weight in the centre of beams that are fixed or supported at the two ends; and, 2nd, when the load is distributed equally all over the length. Similarly, with a cantilever, the load may be, 1st, a single one at the remote end, and, 2nd, it may be equally distributed all over the length.

(431.) The ratios of the loads in these various cases are as follow::

Supported at two ends, and loaded in the centre
Supported at two ends, load spread equally all over
Fixed at two ends, and loaded in the centre..
Fixed at two ends, load spread equally all over
Fixed at one end, and loaded at the other
Fixed at one end, load spread equally all over

[ocr errors]

Ratio of Loads.

1.0

2.0

1.5

..

3.0

0.25

0.50

[ocr errors]

Of course the distribution of the load may be varied endlessly; in (419), &c., the whole matter is fully investigated, and the effect on the sizes and forms of beams is considered in detail on a principle that admits of universal application.

The ratios given above are easily applied in practice:- Say that we require the depth of a cantilever of Riga Fir, projecting 5 feet from the wall, to carry safely a load of 1900 lbs. distributed all over, the thickness being 3 inches. We find first from the ratio 0.50 given above that 1900 lbs. equally distributed over a cantilever is equivalent to 1900 ÷ 5 = 3800 lbs. in the centre of a similar beam of the same length supported at the two ends as in the rules in (323), and the value of MÃ for safe load being for Riga Fir, 78 lbs. by col. 3 of Table 67, the

rule d = √{W x L) ÷ (Mb) becomes in our case {3800 × 5) ÷ (78 × 3}√ = 9 inches, the depth required, &c.

LATTICE GIRDERS.

(432.) The investigation of the strains on the several parts of lattice girders is an interesting study on its own account, and is also instructive as illustrating the internal strains in girders of other kinds, where the phenomena are often very obscure. In lattice girders the tensile and compressive strains are confined to certain definite lines formed by the different members of the structure, and this fact enables us to estimate the force, direction, and resultants of those strains with a facility and precision not attainable with girders of other kinds.

The forms of lattice girders are so very variable, that in most cases the strains on the various parts must be found by analysis and reasoning rather than by set rules; but for the main question of the Load which can be borne, the Rules for Plate-iron Girders in (386), &c., will equally apply to Lattice Girders.

(433.) Let A, B, C, Fig. 134, be a triangular frame loaded with a weight W of 1 ton, the angle of the strut B being 45°. The weight W may be resolved into two forces on A and B respectively by the well-known parallelogram of forces; making the diagonal a equal to the weight W by a scale of equal parts, the strain on B will be equal to the length d or e, and will be

« AnteriorContinuar »