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The three last cubes continued to support the weight, although cracked in all directions; they fell to pieces when the load was removed. All began to show irregular cracks a considerable time before it gave way. The average weight supported by these bricks was 33.5 tons per square foot, equal to a column 583.69 feet high of such brickwork. (Fair. bairn, Application, &c., page 192)
1502n. To crush a mass of solid brickwork 1 foot square, requires 300,000 lbs. avoir. dupois, or 134 tons 7 cwt.
15020. Besides compression, stone is subject to detrusion and a transverse strain, as when used in a lintel. Of these strengths in stone little is officially known, but we are perfectly aware of the danger of using any kind of stone for beams where there is much chance of serious or of irregular pressures. Its weakness in respect to this strain is manifest from all experimental evidence concerning it. Gauthey states the value of a constant S, for hand limestone = 78 lbs ; for soft limestone = 69 lbs. Hodgkinson, taking the power of resisting a crushing force as = 1000, notices—
Black marble - - - - - - - 143 and 10:1 Italian marble - - - - - - 85 ,, 10:6 Rochdale flagstone - - - - - - 104 , 9 9 Yorkshire flag - - - - - - O , 9.5 Mean - - - - - - 104 , 10-0
Common bricks, S=64 lbs. (Barlow.)
1502p. The danger above noticed is so great, that it becomes essentially necessary in all rough rubble work to insert over an opening either an iron or timber lintel, or a brick or stone arch, to carry the superincumbent weight, and thus prevent any pressure upon the stone. This must be done more especially when beams or lintels of soft stone are used; the harder stones, as Portland, may in ashlar work support themselves without much danger. In rubble masonry, the stone arch may be shown without hesitation in the face of the work; and also in domestic architecture, the brick arch may exhibit itself in the facework if thought desirable. Portland stone has been constantly used to extend over a comparatively wide opening. All blocks set upon it should have a clear bed along the middle of its length. Thus cills to windows should always be set with clear beds, or, as the new work settles, they are certain to be broken. Lintels over even small openings worked in Bath or some of the softer stones, are very likely to crack across by very slight settlements, especially when supported in their length by a mullion or small pier, as is often introduced. We need hardly add that where impact or collision is likely to occur, no lintel of stone should be used.
15027. Marble mantles may sometimes be seen to have become bent by their own weight. Beams of marble have been employed in Grecian temples as much as 18 feet in the clear in the pro ylaea at Athens; and marble beams 2 feet wide and 13 inches deep were hollowed out, leaving 4% inches thickness at the sides and 3 inches at the bott m; these beams were about 13 feet in the clear in the north portico of the temple at Bassae near Phigaleia.
1502r. The cohesive power of stone is seldom tested. The subject of crushing weights, or the compression of timber and metals, will be treated in a subsequent section (1631e. et seq.); and the strength of some other materials will be given in the chapter MATERIALs.
OF THE STAbility of WALLs.
1503. In the construction of edifices there are three degrees of stability assignable to walls. I. One of undoubted stability; II. A mean between the last; and the III. The least thickness which they ought to possess.
1504. The first case is that in which from many examples we find the thickness equal to one eighth part of the height: a mean stability is obtained when the thickness is one tenth part of the height; and the minimum of stability when one twelfth of its height. We are, however, to recollect that in most buildings one wall becomes connected with another, so that stability may be obtained by considering them otherwise than as independent walls.
1505. That some idea may be formed of the difference between a wall entirely isolated and one connected with one or two others at right angles, we here give figs. 591, 592, and 593. It is obvious that in the first case (fig. 591.), a wall acted upon by the horizontal force MN, will have no resistance but from the breadth of its base; that in the second Fig. 591 Fig. 592. ease (fig. 592.) the wall G F is opposed to the force M.N. so that only the triangle of it III F can be detached; lastly, in fig. 593. the force MIN would only be effective against
ig. 595 the triangle CGH, which would, of course, be greater in proportion to the increased distance of the walls CD, HI. 1506. In the first case, the unequal settlement of the soil or of the construction may produce the effect of the force MN. The wall will fall on the occurrence of an horizontal disunion between the parts. 1507. In the second case the disunion must take place obliquely, which will require a greater effort of the power MN. 1508. In the third case, in order to overturn the wall, there must be three fractures through the effort of MN, requiring a much more considerable force than in the second case. 1509. We may easily conceive that the resistance of a wall standing between two others will be greater or less as the walls CD, HI are more or less distant; so that, in an extreme approximation to one another, the fracture would be impossible, and, in the opposite case, the int. rmediate wall approaches the case of an isolated wall. 1510 Walls enclosing a space are in the preceding predicament, because they mutually tend to sustain one another at their extremities; hence their thickness should increase as their length increases. 1511. The result of a vast number of experiments by Rondelet, whose work we are still using, will be detailed in the following observations and calculations. 1512. Let ABCD (fig. 594.) be the face of one of the walls for enclosing a rectangular
* ~~ ~~~~~~~~ 2:
Fig. 594, D' D **, *.
space, EFG II (fig. 595.). Draw the diagonal BD, and about 13 make Bd equal to one eighth part of the height, if great stability be required; for a mean stability, the ninth or tenth part; and, for a light stability, the eleventh or twelfth part. If through the point d a parallel to A B be drawn, the interval will give the thickness to be assigned to the great walls EF, GH, whose length is equal to AD. 1513. The thickness of the walls EG, FH is obtained by making AD equal to their length, and, having drawn the diagonal as before, pursuing the same operation. 1514. When the walls are of the same height but of different lengths, as in fig. 596.,
the operation may be abridged by describing on the point B (fig. 597.) as a centre with a radius equal to one eighth, one tenth, or one twelfth, or such other part of the height as may be considered necessary for a solid, mean, or lighter construction, then transferring their lengths, E.F, FG, GH, and HE from A to D, D, D", and D’”; and having made the rectangles AC, AC, AC", and AC", draw from the common point B the diagonals BD, BD, BD", and BD”, cutting the small circle described on the point B in different points, through which parallels to AB are to be drawn, and they will give the thickness of each in proportion to its length. 1515. In figs. 598. to 602. are given the operations for finding the thicknesses of wails
enclosing polygonal areas supposed to be of the same height; thus AD, represents, the side of the hexagon (fig. 602.); AD that of the pentagon (fig. 601.): AD" the side of the square (fig. 599.); and AD" that of the equilateral triangle (fig. 600.). 1516. It is manifest that, by this method, we increase the thicknesses of the walls in proportion to their heights and lengths; for one or the other, or both, cannot increase or diminish without the same happening to the diagonal. 1517. It is obvious that it is easy to calculate in numbers the results thus geometrically obtained by the simple rule of three; for, knowing the three sides of the trial ge. ABD. similar to the smaller triangle Bde, we have BD : Bd:: AD : ed. Thus, suppose the length of wall represented by AD=28 feet, and its height AB = 12 feet, we shall have the length of the diagonal = 30 feet 5 inches; and, taking the ninth part of AB, or 16 inches, as the thickness to be transferred on the diagonal from B to d, we have 30 ft. 6 in. : 16 in. ::28 ft. : 14 in. : 8 lines (ed). The calculation may also be made trigonometrically: into which there is no necessity to enter, inasmuch as the rules for obtaining the result mav be referred to in the section “Trigonometry,” and from thence here applied.
Method of enclosing a given Area in any regular Polygon.
1518. It is manifest that a polygon may be divided by lines from the centre to its angles into as many triangles as it has sides. In fig. 601., on one of these triangles let fall from C (which is the vertex of each triangle) a perpendicular CD on the base or side A B which is supposed horizontal. The area of this triangle is equal to the product of DB (half A B} by CD, or to the rectangle DCF B. Making DB =r, CD =y, and the area given = p, w. shall have, For the equilateral triangle, r x yx 3=p, or ry== #; For the square, ry x 4=p, or xy=';
Each of these equations containing two unknown quantities, it becomes necessary to as
certain the proportion of x to y, which is as the sines of the angles opposite to the sides IDB and CID.
1519. In the equilateral triangle this proportion is as the sine of 60 degrees to the sine of 30 degrees; that is, using a table of sines, as 86603 : 50000, or 83 : 5, or 26 : 15, whence
Supposing the area given to be 3600, we shall therefore have
1520. Suppose the case that of a pentagon (fig. 601.) one of whose equal triangles is ACB. Let fall the perpendicular CD, which divides it into two equal parts; whence its area is equal to the rectangle CDBF.
1521. Upon the side AB, prolonged, if necessary, make DE equal to CD, and from the middle of BE as a centre describe the semi-circumference cutting CD in G, and G D will be the side of a square of the same area as the rectangle CDBF. The sides of similar figures (Geometry, 961.) being as the square roots of their areas; find the square root of the given area and make Dg equal to it. From the point g draw parallels to GE and GB, which will determine on AB the points e and b, and give on one side Db equal to one half of the side of the polygon sought; and, on the other, the radius De of the circumference in which it is inscribed. This is manifest because of the similar triangles EGB and eyb, from which BD: DE :: b1) : De.
1522. From the truth that the sides of similar figures are to each other as the square roots of their areas we arrive at a simple method of reducing any figure to a given area. Form an angle of reduction (fig. 603.) one of whose sides is equal to the square root of the greater area, and the chord of the arc, which determines the size of the angle equal to the square root of the smaller area. Let, for instance, the Z" larger area = 1156, and that of the smaller, to which the figure & is to be reduced, =529. Draw an indefinite line, on which make AB =34, the square root of 1156. Lastly, from the point A, as a centre, having described an indefinite arc, with a length equal to the square root 23 of 529, set out Bg; through
g draw Ag, which will be the angle of reduction g AB, by means à of which the figure may be reduced, transferring all the mea- d sures of the larger area to the line Al), with which arcs are Fig. 503.
to be described whose chords will be the sides sought.
1523. If it be not required to reduce but to describe a figure whose area and form are given, we must make a large diagram of any area larger than that sought, and then reduce it.
1524. The circle, as we have alreadv observed in a previous subsection (93.3.), being but a polygon of an infinite number of sides, it would follow that a circular enclosure would be stable with an in initely small thickness of wall. This property may be easily demonstrated by a very simple experiment. , Take, for instance, a sheet of paper, which would not easily be made to stand while extended to its full length, but the moment it is bent into the form of a cylinder it acquires a stability, though its thickness be not a thousandth part of its height.
#. But as walls must have a certain thickness to acquire stability, inasmuch as they are composed of particles susceptible of separation, we may consider a circular enclosure as a regular polygon of twelve sides, and determine its thickness by the preceding process. Or, to render the operation more simple, find the thickness of a straight wall whose length is equal to one half the radius.
1526. Suppose, for example, a circular space of 56 ft. diameter and 18 ft. high, and the thickness of the wall be required. Describe the rectangle ABCD (fig. 594.), whose base is equal to half the radius, that is, 14 ft., and whose height AB is 18 ft.; then, drawing the diagonal BD, make Bal equal to the ninth part of the height, that is, 2 ft. Through d draw ad parallel to the base, and its length will represent the thickness sought, which is 144 inches.
1527. By calculation. Add the square of the height to that of half the radius, that is, of 18 =324, and of 14= 196 (=520). Then extract the square root of 520, which will be found = 22.8, and this will be the value of the diagonal B.D. Then we have the following proportion: 22:8: 14 :: 2 ft. ( the height): 14.74.
1528. The exterior wall of the church of St. Stefano Rotondo at Rome (Temple of Claudius) incloses a site 198 feet diameter. The wall, which is contructed of rubble masonry faced with bricks, is 2 ft. 4 in. (French) thick, and 223 ft. high. In applying to it the preceding rule, we shall find the diagonal of the rectangle, whose base would be the side of a polygon, equal to half the radius and 224 ft. high, would be v494 x 49' 4 22 x 22}=541%. Then, using the proportion 54:37: 49.5 :: # : 2 ft. 3 in. and 4 lines, the thickness sought, instead of 2 ft. 4 in., the actual thickness. We may as well mention in this place that a circle encloses the greatest quantity of area with the least quantity of walling; and of polygons, those with a greater number of sides more than those with a lesser: the proportion of the wall in the circle being 31416 to an area of 78540000; whilst in a square, for the same area, a length of wall equal to 35448 would be required. As the square falls away to a flat parallelogram, say one whose sides are half as great, and the others double the length of those of the square, or 17724 by 4431, in which the area will be about 7854.0000, as before; we have in such a case a length of walling = 44310.
On the Thickness of Walls in Buildings not vaulted.
1529. The walls of a building are usually connected and stiffened by the timbers of the roof, supposing that to be well constructed. Some of the larger edifices, such as the ancient basilicae at Rome, have no other covering but the roof; others have only a simple ceiling under the roof; whereas, in palaces and other habitations, there are sometimes two or more floors introduced in the roof.
1530. We will begin with those edifices covered with merely a roof of carpentry, which are, after mere walls of enclosure, the most simple.
1531. Among edifices of this species, there are some with continued points of support, such as those wherein the walls are connected and mutually support each other; others in which the points of support are not connected with each other, such as piers, columns, and pilasters, united only by arcades which spring from them.
1532. When the carpentry forming the roof of an edifice is of great extent, instead of being injurious to the stability of the walls or points of support, it is useful in keeping them together.