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A different Application of the preceding Example.
1422. The model (fig. 580.) is an arch similar to that of the preceding example, having a story above it formed by two walls, whose height is 100, and the whole covered by a timber roof. The object of the investigation is to ascertain what change may be made in the thickness of the piers which are strengthened in their resistance by the additional weight upon them.
1423. The simplest method of proceeding is to consider the upper walls as prolongations of the piers.
1424. In the model the walls were made of plaster, and their weight was thus reduced to 3 of what they would have been if of the stone used for the models hitherto described. The roof weighed 12 ounces. We shall therefore have that 100, which in stone would have represented the weight of the walls, from the difference in weight of the plaster, reduced to 75. In respect of the roof, which weighed 12 ounces, having found by experiment that it was equal to an area of 576 lines of the stone, both being reduced to equal thicknesses, we have 12 ounces, equal to an area of 13.82 whose half
6'91 must be added to that of the vertical efforts represented by b in
: and # Changing these terms into: and #. the formula becomes
The height of the piers or a in the formula = 183 + 75 =258.
p does not change its value, therefore 2p (as in the preceding example)=265.86.
d, the difference between the height of the pier and the arm of the lever, will =75.
Again,'-3122. Substituting these values in the formula, we shall have a = x/265.86–77-28 +31-22-5-58-9-15.
In the model a thickness of 11 lines was found sufficient to resist the thrust, and taking the root of double the thrust the result is 13 lines.
1425. By the geometrical method, given in the last, taking from the result 17+ lines, there found, the value of #. that is, 5:58, the remainder 113 lines is the thickness sought.
1426. It may be here observed, that in carrying up the walls above, if they are set back from the vertical BF in hf. the model required their thickness to be only 6 lines, because this species of false bearing, if indeed it can be so called, increases the resistance of the piers.
This was a practice constantly resorted to in Gothic architecture, as well as that of springing pointed arches from corbels, for the purpose of avoiding extra thickness in the walls or piers.
Another Application of the Principles to a differently constructed Arch.
1427. The model (fig. 581.) represents an arch of 11 voussoirs whereof 10 are with crossettes or elbows, which give them a bearing on the adjoining horizontal courses; the eleventh being the keystone. The opening is 9 inches or 108 lines, as in the preceding examples.
1428. Having drawn the lines BF, FC, the secant FO, and the horizontal line IKL, independent of the five courses above the line FC of the extrados, we have
P R Fig. 31.
a, the height of the pier, = 198'00. The area KFCL of the upper part of the arch will be 1223-10, from which subtracting that of the triangle FKG, which is 590-82, the remainder 832.28 being multiplied by 30-73 and divided by 32.7 makes the effort of this part 782:44. 1429. The area of the lower part is 697.95, to which adding the triangle FKG = 390-82, we have 1088-77, which multiplied by 23.27 and divided by 52.15, gives 485-82 for its effort. The expression of the thrust, represented by p in the formula,
b, representing the sum of the efforts of the semi-arch, will be **- 1762-03 b_1762.8 g. bb #=''-89 and : - - - = 79-21 Substituting these values in the formula, we have the equation * = x/593:24-100-64 + 79.21 – 8-9-15-ol.
By taking double the square root of the thrust the result is 2391, a thickness evidently too great, because the sum of the vertical efforts, which are therein neglected, is considerable. 1430. The geometrical method gives 19 lines. The least thickness of the piers from actual experiment was 16 lines. 1431. Rondelet gives a proof of the method by means of the centres of gravity, as in some of the preceding examples, from which he obtains a result of only 13:26 for the thick
ness of the piers.
Consideration of an Arch whose Voussoirs increase towards the Springing.
1432. The model (fig. 582.) has an extrados of segmental form not concentric with its intrados, so that its thickness increases from the crown to the springing. The opening is the same as before, namely, 9 inches, or 108 lines. The thickness at the vertex is 4 lines, towards the middle of the haunches 7 lines, and at the springing 14 lines. The centre of the line of the extrados is one sixth part of the chord AO below the centre of the intrados; so that
The radius DN = 68.05
IK = 15.82
The arc BK =KC=42-43
1433. The area KHDC of the upper part of the arch is 258-75,
that of the lower part BAHK 486.5; hence the effort of the upper part is represented by the expression *#"- 232°47.
1434. The half segment ABebeing supposed to be united to the pier; BeBK, whose area is 178, is the only part that can balance the upper effort; its expression will be '*=6624. The difference of the two efforts 166-23 will be the expression of the thrust represented by p in the
These values being substituted in the formula, will give
1435. The smallest thickness of pier that would support the arch in the model was 17] lines.
1436. With the geometrical method, instead of the double of CD, make Bh double the mean thickness HK, and Bn equal to mL, and on nh as a diameter describe the semicircumference cutting OB produced in E; then EB= 18 lines will be the thickness sought.
1437. If the pier is continued up to the point e where the thickness of the arch is disengaged from the pier, the height of the pier represented in the formula by a will be 151-5
instead of 120, and the difference b, instead of being *::::A; will be only *:::: =277.46.
1438. d, expressed by Ie, will be 6'5, all the other values remaining the same as in the preceding article, the equation is
1439. Using the method by means of the centres of gravity, Rondelet found the result for the thickness of the piers to be 15'84. So that there is no great variation in the different results. 1440. In the preceding examples arches have been considered rather as arcades standing on piers than as vaults supported by walls of a certain length. We are now about to consider them in this last respect, and as serving to cover the space enclosed by the walls. In respect of cylindrical arches supported by parallel walls, it is manifest that the resistance they present has no relation to their length; for if we suppose the length of the vault divided into an infinite number of pieces, as C, D, E, &c. (fig. 584. No. 2.), we shall find for each of these pieces the same thickness of pier, so that all the piers together would form a wall of the same thickness. For this reason the surfaces only of the arches and piers have been hitherto considered, that is, as profiles or sections of an arch of any given length. Consequently it may be said that the thickness of wall found for the profile in the section of an arch would serve for the arch continued in length infinitely, supposing such walls isolated and not terminated or rather filled by other walls at their ends. When cylindrical walls are terminated by walls at their extremities, after the manner of gable ends, it is not difficult to imagine that the less distant these walls are the more they add stability to those of the arch. In this case may be applied a rule which we shall hereafter mention more at length under the following section on Walls. 1441. If in any of the examples (fig. 582. for instance) PR be produced indefinitely to the right, and from R on the line so produced the length of the wall supporting the arch be set out, and if from the extremity of such line another be drawn, as TB produced through B, indefinitely towards a, and Babe made equal to the thickness of the pier first found, a vertical line let fall from a will determine the thickness sought. When arches are connected with these cross walls, the effect of the thrust may be much diminished if they are not very distant. If there be any openings in the walls, double the length of them must be added to that of the wall as well as of any that may be introduced in the gable wall. 1442. Fig. 583. represents the mode in which an arch fails when the piers are not of sufficient strength to resist the thrust: they open on the lower part of the summit at DM and on the upper part of the haunches at HN; from which we may infer that the thrust of an arch may be destroyed by cramping the under side of the voussoirs near the summit and the upper side of those towards the middle of the haunches; and this method is greatly preferable to chains or iron bars on the extrados, because these have no effect in preventing a failure on the underside. Chains at the springing will not prevent failure in arches whose voussoirs are of equal depth but that too small, inasmuch as there is no counteraction from them against the bulging
that takes place at the haunches, like a hoop loaded when its ends are fixed. The most advantageous position for a chain to oppose the effort of an arch is to let it pass through the point K where the efforts meet. PC is the tangent before failure, and O the centre; R. being the inner point of the pier.
1443. M. Frezier, in speaking of the thrust of this sort of arches, proposes, in order to find the thickness of the piers which will support them, to find by the ordinary manner the thickness suitable to each part of the cylindrical arch BN, BK (No. 3. fig. 584.) by which the groin is formed, making BE the thickness suitable to the arch BN, and BF that which the arch BK requires; the pier BEHF would thus be able to resist the thrust of the quarter arch OKBN. According to this method we should find the bay of a groined arch 9 inches opening would not require piers more than 21 lines square and 120 lines high; but experience proves that a similar arch will scarcely stand with piers 44 lines square, the area of whose bases are four times greater than that proposed by M. Frezier.
Method for groined Vaulting.
- 1444. The model in this case (see the
* = V2p+---aa-aLetting a always stand for the height, and d for TI of the profile, the arm of the lever of the thrust will, as before, be a +d, and its algebraic expression be pa: 1450. The pier resists this effort by its cube multiplied by the arm of its lever. If the lines KB and OB of the triangle BKO,(which represents the projection of that part of the vault for which we are calculating) be produced, it will be seen that the base of the pier to resist the thrust will be represented by the opposite triangle BHF, which is rectangular and isosceles;
therefore, letting a represent its side BF, the area of the triangle will be expressed by #. the
height of the pier being a, its cube will be #. The arm of the lever of this pier will be
determined by the distance of the vertical let fall from its centre of gravity on the line HF = 5. which gives for the pier's resistance 2:. 1451. This resistance will be increased by the vertical effort of each part of the vault multiplied by the arm of its lever. That of the upper part will be expressed by its cube multiplied by the vertical KM, and the product divided by the mean arc KG. The cube of this part will be equal to the mean area; that is, the arc KG multiplied by the thickness of the vault. 1452. To obtain the mean area, multiply KG less KM by the length GO taken on the plan. The length of the arc KG being 46 and KM 17), we shall have KG-KM=28}; G0 being 54, the mean area will be 28' x 54=1558. This area multiplied by 9, the thickness of the vault, makes the cube of the upper part 140244, which multiplied by KM=17} and divided by the arc KG=46, makes 5226; the value of the vertical effort of the part of the arch m in the formula; and the arm of its lever is IK+ i H. 1453. IK being = c and iH = x, its expression will be ma' + mc. The vertical effort of the lower part will be represented by its cube multiplied by TI, and the product divided by the length of the are TK. This cube will be found by multiplying the mean area by the thickness of the vault. The area being equal to the arc TK-TIx GO, that is, 46–41 fix 54=250 for the mean area and 250 x 9=22563 for the cube of the lower part of the vault. This cube
multiplied by TI and divided by the arc TK gives **-*. for the value of
the vertical effort of the part n of the formula. And it is to be observed, that this effort
acting against the point B, the arm BF of the lever will be x and its expression na. 1454. Bringing together all these algebraic values we obtain the equation pa+pd =''
Lastly, multiplying all the terms of the equation by : for the purpose of eliminating x3,
we shall have instead of the preceding formula 6p + Se:-- + "#, which is an equation
of the third degree, whose second term is wanting. For more easily resolving this equation, let us find the value of 6p + 6p* and that of #, by which x is multiplied in the second part of the equation.
= 3/448894+4490944, 3/448894–449093, from which extracting the cube roots, we have x=444–2|=42 for the length BF of one of the sides of the triangular pier BAF; the other fa may be determined by the production of the diagonal or line of groin OB. The part of the pier answering to the part of the vault, BNO is determined by drawing from the points B and A the parallels BM and MA to FA and FB. These two triangles will form a square base, each of whose sides will be 42 lines, answering to one quarter of the vault KBNO; thus, to resist the thrust of the vault, four piers, each 42 lines thick, are necessary. • 1455. The above result corresponds in a singular manner with the experiments which were made by Rondelet, from which he deduced a thickness of 481 lines. In his investigation of the example by means of the centres of gravity 40:53 lines was the result. Our limits prevent further consideration by other examples: we will merely therefore observe, that c