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 LINKAGES ; THE DIFFERENT FORMS AND 
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 -THEORY OF SOLID AND BRACED ARCHES 
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 IMAGINARY QUANTITIES. Translated from the 
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 -INDUCTION COILS: HOW MADE AND HOW 
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 -KINEMATICS OF MACHINERY. By Prof. Ken- 
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 SEWER GASES : THEIR NATURE AND ORIGIN. 
 By A. de Varona. 
 
 THE ACTUAL LATERAL PRESSURE OF EARTH- 
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 INCANDESCENT ELECTRIC LIGHTING. A 
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BRIDGE 
 
 AND 
 
 TUNNEL CENTRES. 
 
 JOHN B. McMASTEE, 0. E. 
 
 OF THB 
 
 TJHI7ER 
 
 YORK: 
 
 D. VAN NOSTRAND, PUBLISHER, 
 
 23 MURRAY AND 2T WABREN STREET. 
 
 1875. 
 
Page 77, lines 4 and 7 from bottom, and 
 78 ' "2-6 " top, for 
 yard read foot. 
 
 nor is it intended to furnish a variety of designs 
 likely to be useful to the carpenter and bridge- 
 builder ; it does not claim to be analytical ; it 
 is purely practical. 
 
 Very much, therefore, of what, under other 
 circumstances, might most fittingly have been 
 introduced, has been carefully omitted, and 
 nothing set down which does not bear directly 
 
<0 
 
 Y 
 
It is the purpose of the following essay to 
 present in as brief a manner as the nature of 
 the subject will allow, the rules and principles, 
 the application and observance of which is of 
 really vital importance in the planning and con- 
 struction of Bridge and Tunnel Centring. It 
 is not offered as a highly elaborated and ex- 
 haustive treatise on this branch of engineering, 
 nor is it intended to furnish a variety of designs 
 likely to be useful to the carpenter and bridge - 
 builder ; it does not claim to be analytical ; it 
 is purely practical. 
 
 Very much, therefore, of what, under other 
 circumstances, might most fittingly have been 
 introduced, has been carefully omitted, and 
 nothing set down which does not bear directly 
 
on the subject in hand, and had not been veri- 
 fied, time and again, by actual experiment. It 
 will be observed, for instance, that what may 
 be termed the mathematics of the subject finds 
 no place here. There are no mathematical de- 
 monstrations, no lengthy discussions of the 
 various formulae introduced ; they are simply 
 set down as expressing established truths, the 
 proof of their correctness in many cases sup- 
 pressed, and the reader requested to accept 
 them as true. In the form in which this essay 
 first appeared, this was done to save space ; in 
 the present form it has been strictly adhered 
 to, because it is believed that those to whom 
 the work will be of the most use, are precisely 
 those who will be content to take as true the 
 formulae given, caring very little for the steps 
 by which it has been reached. 
 
 In connection with the matter of estimating 
 the load on a centre, four methods have been 
 selected, either of which will give results close 
 enough to the absolute truth for all practical 
 purposes. The first, that of M. Couplet, is 
 extremely simple^ and if it errs at all, does so 
 on the side of safety. The second or " graphic- 
 
al" is constantly growing in favor, and most 
 deservedly so ; the third, that by calculus, dis- 
 regards friction and give results greatly in ex- 
 cess of the truth, while the fourth, or trigono- 
 metrical, is perhaps the most exact of all, and 
 admits the application of logarithms. 
 
 The remarks on the subject of uncentring are 
 believed to be sufficiently extended, though the 
 subject is one of great importance. The prin- 
 ciples, however, to be observed in striking 
 centres are quite few and simple, the observance 
 being all that is necessary to secure success. 
 The sand method cannot be too highly com- 
 mended. 
 
 The remarks coming under the head of tun- 
 nel centres, have been limited to pointing out 
 the essential difference between the centre 
 proper for bridging, and that suitable for tun- 
 nelling, to calling attention to the peculiar 
 variability of the strains, and to the care to be 
 observed in guarding against the accidents so 
 liable to produce injury to the ribs, and to of- 
 fering a few practical suggestions as to econo- 
 my. A few designs have also been added as 
 illustrative of the principles 1 aid down, and as 
 
6 
 
 affording examples of cheap and durable 
 frames. The patent centre of Mr. Frazer is 
 worthy of some attention. 
 
 JOHN B. McMASTER. 
 NEW YORK, November, 1875. 
 
BRIDGE AND TUNNEL CENTRES. 
 
 IN the construction of stone and brick 
 arches, of whatever shape and span, and 
 to whatever use applied, whether as sup- 
 ports for roadways or roofs of tunnels, 
 there is nothing which requires more 
 careful attention on the part of the con- 
 structing engineer, than the centres. 
 Independent of the choice of material, 
 of the exactness with which each stone 
 is cut, and the care with which it is laid 
 in place, the success of arches of great 
 span, their settlement and ultimate sta- 
 bility depends essentially on the care 
 given to the framing, setting up and 
 striking of the centres. The slightest 
 change in the shape of the frame caused 
 by the shrinking of an ill-seasoned tim- 
 
8 
 
 ber, or the yielding to compression of a 
 badly proportioned brace, will assuredly 
 be followed by a change in the curve of 
 the intrados, which may possibly result 
 in the ruin of the arch itself. 
 
 Well constructed centring, therefore, 
 is indispensably necessary to a well con- 
 structed arch, and in the following papers 
 it is our intention to offer a practical in- 
 vestigation of the principles which must 
 be followed out in the planning and me- 
 chanical execution of all such centre 
 frames ; to determine wKat strains must 
 be withstood, at what point they act 
 with most vigor, and by what combina- 
 tion of beams and by what system of 
 bracing, the greatest strength and stiff- 
 ness may be combined with the utmost 
 lightness and the strictest economy of 
 material. 
 
 BRIDGE CENTRES. 
 
 Of all classes of centres, the most com- 
 plicated in structure is, beyond doubt, 
 that of a large span stone bridge. Like 
 a roof frame, it consists of a number of 
 vertical pieces, placed in the direction of 
 
9 
 
 the span, from 5 to 7 ft. from centre to 
 centre, and known as the ribs, upon which 
 are placed horizontal pieces or laggings, 
 and on these latter rest the voussoirs till 
 the key stone course is driven and the 
 arch becomes self-supporting. 
 
 THE FKAME in its turn is composed of 
 back pieces, or short beams cut on the 
 outer edge to the same curve as the in- 
 trados of the arch, a horizontal tie beam, 
 and a number of struts, ties and braces, the 
 arrangement, number and dimensions of 
 which, will depend on the shape and span 
 of the arch, and the number and position 
 of the points of support. Whatever may 
 be the span and curve of the arch, and the 
 points of support afforded, experience 
 has amply proved that the ribs should be 
 polygonal in shape, with short sides ; 
 this shape being given by forming the 
 back-pieces, on which rest the laggings, 
 of two or more courses of planks, placed 
 in the form of a polygon and firmly nail- 
 ed together ; the planks in each course 
 abutting end to end by a joint in the di- 
 rection of the radius of curvature of the 
 
JO 
 
 arch, and breaking joints with those of 
 the other course. 
 
 For light arches of moderate span, or 
 indeed for heavy arches of wide span 
 when firm intermediate points of sup- 
 port can be had between the abutments, 
 the back pieces may be strengthened by 
 struts or ties placed under them, well 
 braced, and abutting against a horizon- 
 tal tie beam. This beam spans the arch 
 a little above the springing line, is bolt- 
 ed to the back-pieces at either side, thus 
 preventing them from spreading later- 
 ally, and if well sustained by props from 
 beneath, affords a firm support to the 
 struts and braces of the rib. In by far 
 the greater number of cases, however, 
 where headway is required under the 
 centring during the construction of the 
 arch, as is the case with stone bridges 
 spanning a river whose navigation can- 
 not be impeded, or whose current is too 
 swift and depth too great to give firm 
 points of support to the props of the 
 tie beam, it becomes necessary to do 
 away with the latter, and supply its place 
 
11 
 
 by such an arrangement of beams as will 
 transmit the strains received to points 
 of support at the abutments. This lat- 
 ter class of centring is known as "re- 
 troussee" or " cocket" and requires a 
 much more careful and elaborate arrange- 
 ment of its parts than the former. 
 
 We have therefore two classes of 
 bridge centres to deal with ; one in 
 which the frame is constructed without 
 regard to headway beneath it, and is 
 supported from firm points of support 
 between the abutments, and one arrang- 
 ed to leave headway under the frame, 
 and upheld by framed supports at the 
 abutments. 
 
 Before attempting to determine the 
 most advantageous arrangement of the 
 pieces which must compose the frame, 
 their number and the dimensions it is 
 necessary to give them in order that they 
 may offer a solid support to the arch 
 stones, it is fitting to consider the effect 
 of the load the ribs are expected to up- 
 hold, the strains it produces, the points 
 where and the directions in which the 
 strains act and their intensity. 
 
12 
 
 THE STRAINS. 
 
 The strains to which centre frames are 
 subjected arise solely from the pressure 
 upon the back-pieces and laggings, due 
 to the weight of the voussoirs laid upon 
 them, and are therefore extremely vari- 
 able, depending on the span and curve 
 of the arch, and the thickness and weight 
 per cubic foot of the voussoirs which 
 press upon the centring. It is not, 
 however, to be supposed that all the 
 voussoirs from springing line to spring- 
 ing line do press upon the frames, this 
 depending to a very great degree on the 
 curve of the arch. If, for example, we 
 take the case of a full centre arch and 
 starting at the springing line on either 
 side pass towards the crown, we shall 
 find that for a considerable distance 
 above the springing line the stones do 
 not exert any pressure upon the ribs, 
 but that, as soon as this point is passed, 
 the pressure begins and increases rapidly, 
 reaching its maximum intensity just be- 
 fore the keystone course is driven into 
 place. When this is done the pressure 
 
13 
 
 is almost entirely removed, and were it 
 not for the slowness of the mortar in 
 drying, the frame work of the arch 
 might be done away with. 
 
 And, here, I would mention that, al- 
 though it is generally held that when 
 the keying course is placed, the vous- 
 soirs, with the exception of a few courses 
 at the crown, cease to press, I have found 
 by the most careful experiments with 
 large, well-framed models, that the thin- 
 nest Chinese paper when coated with 
 black lead and placed under the blocks 
 of arch stone, could not be drawn out, 
 even when the arch was keyed, without 
 considerable resistance. 
 
 Upon further examination it will be 
 found that these voussoirs which lie near 
 the springing line and exert no pressure 
 upon the laggings and back-pieces, are 
 all of them contained within the angle 
 of repose ; that is to say, the voussoirs 
 do not begin to press upon the centring 
 until we reach one whose lower joint 
 makes so great an angle with the hori- 
 zon, that the stone is caused to slide 
 
14 
 
 along its bed under the action of gravi- 
 tation. This angle for full centre arches 
 has been fixed at from 28 to 30, but 
 the quality of the stone and mortar used, 
 will cause it to vary greatly. For ordi- 
 nary cut stone, we may with safety as- 
 sume the angle of friction at 30 with the 
 horizon : when laid in thin tempered 
 mortar it is increased to 34 or 36, and 
 with very porous stone, such as free- 
 stone, laid in full mortar it will reach 
 almost 45. 
 
 It is to be observed, however, that this 
 is not strictly true unless the arch is of 
 sufficient thickness at bottom to prevent 
 all tendency to upset inwards. A thick- 
 ness of TV the radius of curvature is usu- 
 ally adopted as sufficient for this pur- 
 pose. 
 
 Adopting 30 as the angle of repose 
 for cut stone, the number of voussoirs 
 which load the centre will depend on 
 the curve given to the intrados. If we 
 take, for instance, a full centre, an oval 
 and a flat segmental arch, and give to 
 each the same number of voussoirs, it is 
 
15 
 
 evident that the number of stones which 
 do not press on the laggings will be 
 greatest in the full centre, less in the 
 oval, and least of all in the flat segmen- 
 tal arch, because in this latter case the 
 stone whose lower joint makes an angle 
 of 30 with the horizon will be found 
 nearer the springing line. We should 
 expect, therefore, the number and weight 
 of the stones being the same, that the 
 segmental arch could give the greatest 
 load to the centres, and the full centre 
 arch the least ; and this is strictly the 
 case. 
 
 In estimating the load upon the centres 
 in any case, it is to be remembered that 
 none of the stones bear upon the ribs 
 with their entire weight, a part of this 
 latter being consumed in overcoming 
 friction. The determination of the 
 amount of weight thus expended is a 
 matter of some mathematical intricacy, 
 and we are indebted for its solution to 
 M. Couplet.* By his calculation he 
 
 * M6moire de 1'Academie Ann6e, 1792. 
 
16 
 
 found that the total weight of the vous- 
 soirs which do press on the laggings, is 
 to the weight with which they actually 
 load the frame, as an arc of 60 is to 
 twice its sine less the same angle ; or, to 
 express it algebraically, denote by P the 
 total weight of the voussoirs which rest 
 on the centring, and by p, the weight 
 with which they load the centres, and 
 we shall have the expression 
 
 P : p\ iarc 60 : 2 sin 60 arc 60 (1) 
 
 or 
 
 P (2 sin 60 arc 60) .. 
 
 *= Arc 60. ' < 2 > 
 
 If, therefore, we suppose the radius of 
 a circle to be divided into 10,000 equal 
 parts, the circumference will contain 
 62,832, and the arc of 60 10,472, and 
 its sine is equal to i/\/3,8660. Substi- 
 tuting these values in the above equa- 
 tion (1), we shall have 
 
 P :/>::i0472; ;2X8660 10472 or 
 P :p\ i 10472 : 6848 
 
 which gives us a ratio of 3 to 2 very 
 
17 
 
 nearly. Whence we see that the vous- 
 soirs in a full centre arch which press 
 upon the laggings will do so with but f 
 of their weight, and, taking the angle 
 of repose on each side at 30, only on 
 of the surface of the centring. We 
 may, therefore, without any sensible 
 error take J of the gross weight of the 
 voussoirs of the arch to express the load 
 on the centres. 
 
 With an arch which is not full centre 
 the case is quite similar. We will take 
 an oval of three centres fulfilling the 
 conditions that each of the three arcs 
 composing it shall be 60. This oval 
 being drawn, it is at once apparent that 
 the arcs of 60 at each end of the oval 
 do not differ materially from that of 30 
 in the full centre arch. We may, there- 
 fore, to facilitate calculation, safely as- 
 sume that the stones forming these two 
 arcs of 60 do not press on the centres, 
 when the arch is all up except the key- 
 stone, and are held in place by the weight 
 of the voussoirs above them. There re- 
 mains then but thrrrptrftl arcf 60 to 
 
18 
 
 load the framing. But from equation 
 (1) P : p as the arc of 60 is to twice its 
 chord less the arc of 60 ; and since 60 
 is to its chord very nearly as 22 to 21, 
 we may without sensible error express 
 the relation of P to p by the ratio of 1 1 
 to 10. When we have found the gross 
 weight of the voussoirs in this arc of 60 
 it follows that we must take 11 of their 
 weight to express the load on the fram- 
 ing. 
 
 The chord of an arc of 60 is equal to 
 the radius, and the radius in this case 
 being 10000, the chord will equal 10000, 
 and the arc of 60, 10472. Hence we 
 have the relation 10000 : 10472; ;21 : 22 
 nearly. 
 
 These values may also be obtained 
 from the integral calculus, in which case 
 no regard is taken of friction, and the 
 formulae are therefore a little uncertain. 
 This uncertainty, however, is on the side 
 of safety, for when we leave out of con- 
 sideration the pressure expended in over- 
 coming friction we are forced to give the 
 ribs and laggings unnecessary strength. 
 
19 
 
 % 
 
 Referring to Figure 1, we wish to find 
 the load which the voussoirs between 
 A B C D give to the centre's rib. 
 
 Let w equal the weight per lineal foot 
 of intrados of the arch resting on 
 the rib. 
 
20 
 
 r 
 
 By x and y the horizontal and vertical 
 
 co-ordinates respectively of the 
 
 point A. 
 
 By x' and y' the co-ordinates of D. 
 By r the radius of the curve of intra- 
 
 dos at A, and 
 By x the angle it makes with the hori- 
 
 zon. 
 By P the gross vertical pressure on 
 
 the rib of A B C D. 
 By p the pressure per lineal foot of in- 
 
 trados at A. 
 
 Then we shall have 
 
 From this we may compute the load on 
 any vertical post at this point, or the 
 vertical component of the load on the 
 back pieces. 
 
 If the arch had been completed up to 
 the keystone course, equation (3) would 
 have been 
 
 /V 
 
 77 o 
 
 pdx . . . (4) 
 
21 
 
 P' being the greatest vertical load on 
 the half rib, and x* the horizontal dis- 
 tance from the middle of the span to a 
 point where p is zero. 
 
 The value of p is found from the equa- 
 tion 
 
 p= W. Cos x - - 1 - / y W. dy . (5) 
 
 which will evidently be greatest when 
 p=W. Cos oc . . . . (6) 
 
 When the arc is a segment of a circle 
 not greater than 120, we shall have 
 from the relation between the sides and 
 angles of a triangle, the following values 
 of the co-ordinates : 
 
 xr sin cc x'=r sinoc' x"= r(sin oc*) 
 y r (J cos cc ) y'r (1 cos oc') 
 
 4 
 
 Substituting these values in equation (5) 
 we shall have 
 
 pw (2 cos oc cos oc'). . . (7) 
 
 And from the expression y=r(l cos X ) 
 we have 
 
 ~ r y 
 
 Cos x = * ; 
 
 r 
 
22 
 and from y'=r (1 cos oc') 
 
 Cos oc'=^' 
 r 
 
 Substituting this in equation (7) we shall 
 have 
 
 r 
 Equation (6) then becomes p=w cos oc 
 
 T I/ 
 
 =w ---- - the greatest value for a given 
 
 point of the arch. 
 
 Substituting in equation (3) the value 
 of p found in eq. (8), and, reducing, we 
 obtain 
 
 (9) 
 ]*=wr[cc GC' sin oc (cos x ' cos o>)] 
 
 r 
 
 or 
 
 in which I and V represent the length in 
 feet of the arcs from the crown E (Fig. 
 1) to the points A and D respectively. 
 Equation (4) then becomes 
 
23 
 
 P'=wr (oc"-sm <x" [l cosoc"] ) (10) 
 
 To find the gross weight of that por- 
 tion of the arch which presses on the 
 back pieces and laggings, it is necessary 
 to know the number of the voussoirs, 
 their volume and weight per cubic foot. 
 
 The weight of stone generally used 
 in arches varies from 120 to 180 pounds 
 per cubic foot. The following results 
 were obtained from the examination of 
 a number of - specimens of American 
 granite, sandstone and limestone, taken 
 from the best known quarries in the 
 country. Of seventy-two specimens of 
 granite examined, the greatest weight 
 per cubic foot was 182.5 Ibs., the least 
 161.2, and the average 167.09 Ibs. Of 
 fifty-three specimens of sandstone exam- 
 ined, the greatest weight per cubic foot 
 was 164.4 Ibs, the least 127.5, and the 
 average 140.9 Ibs. Of thirty -eight 
 s r 'e~:::: :ns of limestone, the greatest 
 weight per cubic foot was 173.8, the least 
 143.2, the average 162.9. 
 We may therefore without sensible 
 
24 
 
 error assume the average weight of these 
 three classes of stone as follows : 
 
 Average weight 
 per cubic foot . 
 
 Granite 167.09 Ibs. 
 
 Sandstone 140.9 Ibs. 
 
 Limestone 162.9 Ibs. 
 
 Brick (well burnt) 92.0 Ibs. 
 
 From the moment the angle of repose 
 is passed and the first voussoir begins to 
 press on the frames, the centring be- 
 comes subjected to a series of strains 
 which increase rapidly up to the time 
 the keystone is laid, and are produced 
 by the yielding of the ribs under the 
 weight of the stones. No matter how 
 well seasoned and admirably proportion- 
 ed the timbers may be, or how evenly 
 the load may be distributed, the centre, 
 pressed more and more severely on each 
 side by the successive courses of vous- 
 soirs laid upon it, will bend in on the 
 sides, and as a consequence bulge out at 
 the crown, to be in turn followed by a 
 bending in of the crown when the arch 
 is all but completed. This movement of 
 
25 
 
 the ribs can be greatly checked and the 
 severity of the resulting strains much 
 lessened by loading the centres at the 
 crown with the spare voussoirs and in- 
 creasing the load as the arch progresses. 
 In the case of a full centre arch of 90 
 feet, and composed of four hundred and 
 eighty courses of voussoirs, the cen- 
 tring, when the fifteenth course of vous- 
 soirs on each side were laid in place, had 
 risen three inches at the crown. When 
 loaded with 325,000 Ibs., it settled under 
 it two inches ; but when the twentieth 
 course was completed the pressure was 
 so great that it again rose one inch. 
 When the arch was three-quarters com- 
 pleted it had again sunk one inch and 
 three-quarters in consequence of the ad- 
 ditional load and the compression of the 
 wood, still leaving a rise of one quarter 
 of an inch. This yield caused the 
 joints at the twenty-second course to 
 open a fraction of an inch, but closed 
 when the keystones were driven. This 
 distortion of the centring is always 
 greatest for full centre arches, and pro- 
 
26 
 
 portionally less as the arch becomes 
 nearer and nearer to the segmental. 
 
 DIRECTION OF THE STRAINS. 
 
 To find the direction and intensity of 
 the strain at any point of the rib, we re- 
 sort to the usual method of the " paral- 
 lelogram of forces." Returning to Fig. 
 1, let it.be required to find the direction 
 of the strain caused by the voussoirs 
 A B C D. Denote by F the centre of 
 gravity of this part of the arch, and 
 through it draw a vertical line G I of in- 
 definite length, and cut it at I by a per- 
 pendicular from the point E at which 
 the curve drawn through the centres of 
 gravity of the voussoirs supposed in- 
 definitely small cuts the line AB. 
 Complete the parallelogram by drawing 
 ; the line IM to the centre of arch, and 
 NL parallel to it. The diagonal IN 
 will then express the weight of the vous- 
 soirs A B C D, the side I L the pressure 
 they exert upon the lower part of the 
 arch, and the side I M the pressure upon 
 the backpieces o the rib. 
 
The strains, then, upon the centring 
 take the direction of the radius of cur- 
 vature of the intrados, and it now re- 
 mains to consider the position which 
 should be given to the beams which are 
 to withstand the strains, their number 
 and dimensions. 
 
 THE PRINCIPAL BEAMS AND THEIR 
 POSITION. 
 
 As the sole object of the framing is to 
 uphold the voussoirs and transmit the 
 strains it receives as directly as possible 
 to firm points of support, the beams must 
 be so arranged as to do this with the 
 least tendency to change the shape of 
 the rib, by their bending or breaking. 
 The condition will be best fulfilled by 
 giving each beam a position such that it 
 shall offer the greatest possible resist- 
 ance, and this will be accomplished when 
 the direction of the fibres of the beam 
 and the direction of the strain are one 
 and the same. 
 
 If, for instance, we support a horizon- 
 tal beam at its two ends and load it in 
 
28 
 
 the middle it will offer its least resist- 
 ance to the load. If now we raise one 
 end so that the direction of the strain is 
 oblique to the fibres of the beam, the re- 
 sistance of the beam to bending will be 
 found to have increased largely, and the 
 resistance in this latter case, will be to 
 that in the former case, as the cosine of 
 the angle made by the direction of the 
 strain and the fibres of the wood is to 
 the sine of 90 or 1. 
 
 It should follow from this that, when 
 the angle between the beam and the 
 strain is zero, the resistance becomes in- 
 finite, and such would indeed be the case 
 were it not for the compressibility of the 
 wood and other physical causes which 
 weakens its strength. It is sufficient, 
 however, for us to know that when the 
 strain is carried through the axis of the 
 beam, it is then strongest, and that as 
 the force becomes more and more oblique 
 to the fibres its strength decreases. 
 
 Applying this fact to the framing of 
 the ribs, it follows that the greatest stiff- 
 ness and strength will be gained when 
 
29 
 
 the principal pieces are placed in the 
 direction of the strains, or in the direc- 
 tion of the radii of curvature of the 
 arch to be upheld. This deduction, un- 
 fortunately, is under certain restrictions 
 placed upon it by the imperfections of 
 the timber, and demands of economy 
 and the circumstances of construction, 
 which make its practical application quite 
 limited. 
 
 To illustrate, we will once more return 
 to Fig. 1. The direction and intensity 
 of the strain on the backpieces resulting 
 from the weight of the voussoirs ABCD, 
 will then be represented, as we have 
 just seen by the line V M, and that of 
 the voussoirs P Q by the line V S. The 
 beams, therefore, which are to support 
 these stones, in order that they may of- 
 fer the utmost resistance, must take the 
 direction of the lines V S and V M, or 
 radiate from the centre V like the spokes 
 of a wheel. For small span arches, such 
 an arrangement of beams undoubtedly 
 answers all purposes of stiffness and 
 economy, but for arches of larger span 
 
30 
 
 where timbers of thirty, fifty, or even a 
 hundred feet in length would be requir- 
 ed, it fails most signally ; for while a 
 beam of ten feet will offer great resist- 
 ance to compression when loaded in the 
 direction of the fibres, a beam of fifty 
 feet will be almost sure to bend under 
 the action of the strain, and hence re- 
 quire bracing. This system, therefore, 
 cannot be successfully carried into prac- 
 tice in large span centres. 
 
 To overcome this difficulty we are 
 forced to resolve the force represented 
 by the line S Y into two components, one 
 vertical and represented by the line S T, 
 and one horizontal represented in direc- 
 tion and intensity by S R. By a similar 
 treatment of the force represented by 
 V M, we shall obtain two other similar 
 lines, all four of which will represent the 
 direction of three beams, which can be 
 made to take the direction of the two 
 V S and V M, namely, a long horizontal 
 beam spanning the arch and supported 
 at each end by a vertical beam. This 
 horizontal beam is the tie beam to which 
 
31 
 
 we have already alluded, and is gener- 
 ally placed at points about 45 up the 
 arch. The voussoirs above this beam 
 are then supported by another horizon- 
 tal tie upheld by small vertical beams 
 abutting on the lower tie. An excellent 
 illustration of this system of framing is 
 found in centres of London Bridge over 
 the Thames, built in 1831 by Rennie. 
 
 There will frequently arise cases in 
 which ribs framed in this manner either 
 on account of the quantity of material 
 they consume, or the difficulty of finding 
 firm points of support between the abut- 
 ments, cannot be used to advantage. It 
 then becomes necessary to change the 
 point of support T of the beam ST 
 (Fig. 1) to a point t nearer the abutment, 
 and for the sake of economy we may do 
 away with the horizontal and vertical 
 beams tg, s S, T S, c a and a b, supply- 
 ing their place by two beams t S and S e. 
 These two beams, therefore, will sustain 
 the strain represented by the line S V, 
 and the efforts they resist will be repre- 
 sented in direction and intensity by the 
 
32 
 
 sides S X and S Y of the parallelogram 
 XY constructed on S V as a diagonal. 
 
 In "cocket" centres, therefore, what- 
 ever the span of the arch, whether large 
 or small, whatever the shape, whether 
 full centre, oval or segmental, a great 
 saving of material may be made, and 
 abundance of strength may be secured, 
 by placing the principal beams in the 
 direction of the chords of the curve of 
 the intrados. 
 
 The length that should be given to 
 beams thus placed, the angle they should 
 make with each other at their point of 
 junction, the manner of supporting, and 
 when necessary bracing them, are points 
 we shall reserve for future consideration. 
 
 There are, therefore, three methods of 
 arranging the principal pieces or struts 
 of a centre frame. 
 
 1. They may be placed in the direc- 
 tion of the radii of curvature of the 
 arch, thus giving a figure of invariable 
 form as the strain at any one point is re- 
 ceived by the beam in the most favor- 
 able position, and transmitted through 
 
33 
 
 its axis directly to the fixed point of sup- 
 port. 
 
 2. They may be placed in a vertical, 
 or in vertical and horizontal directions. 
 
 3. The curve of the arch may be di- 
 vided into a number of arcs, and the 
 beams placed in the direction of the 
 chords of these arcs. 
 
 4. To these three we may add a fourth, 
 which embraces by far the largest num- 
 ber of centre frames, and is based on 
 two or all of the preceding methods. In 
 this class the beams are not arranged in 
 accordance with any one system, but 
 several ; as, for instance, the second and 
 third, in which case, as we shall see here- 
 after, several straining beams span the 
 arch at different points, and are sustain- 
 ed by inclined struts ; or if all three sys- 
 tems are used, we may use the straining 
 beam and inclined struts, and strengthen 
 them by bridle pieces in the direction of 
 the radii. 
 
 It would, indeed, be quite a hopeless 
 task to attempt to lay down, in more 
 than a general way, the principles which 
 
34 
 
 ought to rule in making a selection of 
 one of these methods to the exclusion of 
 the remaining three. In every case the 
 choice must be determined largely by the 
 circumstances of the case, the points of 
 support, the shape and span of the 
 frame, and the strength required. If the 
 centre is to be " cocket," the arch heavy, 
 the span large, and considerable head- 
 way required beneath the frame, the 
 third or fourth arrangement will undoubt- 
 edly afford the best results whatever 
 may be the shape of the arch. If the 
 arch is light, the span moderate, and little 
 or no headway is wanted, then the sec- 
 ond or first will generally be most con- 
 venient. 
 
 Theoretically, the first method will in 
 all cases afford the greatest amount of 
 strength and stability with the least 
 amount of material, since the beams are 
 then capable of resisting the most se- 
 vere strains. Nor can there be any 
 doubt that, within moderate limits, this 
 result actually is attained in practice, 
 and that of two ribs constructed with 
 
35 
 
 the same number of beams, of the same 
 quality of wood and similar dimensions, 
 in one of which the pieces are placed 
 radially, and in the other vertically or 
 inclined, the rib arranged on the former 
 plan will be decidedly the stronger of 
 the two. But, unfortunately, the im- 
 possibility of always obtaining firm 
 points of support at the centre of curva- 
 ture, the difficulty of finding sound, well 
 seasoned timber of such length as would 
 be required in arches of large span, and 
 the relation which exists between the 
 length and strength of beams under 
 longitudinal compression the strength 
 varying inversely as the square of the 
 length restricts its application to cen- 
 tre frames of very small span and rise. 
 In semi-circular arches of twelve, fifteen 
 or even twenty feet span, when a hori- 
 zontal beam can be used at the springing 
 line this arrangement can be used with 
 great success. The frame then consists of 
 the tie beam and two, or if great strength 
 is required, three radial struts which sup- 
 port the backpieces and abut against the 
 
36 
 
 horizontal beam at the centre of curva- 
 ture. These struts, when two are used, 
 should be inclined on the right and left 
 at a little less than 45 to the horizon, so 
 as to meet the backpieces at the point 
 where the voussoirs first begin to press 
 on the rib. A vertical strut is in such 
 an arrangement of little or no use, as no 
 strain of any consequence can possibly 
 reach it ; the voussoirs almost ceasing to 
 press on the frame when the keystone 
 is driven down. As these supports are 
 struts and not bridle pieces clamping the 
 backpieces and tie beam between them, 
 the joints, especially in the larger and 
 heavier arches, must be secured by pieces 
 of iron placed across them and bolted to 
 the backpieces and struts, to prevent the 
 joints opening in consequence of the 
 bulging at the crown as course after 
 course of stone is laid on the frame. 
 
 In frames for flat segmental arches of 
 a span as great as sixty or seventy feet 
 and rise of about one-fifth the span, as 
 also for ovals of several centres, this 
 radial arrangement may be slightly modi- 
 
38 
 
 fied and a frame produced (Figure 2), 
 which shall meet all the requirements of 
 strength, lightness and economy. The 
 rib in this case again consists of a hori- 
 zontal tie beam spanning the arch a little 
 above the springing line, generally at 
 the first voussoir that presses on the 
 backpieces, and struts placed in the di- 
 rection of the radii of curvature and 
 from eight to ten feet apart depending 
 on the weight of the arch. These struts, 
 as it would be impossible to have them 
 actually meet at the centre of curvature, 
 which, for an arch of seventy feet span 
 and fifteen feet rise, would be about 
 forty-five feet from the circumference, 
 go no further than the tie beam and are 
 fastened to it and the backpieces by the 
 iron bands shown in the figure. 
 
 When great stiffness is required in the 
 rib, additional braces may be added, as 
 shown in Fig. 2, dividing the rib into a 
 number of triangles. The strains re- 
 ceived will then be transmitted through 
 the axes of the beams, and as all unnec- 
 essary transversal strains will be avoid- 
 
39 
 
 ed, the resistance offered by the braces 
 will be the greatest possible. In all 
 centre ribs, the normal pressure being 
 in the direction of the radii of curvature, 
 the laggings, backpieces and tie beam, 
 when used, will of necessity be subject- 
 ed to transversal strain. 
 
 Before, however, we proceed to con- 
 sider the strains to which the beams in 
 centre frames are subjected, and the di- 
 mensions we must give them in order 
 that they may withstand the pressure 
 put upon them, we would offer the fol- 
 lowing practical rule for estimating the 
 pressure of any arch stone in any part 
 of the arch, upon the centre rib, or the 
 pressure upon the rib at any stage of the 
 construction of the arch, as also the 
 pressure when the arch is completed up 
 to the key stone. 
 
 It has been well established by the ex- 
 periments of Rondelet, that a stone 
 placed upon any inclined plane does not 
 begin to slide on that plane until it has 
 reached an angle of inclination to the 
 horizon equal to 30. It is obvious, 
 
40 
 
 therefore, that if the arch stones were 
 placed upon one another they would not 
 begin to press on the centre rib till the 
 plane of the lower joint of one of them 
 reached an angle of 30 with the horizon. 
 It has been found, moreover, that the 
 mortar increases this angle, for hard 
 stone to 34 or 36, and for soft, porous 
 stone (in semi-circular arches) to 42. 
 We may, then, consider the pressure to 
 commence in general at the joint which 
 makes an angle of 32 with the horizon. 
 If we suppose the radius to represent 
 the pressure the tangent will then repre- 
 sent the friction, and making the radius 
 unity the friction will be 0.625. The 
 next stone will press a little more, the 
 third still more, and the pressure will 
 thus continue to grow larger and larger 
 with each succeeding course. The rela- 
 tion between the weight of an arch stone 
 and its pressure upon the rib in the di- 
 rection perpendicular to the curve is 
 given by equation : 
 
 Q W (cos a- /sin a) . . (11) 
 in which Q is the pressure, W the weight 
 
41 
 
 of the arch stone,/* the friction =: 0.625, 
 and a the angle the lower joint makes 
 with the vertical. The following table 
 calculated from eq. 11, gives the value 
 of Q for every 2 of curve from the 
 angle of repose = 32 up to 60 : 
 
 When the angle which the joint makes 
 
 with the horizon is 
 
 34 then Q=.04 W 
 
 When 36 " Q = .08 W 
 
 " 38 " Q=.12W 
 
 " 40 " Q = .17W 
 
 " 42 " Q=.21 W 
 
 " 44..... " Q=.25W 
 
 " 46 " Q=.29W 
 
 " 48 " Q = .33W 
 
 . " 50 " Q = .37 W 
 
 52 " Q=.40W 
 
 " 54 " Q=.44W 
 
 " 56 " Q=.48W 
 
 " 58 " Q=.52W 
 
 " 60 " Q = .54W 
 
 To take an example : What is the 
 pressure on a backpiece of 20 in length 
 from the angle of repose, the ribs of the 
 
42 
 
 frame being placed 5 ft. from centre to 
 centre, and the arch stones 3 ft. in depth 
 and weighing 160 Ibs. per cubic foot. 
 We take from the above table the sum 
 of the decimals from 32 52= 2.26, and 
 multiply this by the weight upon 2 and 
 the product will equal the pressure. 
 The volume of the stones which cover 
 
 The number of feet contained in 2 is 
 found from the expression 2 X. 01 745 329 
 Xr', in which r' is equal to the radius 
 of the arch plus one half the depth of the 
 arch st,one. If we take the radius = 25 
 ft., then the depth of the stones being 3 
 ft, r' = 26.5 and number of feet in 2 
 equals .88 ft., whence the volume of the 
 stones which press on the 2 equals 
 5X3X.88 = 13.4 cubic feet, and the 
 quantity W=2144 Ibs. and Q, or the 
 pressure on the backpiece equals 4845 
 Ibs. 
 
 If we denote by a the angle included 
 between the upper and lower joints of 
 an arch stone, and suppose every stone 
 in the arch to have the same weight and 
 
43 
 
 equal angle a, then the pressure of any 
 number n of such stone upon the rib will 
 be given from the expression 
 
 n+I (12) 
 
 W + sin - a. , , 
 Q= 2 X( cos 
 
 sin J a 
 
 which gives the total pressure on one 
 half of the rib. 
 
 This equation is found as follows : 
 The pressure perpendicular to the soffit 
 is W (sin a f cos a), or W (cos af 
 sin a), according as the angle a is mea- 
 sured from the horizon or from the ver- 
 tical drawn through the crown. If now 
 we denote by a the angle included be- 
 tween the joints of one stone, and sup- 
 pose each stone alike in size and weight, 
 the pressure of any number n of such 
 stones will evidently be found by getting 
 the sum of the sines and cosines of n a, 
 or expressed in formula, 
 
 Pz=W (sum of cosines of na /xsum 
 of sines of na) . , ; . Eq. A. 
 
 By trigonometry we obtain two expres- 
 sions for the sum of the sines and cosines 
 
44 
 
 of a number of angles in arithmetical 
 progression, viz. : 
 
 Sin A + sin (A + B)-fsin (A + ^B) 
 __Cos (A JB) cos (A-f rc + B) 
 
 2 sin i B 
 Sin A + j^BXsin M + ! B 
 
 sin -g B. 
 Also 
 
 Cos Af cos (A + B) + cos (A + wB) 
 
 cos A + |^BXsin (w+1 B 
 
 sin -J B. 
 
 Applying these two equations to the 
 above case, we shall have from eq. A, 
 
 P=W Eq. B. 
 
 n n+1 -, . n . n + I 
 
 cos - a X sin - a /(sin - aX sin a) 
 
 22 22 
 
 sin a. 
 Or taking out the common factors W and 
 
 K+l 
 
 sin 2 we shall have equation B in 
 
 sin^ a 
 the form. Eq. (12) 
 
 ^ WXsin ^ x / w . n 
 
 P= 2 aXl cos p a ~c/ sm ~ a 
 
 sin -J a 
 
45 
 
 The value of Q may also be obtained 
 from eq. 11 by considering that when 
 the depth of the arch stone is nearly 
 double its thickness ; its weight rests on 
 the rib at the angle of 60. Equation 12 
 is, however, the best, and may be readily 
 solved by logarithms. 
 
 For example : let the arch be semi- 
 circular and a2 y then na=29 and 
 f=.625. Put equation 12 in the form 
 
 COS 
 
 >...' "i . w + 1 
 
 / sm ^ /I aXsm a 
 
 _____________^_____ 
 
 sin a x R 
 
 log cos na =log cos 29 = 9.941819 
 
 , w-f 1 
 log sm a log sm 30 =9.698970 
 
 19.639789 
 
 log sin | a = log sin 1= 8.241855 
 R =10.000000 
 
 18.241855 
 
46 
 
 Difference 1.397934 
 = log 24.68 
 
 log/ = log .625= -- 1.795880 
 log sin na=^\og sin 29 = 9.685571 
 
 n-\- 1 
 log sin a log sin 30=9.698970 
 
 19.180421 
 
 log sin ^ a = log sin 1= 8.241855 
 R =10 
 
 18.241855 
 
 Difference^ 0.938669 = 
 log 8.55 
 
 Hence the weight on the half rib is 
 24.68 8.55 = 16.13 W. 
 
 In a frame constructed, as that shown 
 in Fig. 2, the determination of the 
 strains is a matter of great simplicity, 
 and may be had either from arithmetical 
 calculation or by constructing the paral- 
 lelogram of forces. The strain on any 
 radial strut as B G would be found by 
 calculating from eq. 11 the pressure on 
 D E, taking half of it and supposing it to 
 
47 
 
 act at B in the direction B G. The strain 
 on any inclined strut, as E G or E H, may 
 be found by estimating from eq. 11, the 
 strain on B H taking one half of it, and 
 supposing it to act at E in the direction 
 of the radius at that point, and denote 
 by b and b' the angles these pieces make 
 with the direction of the force. Then, 
 if these angles are unequal 
 
 8= P ; ln S ' and S'=J^*v (13) 
 
 sm (b + b') sin (b + b') v 
 
 And if the two beams make equal angles 
 with the direction of the force, then the 
 strain in the direction of each is the same 
 and expressed by 
 
 Of all methods of calculating the strain 
 on the different beams, by far the sim- 
 plest, is to actually construct the dia- 
 gram of forces to a given scale and find 
 the pressure by measurement. In above 
 case, for example, draw E e parallel to 
 the direction of the force to any con- 
 
 faHIVEESITT) 
 
48 
 
 venient scale, say i 1 * inch equal 1,000 
 Ibs., which, supposing the pressure at 
 E=10,000 Ibs. will make Ee=one inch. 
 From E draw E# parallel to E G; 'also 
 E h parallel to EH, and eg to EA and 
 eh to E#. Then E# being measured 
 will give the pressure on the beam E G 
 to which it is drawn parallel. 
 
 When we have once ascertained the 
 strain which any beam in a frame will 
 have to undergo and resist, the next step 
 is to determine the dimensions, or rather 
 the area of cross section, the beam must 
 have to withstand this pressure without 
 injury. Whatever may be the length of 
 the beam, this section may be obtained 
 from the following formulae : If the 
 strain is one of compression in the direc- 
 tion of the length, then 
 
 in which A is the section required in 
 square inches, F the crushing force to 
 which the beam is subjected, and K the 
 resistance to crushing. When the strain 
 is a transverse or breaking strain, then 
 
49 
 
 in which K' is the modulus of rupture of 
 the beam. 
 
 In place of K and K', however, which 
 are the ultimate resistance to crushing 
 
 K K' 
 
 or rupture, we must use and , in 
 
 n n 
 
 which n is the factor of safety, usually 
 taken as 10 for wood. The values of K 
 and K' are variously stated by different 
 writers on the strength of materials. 
 Those given below for the woods mostly 
 used in centre frames are from Rankine: 
 
 Wood. 
 
 Value of 
 K in Ibs 
 
 Value of K' 
 in Ibs. 
 
 Ash 
 
 9,000 
 
 12,000 
 
 Pine yellow 
 
 5,400 
 
 9,900 
 
 Pine, red 
 
 6,200 
 
 7,100 
 
 Oak English 
 
 10,000 
 
 10,00013,000 
 
 Oak, American 
 
 6,000 
 
 10,600 
 
 If it is not always possible to obtain 
 these values of K and K', a very safe 
 method, and one easily remembered, i& 
 
50 
 
 to find from the diagram of forces the 
 strain on a beam in Ibs., and divide this by 
 1,000; the result will be the cross section 
 of the beam in inches. Thus, if a tim- 
 ber is loaded with 36,000 Ibs., ' 36 
 in., and the beam should be 6 in.X6 in. 
 
 Example. 
 
 Required the proper dimension of 
 the scantling of a centre rib of a seg- 
 mental arch of 60 feet span and 9 feet 
 rise ;' the arch stones to consist of old 
 quarry granite, weighing 165 pounds per 
 cubic foot, and three feet in depth ; the 
 rib to be of the pattern shown in Fig. 2. 
 The frames to be placed 5 ft. from centre 
 to centre. 
 
 The first step is to find the 
 weight of the arch stone for 1 of the 
 curve. The span is 60 ft., the radius is 
 50 ft., and the arch stones being 3 ft. 
 thick the radius of the arch passing 
 through their centre is 51.5 ft. The 
 length of 1 is, therefore, .01745329X 
 -51.5 =.89 ft. Then 5X3X-89 = 13.3 
 
51 
 
 cubic ft., the solid contents of 1 of the 
 arch ring, and this multiplied by 165 
 gives the weight of 1= 13.3X165 = 
 2195 pounds. Now the arch being a 
 very flat segmental, it is evident that all 
 the arch stones will press upon the rib. 
 If then we calculate the weight of the 
 stones between E E', and suppose them 
 to act with one half their entire weight 
 at C in the direction CH, it is evident 
 that this will be the greatest pressure 
 that C H will be required to support. 
 The arc EE'=:20 , and the weight for 
 1 being 2195 Ibs., the pressure at C i 
 21950 Ibs., and the beam CH should be 
 
 21950 
 
 = 21.9 inches or4^X^ in. To find 
 
 the dimensions of E G and E H take eq. 
 12. Then a=l, w =20, /=.625, W 
 = 2195. 
 
 173648) = 18301 Ibs. 
 
 Take this and lay it off to any conven- 
 ient scale on the line E 0, and from E 
 
52 
 
 draw EC/ parallel to EG, and EA to 
 EH and as before eg and eh. Then 
 measuring E h by the same scale it will 
 be found to equal 10250 Ibs. ; the beam 
 EH then must be 3j in. by 3 in. In the 
 same manner the pressure on B G is found 
 to be 18301 Ibs., and the beam must be 
 4 in.X 4 in. To find the strain on the 
 inclined strut, estimate from eq. 12 the 
 weight of the arch stones between A 
 and C, add to this half the weight of 
 the rib and let the gross weight act ver- 
 tically at the point K, and lay it off to 
 any scale on the vertical line KK', and 
 draw K' I/ parallel to the horizontal tie 
 beam. The line K L' being measured 
 will give the strain on the beam K L'. 
 
 Frames arranged on the second meth- 
 od, with the principal pieces all vertical, 
 afford centres of great simplicity of 
 structure and of almost as much strength 
 as one with radial struts supposing, of 
 course, that the number and dimensions 
 of the struts are the same in each case 
 and of much greater strength than one 
 constructed with inclined beams, since 
 
53 
 
 the nearer the aiigle the direction of the 
 strain makes with the fibres of the wood 
 approaches a right angle the less be- 
 comes the resistance of the beam. In 
 segmental and oval arches of large span, 
 the difference in the strength of ribs ar- 
 ranged on the vertical and radial plan 
 is comparatively insignificant, as the 
 radius being very large, the vertical 
 beams, especially near the crown where 
 the strain is severe and most strength 
 is required, do not depart much from the 
 direction of the radius. 
 
 The objection to this vertical bracing 
 of the frame is that it requires the use 
 of a horizontal tie beam, unless the rib 
 is constructed as a girder resting upon 
 framed abutments of its own. If the 
 former arrangement is used, the struts 
 should be placed from five to eight feet 
 apart, depending on the strength requir- 
 ed, and mortised to the tie beam and 
 backpiece. When the beams are of such 
 length that there is danger of their bulg- 
 ing or curving under the load laid on 
 them, they may be strengthened by di- 
 
54 
 
 agonal braces or horizontal wales. Of 
 the two, the diagonal braces are to be 
 preferred as they not only give stiffness 
 to the posts, but sustain a portion of the 
 load on the backpieces in case any of the 
 piles under the horizontal tie beam should 
 give way. Figure 3 represents the rib 
 of a full centre arch of 75 ft. span ar- 
 ranged with the principal pieces placed 
 vertically and strengthened with a hori- 
 zontal waling piece made double, and 
 braces abutting under the backpieces. 
 The strains on the different beams com- 
 posing such a frame, and their necessary 
 dimensions may be computed with ease 
 by the method just explained. It should, 
 however, be remembered that beams 
 which are to be notched must have their 
 dimensions increased beyond those given 
 by calculation, in as much as notching 
 will, even when not very deep, cut down 
 the strength of a beam from one third 
 to one half. In computing the strains 
 on the braces a, a, we may consider the 
 pressure at their abutting point to be 
 the sum of the pressures on the vertical 
 
55 
 
56 
 
 and two inclined braces which meet 
 there, and make no allowance for the 
 resistance of the horizontal beam. 
 
 The third and fourth systems of ar- 
 ranging the principal pieces, afford an 
 almost unlimited number of designs for 
 centre ribs, which are especially worthy 
 of notice, in that they are applicable to 
 every possible shape and span that can 
 be given to stone arches, and may be 
 constructed with or without intermediate 
 points of support, according as circum- 
 stances will admit. The principles which 
 control such arrangements are few and 
 simple. 'The beams should as far as pos- 
 sible abut end to end: they should inter- 
 sect each other as little as may be since 
 every joint causes some degree of set- 
 tlement, and halving destroys fully half 
 the strength of the beams halved. 
 When the framing is composed of a 
 number of beams crossing each other, 
 pieces tending towards the centre should 
 be notched upon and bolted to the fram- 
 ing in pairs : ties should also be continu- 
 ed across the frame at points where 
 
many timbers meet. Particular atten- 
 tion must, furthermore, be given to the 
 manner of connecting the beams so that 
 there shall be no tendency to rise at the 
 crown under the action of the varying 
 load, Figure 4 affords an illustration 
 of a very simple method of arranging 
 the timbers for arches of small span. 
 The inclined struts abut against horizon- 
 tal straining beams placed at different 
 points on the soffit, and to add greater 
 strength to the framing, and to prevent 
 the horizontal beam from sagging, bridle 
 pieces are placed in the direction of the 
 radii of curvature. The chief difficulty 
 with such arrangement as this is, that 
 as they require beams of great length 
 they can be used to advantage only in 
 small span arches. 
 
 The centre frames for the Waterloo 
 Bridge over the Thames were construct- 
 ed on this principle, but in this case no 
 horizontal beams were used. Under the 
 backpieces were placed blocks each sup- 
 ported by two inclined struts which 
 made equal angles with the radius drawn 
 
58 
 
59 
 
 through the centre of the block. In a 
 small span arch, these struts would have 
 rested on framed supports placed at the 
 opposite abutments of the arch ; but in 
 the Waterloo Bridge, to avoid the incon- 
 veniences resulting from crossing the 
 struts, and of building beams where struts 
 of sufficient length could not be obtain- 
 ed from single beams, the ends of sev- 
 eral struts were received into cast-iron 
 sockets placed at their point of crossing 
 and suspended by bridle pieces. 
 
 Figure 5 is a good design for a cocket 
 centre of large span. Here the C F', 
 H F and D d, are placed in the direction 
 of the radii of curvature and made 
 double ; the remaining braces are single. 
 In determining the proper dimensions 
 for the scantling of such a frame, we 
 may take of the total pressure on the 
 arc HH', and suppose it to act at in 
 the direction OF', which will evidently 
 be the greatest load this timber will have 
 to sustain. The strain upon the E DF 
 will then be equal to J the load on B H, 
 and that on H F as \ D C, That on the 
 
61 
 
 beams E F and F F r is to be found from 
 the diagram of forces, Fig. 5. Here hf 
 which is in the direction of H F produc- 
 ed, represents the pressure on this beam; 
 EA is drawn parallel to EF, and ef 
 parallel to F F', which being measured 
 give the strain on EF and FF' respec- 
 tively. If it is desirable to obtain the 
 dimensions of the beams with great ac- 
 curacy we may use the following formu- 
 lae : If we assume the relation between 
 the breadth and depth to be .6 to 1 
 (which is an excellent proportion), then 
 for an inclined beam whose angle of in- 
 clination to the horizon is j3. 
 
 0.6 
 And for a horizontal beam 
 
 (15) 
 
 d = 
 
 0.6 
 
 In which a is to be found from the ex- 
 
 
 
 pression . > '^---=a, in which b is 
 -L W 
 
 the deflection of a beam whose breadth 
 
62 
 
 is b y depth is J, length L, and load W. 
 For pine this quantity a is from .0112 to 
 .0105, and for the best oak .00934. Eq. 
 15 or 16 will give the depth in inches. 
 If it so happens that the value of a, in 
 the above equation, cannot be obtained 
 either by actual experiment or from 
 tables, we may make the square of one 
 side equal to twice the square of the 
 other, which will give a ratio of 7 to 5 
 very nearly, and use the equation 
 
 cos b 
 
 Where w is the load, I the length, and b 
 the angle the beam makes with the ver- 
 tical, and d the dimension of the smaller 
 side, equal f of the larger. In centre 
 frames, however, such a degree of exact- 
 ness is rather unnecessary, since, by al- 
 lowing 1,000 Ibs. to the square inch we 
 can obtain the cross section from the 
 load with all the accuracy desirable in 
 practice. 
 
 The transversal strain on any one back- 
 piece or segment of the rib under the 
 
63 
 
 laggings may be obtained from the ex- 
 pression 
 
 S=P sec ... (17) 
 
 <f> being the angle the backpiece makes 
 with the horizon, and P the vertical com- 
 ponent of the pressure on the same piece 
 found by any of the methods already 
 explained, or from 
 
 . . . (18) 
 
 W being the pressure on each lineal foot 
 of the segment, L its length ; r the 
 radius of curvature at the point in ques- 
 tion, x the distance of the lower end 
 of the backpiece from the vertical 
 through the crown of the arch and the 
 centre of curvature, and h the distance 
 between the two ends of the segment 
 measured vertically. 
 
 The strain upon any one of the lag- 
 gings will depend, independent of the 
 weight of the arch stones, on the dis- 
 tance of the ribs from centre to centre, 
 the place the lagging occupies in the 
 arch and the manner in which the lag- 
 
64 
 
 gings are attached to the backpieces of 
 the frame. As regards the latter point, 
 there are two ways of making them fast 
 to the rib. They may be placed directly 
 on the backpiece and nailed to it, or they 
 may be mounted on folding wedges 
 placed between each bolster or lagging 
 and the rib, which latter arrangement 
 will be considered in detail when we 
 come to speak of the striking plate. 
 The bolsters, moreover, may be placed on 
 the rib in such wise that they touch each 
 other, or may be separated by a space 
 equal to their own breadth. The former 
 method is most usually resorted to in 
 the construction of brick arches, and is 
 illustrated in Fig. 4 ; the latter is used 
 in building stone arches, and is illustrated 
 in Fig. 2. By separating the laggings 
 in this wise a considerable slving of tim- 
 ber is effected, while the air is also given 
 freer access to the joints of the arch and 
 the mortar much sooner dried. When 
 these pieces are separated, it is evident 
 that the cross section of each must be 
 slightly greater than when they are 
 
65 
 
 placed touching each other, and that the 
 section of the laggings placed near the 
 crown should be larger than those near 
 the angle of repose. This latter point 
 is not worth considering in practice un- 
 less the arch stones are very heavy, for 
 in arches of the ordinary span and weight 
 the saving thus effected in the timber is 
 hardly worth the labor of calculation. 
 In determining the proper dimensions of 
 the laggings, it is sometimes customary 
 to insure against any deflection, by sup- 
 posing the entire load on each lagging to 
 act at its middle point and calculate for 
 a beam strained in this manner. 
 
 BRACING. 
 
 It is to be observed in connection with 
 the matter of bracing, that the frames 
 should be arranged in such wise that no 
 piece suffers any strain other than com- 
 pression or extension in the direction of 
 its length. As it is, however, by no 
 means an easy matter to make the dis- 
 tinction, we shall give the following rule 
 to which there is no exception : 
 
Suppose we have two beams abutting 
 Against each other at their upper end, 
 and loaded at their point of intersection 
 with a weight. Take notice of the direc- 
 tion in which this straining force acts,. 
 and from the point at which it acts 
 draw in this direction a line representing 
 by its length the intensity of the strain. 
 From the remote end of this line draw 
 lines parallel to the two pieces on which 
 the strain is exerted. The line drawn 
 parallel to one must of necessity cut the 
 other or its direction produced. If it 
 cut the beam itself the piece is compress- 
 ed y and acts as a strut. If, on the other 
 hand, it cuts the direction of the beam 
 produced^ the piece is stretched and acts 
 as a tie. We may then lay it down as 
 a general rule in framing, that if the 
 piece from which the strain comes lies 
 within the angle formed by the pieces 
 strained, the strains these sustain are of 
 the opposite kind to that of the straining 
 point ; if that inputting, they are push- 
 ing ; if that is compressed they are 
 stretched. Again, if the piece from which 
 
6V 
 
 the strain comes lies within the angle 
 formed by the direction of the two pro- 
 duced, all will have the same kind of 
 strain ; and, finally, if within the angle 
 formed by the direction of one produced 
 and the other piece itself ', the strain will 
 be of the same kind as that of the most 
 remote of the two beams strained, and of 
 the opposite kind to that of the nearest. 
 
 The object of all bracing, then, being 
 to convert all transversal strains into 
 others which act in the direction of the 
 length of the beams, the frame must be 
 divided into a number of triangles ; for 
 as the triangle, or some modification of 
 it, is the only geometrical figure which 
 possesses the property of preserving its 
 figure unaltered so long as the length of 
 its sides remain constant, it is the figure 
 best suited for structures in which rigid- 
 ity is essential for stability. But, again, 
 some forms of triangles are much to be 
 preferred to others ; the strength of the 
 pieces forming the triangle depending 
 very much on the angle they make with 
 each other. Oblique angles are to be 
 
68 
 
 avoided. Acute angles when not accom- 
 panied by oblique are not so injurious, 
 because the strain can, in such pieces, 
 never exceed the straining force ; but in 
 an oblique angle it can surpass it to any 
 degree. 
 
 In all forms of bracing, too much at- 
 tention cannot be given to the joints. 
 Where the beams stand square with each 
 other, and the strains are also square 
 with the beams and in the plane of the 
 frame, the common mortise and tenon 
 is the most perfect joint, a pin usually 
 put through both so as to draw the tenon 
 sight into the mortise, and so cause the 
 thoulder to butt very snugly. Round 
 pins are much better than square ones, 
 as they are not liable to split the bit. 
 Where the beams are very oblique, it is 
 difficult to give the foot of the abutting 
 one such a hold as to bring many of its 
 fibres into actual contact with the beam 
 butted on. It would, in such case, seem 
 proper to give it a deep hold with a long 
 tenon. Nothing, however, can be more 
 injurious, for experience has fully proved 
 
69 
 
 that they are very liable to break up the 
 wood above them and push their way 
 along the beam. For instance, suppose 
 the head of an inclined strut abutting 
 on a horizontal beam to descend a little; 
 the angle with this latter beam is dimin- 
 ished, by the strut revolving round the 
 stress in the tie beam. By this motion 
 the bed of the strut becomes a powerful 
 fulcrum to a very long lever ; the tenon 
 is the other arm and very short. It 
 therefore forces up the wood above it and 
 slides along the horizontal beam. This 
 may be prevented by making the tenon 
 shorter, and giving to its toe a shape which 
 will make it butt firmly in the direction 
 of the thrust, on the solid bottom of the 
 mortise. When the beam is a tie the 
 joint must depend for its strength on the 
 pins or bolts, and the iron straps placed 
 across it. 
 
 STRIKING THE CENTRES. 
 
 Undoubtedly the most dangerous opera- 
 tion connected with the use of bridge 
 centres is the process of striking them. 
 
70 
 
 No matter with how much care the arch 
 may have been constructed, the drying 
 and squeezing of the mortar will cause 
 it to settle in some degree when the cen- 
 tres are removed, and this degree of 
 settlement seems to be very largely af- 
 fected by the time the centres are allow- 
 ed to stand. By some it has been urged 
 that the centring should never be remov- 
 ed until the mortar in the joints of the last 
 course has had ample time to harden ; 
 others going to the other extreme have 
 advocated striking the ribs as soon as the 
 arch is keyed, claiming, not without 
 some reason, that the settlement of a 
 well built arch will never be so great as 
 to become dangerous even though the 
 supporting frames be removed when the 
 mortar is green. But possibly the best 
 practice lies not far from either of these 
 extremes. It has, indeed, time and 
 again, been amply demonstrated that to 
 leave the centring standing till the mor- 
 tar has hardened, and then take away all 
 support, the mortar having become un- 
 yielding, is to cause the courses to open 
 
11 
 
 along their joints. To strike the centre, 
 on the other hand, when the arch is 
 green will, seven cases out of ten, be 
 followed by the fall of the bridge ; but 
 by easing the centring as soon as the 
 arch is keyed in, and continuing this 
 gradual easing till the framing is quite 
 free from the arch, the latter has time 
 to settle slowly as the mortar hardens, 
 and the settlement will be found to be 
 very small. 
 
 It becomes necessary, therefore, to pro- 
 vide some arrangement by which the 
 framing may be slowly lowered from the 
 soffit of the arch, an operation accom- 
 plished in a variety of ways ; by folding 
 or double wedges, by striking plates, by 
 bearing irons and screws, by cutting off 
 the ends of the principal supports, and, 
 finally, by plate iron cylinders filled with 
 sand. The folding wedges are, perhaps, 
 most commonly met with in practice, 
 and are finely suited for arches of small 
 span, as a sill stretching from abutment 
 to abutment may then be used to rest 
 them on. They consist of two hard- 
 
72 
 
 wood wedges, about 15 in. long, right 
 angled along one edge, and placed one 
 upon the other in such wise that the 
 thick end of one shall be over the thin 
 end of the other, thus making their sur- 
 face of contact an inclined plane. These 
 wedges are placed under the tie beam of 
 the rib and on the sill, as is illustrated 
 in Fig. 2. It is evident that by driving 
 the upper wedge up along the inclined 
 surface of the lower, the rib which rests 
 upon the upper one must rise, so that 
 by placing a number of these folding 
 wedges under each rib it may easily be 
 keyed up to the desired level, and by 
 driving the upper down the inclined sur- 
 face of the lower, the rib may gradually 
 be lowered. To keep the under wedge 
 in place, it is usually made fast to the 
 sill and the surface of contact of each 
 wedge well greased with soft-soap and 
 black lead. When the wedges are in 
 place under the rib, it is a good practice 
 to mark each wedge at the point where 
 contact ceases, so that when the centres 
 are being lowered we may be able to 
 
73 
 
 know whether they are lowered uniform- 
 ly or not. For instance, let the lower 
 wedges of three pair of folding wedges 
 project two inches beyond the end of the 
 upper ones, and mark with chalk on the 
 side of each lower wedge the point 
 where contact ceases; namely, two inches 
 from its end. Now, if in striking the 
 centres the upper wedges have all been 
 driven back so that the end of each in- 
 stead of being at the line is one inch be- 
 yond it, then the frame has been uni- 
 formly lowered ; but if some are one 
 inch and some f inch from the line, the 
 frame has not been lowered uniformly, 
 and the difference must be corrected by 
 driving all the wedges till they are one 
 inch from the chalk line. 
 
 It is evident that such an arrangement 
 of folding wedges can be of but little 
 use unless the horizontal beam or sill on 
 which they rest is rigidly supported from 
 beneath, as any yielding of the sill 
 would be followed by a separation of 
 the wedges and rib. In constructing 
 bridges of wide span over creeks or riv- 
 
74 
 
 ers on which there is no navigation to be 
 interrupted, it is usual to make use of 
 the folding wedges and support the sill 
 by a row of piles driven into the river 
 bed, and it then becomes especially ne- 
 cessary to watch the wedges lest by 
 some settling of the piles and sill they 
 have separated in the smallest degree 
 from the tie beam of the rib. 
 
 In cocket centres the folding wedges 
 are replaced by a sriking plate placed at 
 each end of the rib, and sustained by 
 strutting or raking pieces which abut 
 either on off-sets at the foot of the pier 
 or on sills placed on the ground. Each 
 plate consists of three parts, a lower and 
 upper plate and a compound wedge 
 driven between them. The upper of 
 these plates is of wood made fast to the 
 base of the rib, and is cut into a series 
 of offsets on its under surface (see Fig. 
 4). The lower plate is likewise of wood 
 cut into offsets, but on its upper surface, 
 and is firmly attached to the raking 
 pieces which sustain it. The compound 
 wedge consists of a beam cut into offsets 
 
75 
 
 both upon its upper and* lower sides so 
 as to fit those of the two plates, and 
 when driven between them is held in 
 place by keys driven behind its shoul- 
 ders. 
 
 Previous to the time of Hartley, the 
 rib was struck in one piece by the use 
 either of wedges or striking plates. To 
 him, however, we are indebted for an 
 improved system of striking or easing 
 the centres by supporting each lagging 
 upon folding wedges. When this ar- 
 rangement is used the rib is firmly at- 
 tached to its supports, and the laggings 
 rest upon wedges placed between them 
 and the back pieces of the rib. A great 
 advantage gained by this, is that the 
 laggings may be removed course by 
 course from under the arch, and replaced 
 if the settlement prove to be too great 
 at any one part of the soffit. Another 
 method, at one time much in use among 
 French engineers, is to cut off the ends of 
 the chief supports of the rib piece by 
 piece, an operation which cannot be ac- 
 complished with much regularity, nor 
 without much danger. 
 
76 
 
 The least objectionable way of strik- 
 ing centres, and one accomplished with 
 great ease and regularity is by the use 
 of sand, confined in cylinders. A num- 
 ber of plate iron cylinders one foot high 
 and one foot in diameter are placed upon 
 a stout platform sustained by timber 
 framing. The lower end of each cylin- 
 der is stopped by a circular disc of wood 
 of an inch thickness fitting tightly into 
 the cylinder, and at about an inch above 
 this wooden bottom three or four holes 
 an inch each in diameter are drilled 
 through the iron sides of the cylinder 
 and stopped with corks or plugs of 
 wood. 
 
 Into the cylinders thus prepared is 
 poured clean dry sand to a height of 9 
 or 10 inches above the bottom, and on 
 this sand in each cylinder rests a cylin- 
 drical wooden plunger, which fits so 
 loosely as to work with ease, and forms 
 one of the vertical supports of the rib. 
 To prevent moisture getting at the sand, 
 the joint between the plunger and cylin- 
 der is filled with cement. So long as the 
 
77 
 
 sand is dry it remains incompressible to 
 any weight that may press on it, and the 
 rib is thus kept invariably in its place. 
 When the centre is to be lowered, the 
 plugs are taken out of the cylinder, and 
 as the sand runs out of each with uniform 
 velocity the frame is uniformly lowered. 
 This method is of especial value for cen- 
 tres of great weight. 
 
 The distance at which the frames or 
 ribs of centres should be placed apart, 
 measuring from the centre of one rib 
 to that of the next, must be regulated 
 solely by the weight of stone used for 
 the arch, the distance varying inversely 
 with the increase of weight. That is 
 to say, if we assume some distance for 
 stones of a given weight, say 6 feet for 
 stones weighing 150 Ibs. per cubic -yartt, jp*TTf 
 and wish to find the proper distance 
 apart of the ribs when the stones weigh 
 but 120 Ibs. per cubic y#F<L we 
 
 150 : 120*. 15:4. Then making 6 ft. the 
 distance for 150 lb., 
 4 : 5; *.6 : x 4 #=30 x 7 ft. 6 in., 
 
78 
 
 the proper distance for stones of 12o 
 Ibs. per cubic -jSmL The following table 
 has been calculated in this manner : 
 
 Weight of Stone Distance apart 
 
 per i of the 
 
 Cubic <*ed. <T**f Rib of Centring. 
 
 120 Ibs.... 7 ft. 6 in. 
 
 125 Ibs 7 ft. 3 in. 
 
 130 Ibs ; tf ft. 11 in. 
 
 135 Ibs 6 ft. 8 in. 
 
 140 Ibs 6 ft. 5 in. 
 
 145 Ibs 6 ft. 2 in. 
 
 150 Ibs 6 ft. in. 
 
 155 Ibs 5 ft. 10 in. 
 
 160 Ibs 5 ft. 7 in. 
 
 165 Ibs 5 ft. 5 in. 
 
 170 Ibs 5 ft. 3 in. 
 
 175 Ibs .5 ft. Hin. 
 
 180 Ibs 5 ft. in. 
 
 185 Ibs 4 ft. 10J in. 
 
 190 Ibs ...4 ft. 8 in. 
 
 195 Ibs 4 ft. 7 in. 
 
 200 Ibs 4 ft. 1 in. 
 
 It now remains to consider briefly, the 
 subject of centring as used in the con- 
 struction of the arched roofs of tunnels. 
 In work of this description, the span 
 being always small, the arch light and 
 
79 
 
 the facilities for obtaining firm points of 
 support for each rib as great as can be 
 desired, all the hindrances, that so often 
 make the framing of a stone bridge 
 centre a matter of no small difficulty 
 and foresight, are wanting, and the rib 
 admits of a simplicity of arrangement 
 at once favorable to economy of mate- 
 rial and of space. It must, however, be 
 remembered that although the span is 
 small and the arch light, the strength of 
 the rib of a tunnel centre must be much 
 greater in proportion to the burden it 
 has to carry than that of a bridge cen- 
 tre ; since the former has not only to re- 
 sist the weight of the earth above it, 
 but must also withstand the wear and 
 tear of many destructfui causes to which 
 the latter is never exposed. In tunnel- 
 ing through a hill side, no matter how 
 short the distance, more or less rock will 
 invariably be met with, and more or less 
 blasting must therefore be done, and the 
 shock and flying splinters of rock which 
 accompany each explosion do mucli mis- 
 chief to the ribs by disturbing or injur- 
 
80 
 
 ing them. This cause acts strongly on 
 all parts of the centre, but is especially 
 severe with the leading ribs, which, as 
 the brick work must always be kept well 
 up to the heading, are directly exposed 
 to the violence of each explosion. 
 
 A second cause of injury to the ribs, 
 and one quite as damaging and unavoid- 
 able as the first, is the repeated taking 
 down, carrying forward, and putting up 
 of the ribs every time a length of arch 
 is completed. In bridges, unless the 
 structure is composed of a series of 
 arches, the centring is never disturbed 
 from the time it is first put up until it 
 is finally struck on the completion of the 
 works. In tunneling, however, to avoid 
 the foolish expense of building centres 
 from end to end of the tunnel, it is cus- 
 tomary to construct but one length of 
 twelve or fifteen feet of centring, and to 
 move this forward whenever it becomes 
 necessary to turn a new length of arch. 
 Thus, for example, we will suppose that 
 we are driving a tunnel through earth of 
 a moderate degree of heaviness, and are, 
 
81 
 
 therefore, using centres consisting of two 
 sets of laggings and five ribs, two made 
 without and three with a horizontal tie 
 beam. The object in making some of 
 these ribs without the tie beam is that, 
 by so doing, the centring may be brought 
 close up to the heading without interfer- 
 ing with the raking props, which could 
 not be done were the beams to be retain- 
 ed. These five ribs are arranged in 
 practice so that one without the tie 
 beam shall be placed at each end of the 
 length of centring, and between these 
 two are the three with beams. We will 
 suppose this to be the arrangement of 
 the ribs in the present case, and will 
 number them, beginning with that near- 
 est the heading, 1, 2, 3, 4, 5. While the 
 arch is being turned upon this length 
 the excavation for a new one has been 
 made, the invert built, the side walls 
 raised to springing line and all is ready 
 to carry forward the centring. This 
 operation, however, must be done with 
 the utmost caution. If the ribs are 
 taken from under the newly completed 
 
82 
 
 arch before the invert and side walls of 
 the advanced length are built, the whole 
 piece of arch with its side walls will be 
 almost certain to separate from the 
 length just behind, and move forward 
 several inches in the direction the work 
 is progressing. If, on the other hand, after 
 the advanced side-walls are up, all the 
 ribs are taken from under the arch, this 
 latter will be quite certain to come down 
 in ruins, since it has to uphold not only 
 the weight of the earth resting imme- 
 diately upon its bricks, but, in addition, 
 half the weight of the earth which press- 
 es upon the crown bars of the newly exca- 
 vated length, as one end of all these 
 bars rests upon the arch near its end. 
 Rib number 1, then, which is directly 
 beneath the end of the crown bars, can 
 not be removed with any degree of safe- 
 ty. It is also desirable that number 3 
 should be left in place to help support 
 the laggings. Numbers 2, 4 and 5 are 
 the only ribs left, and these are to be 
 taken down and set up forward, taking 
 care that 5, which has no tie beam, is 
 
83 
 
 placed nearest the heading; the order of 
 arrangement then being 5, 4, 2, 1, 3. 
 
 Over the rib thus arranged a second 
 set of laggings is laid, and on them the 
 arch is turned. When this length is 
 completed, and all preparation made to 
 carry forward the centring, the ribs num- 
 bered 4, 1, 3 are taken down and set up 
 forward in the order 1, 3, 4, 5, 2, and so 
 on till the centring reaches the end of 
 the tunnel, or meets that coming from 
 the opposite end of the tunnel, suppos- 
 ing it to be worked both ways. 
 
 Now, it is precisely this continual tak- 
 ing down and setting up of the ribs, 
 that produces so much injury to them, 
 since, in order to pass them under the 
 forward ribs and props which remain 
 standing, it is necessary to take them in 
 pieces. Each rib, therefore, must be 
 framed in such wise that it may be re- 
 peatedly taken apart and put together 
 again without injury to its strength or 
 to the joints of the timbers removed and 
 replaced. Figs. 5 and 6 afford an illus- 
 tration of two centre ribr wamrrvj to 
 
84 
 
 meet these requirements in the simplest 
 manner possible. Fig. 5 is a drawing of 
 a leading or segment rib, which it will 
 be observed is constructed without a 
 complete tie beam at the bottom so as to 
 offer no obstruction to the raking props. 
 It consists of two parts or segments, 
 which, when the rib is placed, join at 
 the crown of the arch and along the line 
 a b, and are made fast to each other by 
 two iron bars placed across the joint at 
 the crown, one on each side of the back- 
 pieces, and bolted through the back- 
 pieces as shown at c c. An additional 
 band is passed around the two vertical 
 beams as shown at d. To prevent any 
 slipping of these beams along the joint 
 a ^ the surface of each beam is notched, 
 as shown at e, and a wedge driven 
 through the notch. When the rib is to 
 be taken down, the band at c c and that 
 at d is removed, and the wedge at e 
 driven out, and the rib thus separated 
 into two segments may be carried 
 through a comparatively small space. 
 As this leading rib is subjected to the 
 
85 
 
 direct effects of the blasts, and to flying 
 fragments of rocks, its joints must be 
 strengthened by irons placed on each 
 side of the rib, over the joint, and bolt- 
 ed through the timbers as shown in the 
 figure. 
 
 This form of rib is finely adapted for 
 tunnel centring, as it may be taken apart 
 without removing a single beam, while 
 its joint is so arranged that the pressure 
 of the arch assists in no small degree to 
 hold its parts in place. Indeed, the only 
 valid reason why this form of rib should 
 not be used in every part of a tunnel 
 centre is the absence of the tie beam, 
 which is certainly a great security against 
 the spreading or contracting of the span. 
 Were this tie beam supplied, and it may 
 easily be supplied by an iron screw rod r 
 this form of frame would probably, in 
 addition to the convenience of taking 
 apart and resetting, sustain any amount 
 of pressure ever likely to occur either 
 vertically or laterally, as also all ordi- 
 nary wear and tear from use. 
 
 Fig. 6 represents one of the interme- 
 
86 
 
87. 
 
 diate ribs called scarf or queen post cen- 
 tres, which, as there are no props to be 
 interfered with, are provided with hori- 
 zontal tie beams. As these ribs are also 
 to be taken apart each time they are 
 shifted, the tie beam is composed of two 
 beams joined by a scarf joint strenthen- 
 ed by a piece of timber placed above it, 
 and bound to the tie by two bands of 
 iron as shown in the figure. The hori- 
 zontal beam joining the queen posts is 
 also movable, and is held in place by 
 the iron placed over its joints and bolt- 
 ed through. In joints thus protected, 
 the holes through which the bolts pass 
 are liable after a time to become so much 
 enlarged, from the repeated driving in 
 and out of the bolts, so as to injure the 
 strength of the joint. This may read- 
 ily be overcome by using a bolt with 
 screw threads at each end in place of a 
 bolt with a head and one nut, so that 
 when once driven through the beam it 
 need not be removed. 
 
 By a comparison of these two forms 
 of ribs, it is evident that while the queen 
 
88 
 
 post centre possesses an advantage over 
 the segment form in that it is not liable 
 to lateral spread, it is at the same time 
 inferior to the former in many important 
 points. It cannot so well resist shocks 
 or side blows, and being so taken to 
 pieces every time it is moved is very lia- 
 ble to be injured especially at the scarf 
 joint. An additional recommendation 
 for centres constructed on the plans of 
 Figs. 5 and 6, is the small amount of ma- 
 terial used, which is quite as small as is 
 consistent with the varying strains the 
 ribs are exposed to, and is so cut that 
 the timbers are almost as valuable when 
 the tunneling is completed as they were 
 when first purchased for the ribs. 
 
 The estimation of the dimensions prop- 
 er to give each tie and brace of the rib 
 is easily determined in so simple an ar- 
 rangement, by any of the methods given 
 for bridge centres. It is, however, to 
 be remembered that, while the bridge 
 centre has to sustain but the weight of 
 the arch stones and bonding mortar, a 
 load which can be calculated to a pound 
 
89 
 
 before one stone is laid, the centring of 
 a tunnel has to resist the pressure not 
 only of the brick roof, but also of the 
 earth above, and that this latter pressure 
 is wonderfully variable. The pressure of 
 the brick work will of course vary when 
 laid in cement and when laid in mortar. 
 From the most careful experiments made 
 to determine the weight of a cubic yard 
 of brick work, we find that when the 
 bricks are laid with cement the weight 
 per cubic yard is 2,897 pounds, or in 
 round numbers 2,900 Ibs.; when laid in 
 mortar beds the weight falls to 2,677, a 
 difference of some 220 Ibs. per cubic yard. 
 It is true that the pressure of the earth 
 does not act to any great extent on the 
 centring, until the arch is turned and the 
 crown bars drawn forward to form the 
 roofing of the newly excavated length, 
 but when this is done, and the three ribs 
 removed to be set up in advance, the 
 pressure on the two ribs remaining under 
 the arch is quite severe. This load is 
 especially variable with the leading or 
 segment ribs, which it will be remember- 
 
90 
 
 ed are placed at the ends of the length 
 of arch, and sustain one end of all 
 the side and crown bars supporting the 
 earth, and the movement which this 
 earth is at any moment liable to take, 
 cannot be foreseen. At times a whole 
 length can be gotten out and the arch 
 turned without any perceptible motion 
 of the earth either at the sides or on top; 
 at others, the earth will of a sudden be- 
 gin to move and throw all its pressure 
 on the side bars ; then, again, the action 
 will take place at the crown and become 
 so great as to press the bars down in the 
 middle through a distance of many 
 inches, or even to break the stoutest 15- 
 inch oak beams. 
 
 This action of the earth, however, 
 seems to be controlled by law, since it 
 depends largely on the depth of the tun- 
 nel below the surface. The pressure on 
 the sides is most severe in those parts of 
 the tunnel which are deepest, and the 
 vertical or crown pressure (and this is 
 always the severer of the two) where 
 the distance below ground is less. At 
 
91 
 
 
 
 first thought this is precisely the reverse 
 of what we should expect to be the case, 
 for it seems but natural tosuppose that the 
 greater the depth of earth the greater the 
 pressure on the arch beneath. The facts 
 are, however, quite the contrary. Thus,for 
 example, in excavating a tunnel through 
 a hill, as we enter the hill side the press- 
 ure is almost exclusively at the crown 
 and very severe; as the work progresses 
 nearer and nearer the centre of the hill 
 where the amount of earth above the 
 arch is greatest, the vertical is changed 
 to lateral pressure, and this latter is in 
 turn changed to vertical as we approach 
 the other end. This is well accounted 
 for, by supposing that in the former case 
 the depth of earth being small, the whole 
 of it gets into motion and acts vertically 
 downwards, while in the latter case the 
 amount of earth being great only a 
 small portion is put in motion. 
 
 The leading rib, then, must be con- 
 structed with no small care, and its joints 
 well strengthened. For tunnels of ordi- 
 nary span, whatever may be the curve 
 
92 
 
 of soffit, we may with safety give the 
 parts the following dimensions. The 
 backpieces two thicknesses of 3 in. plank; 
 the planks breaking joints with each 
 other. For the segment rib make all 
 the braces 6 in. X 6 in. ; the long struts 
 reaching from the half sills to the crown 
 7 in. X 6 in., and the vertical pieces at 
 the crown forming the joint a b also 
 7 in. X 6 i n - F r the queen post centres, 
 make the tie beam 9 in. X 6 i n -> as a ^ 8 
 the short timber placed over the scarf 
 joint ; the queen posts 6 in. X 6 i n > ex> 
 cepting at the upper and lower ends 
 where the braces abut which should be 
 10^ in. X 6 m - 5 tf 16 short piece between 
 the queen posts, and just below the 
 crown 4 in. X 6 m -> an d, finally, the 
 braces 6 in. X H i n - 
 
 The manner of setting the ribs is il- 
 lustrated in Figs. 5 and 6. Under the 
 queen post ribs is placed a long horizon- 
 tal beam, its two ends resting on the 
 side walls and supported immediately 
 under the foot of each queen post by 
 vertical posts. Upon this beam are 
 
93 
 
 placed longitudinally four thick planks, 
 and on these rest the folding wedges. 
 The segment ribs are supported in much 
 the same way, each rib by two short 
 timbers, one end of each resting on the 
 side walls and one on a vertical post 
 under the heel of the rib ; on these rest 
 the longitudinal planks which are placed, 
 however, a little oblique to the tunnel 
 since the heel of the segment rib is not 
 so far from the wall as the foot of the 
 queen post. 
 
 It has already been remarked that it 
 is never wise to strike the centres until 
 the side walls of the newly excavated 
 length are up, as in work of this class 
 there is a strong tendency to move for- 
 ward in the direction of the excavation. 
 If, however, the ribs are struck in the 
 manner already described, with the lag- 
 gings of the back length kept tight up 
 to the arch by the two frames left under 
 them, we shall always have two lengths 
 of completed work remaining with their 
 supports, not only until the next length 
 is excavated but till the side walls are 
 
94 
 
 built and ready for the ribs. Under 
 such circumstances each length is well 
 able to uphold its burden till it receives 
 assistance from the next advancing one, 
 the construction of which to springing 
 line occupies several days, and the ce- 
 ment or mortar has time to harden be- 
 fore the weight comes upon the arch 
 after striking the centring. When, how- 
 ever, from false motives of economy, 
 only three ribs and one set of laggings 
 are used, the entire support of one 
 stretch of arch must be removed before 
 another can be commenced, and this, 
 again, before a third is turned, leaving 
 the green arch unsustained. in which 
 state it is liable to give way, the bricks 
 to crush and the whole arch to come 
 down in utter ruin. Nowhere, indeed, 
 among all the variety of engineering 
 works will a penny wise economy more 
 surely prove a pound foolish one than 
 here ; nowhere else will an unwise sav- 
 ing lead to so profuse an outlay. 
 
 Tunnel centres again differ from those 
 of bridges in that the laggings are very 
 
95 
 
 differently adjusted. In the later case 
 it is the custom in practice to place all 
 the laggings on the ribs before commenc- 
 
 OO O 
 
 ing to turn the arch, by which means no 
 small degree of stability is given to the 
 ribs. In tunneling, however, where only 
 a few inches of space remains between 
 the backpieces of the frame and the pol- 
 ing which sustains the earth, it would be 
 utterly impossible to turn the arch if all 
 the laggings were put in place before 
 the brickwork is begun. To overcome 
 this difficulty, only a few laggings, say 
 five or six are placed at a time. Thus, 
 starting at the springing line, we adjust 
 six laggings on each side of the frame, 
 and carry the arch up equally on both 
 sides. When it has reached the upper 
 bolster, we add six more, and the mason- 
 ry continued as before, and proceed in 
 this way until very near the crown as 
 shown in Fig. 7, where A A! is the brick 
 work. At this stage of the work the 
 two laggings C C' are placed on the ribs, 
 the top of their inner edges being first 
 rabbeted as shown in the figure. In 
 
9(3 
 
97 
 
 these rabbets "cross" or "keying-in" 
 laggings B, consisting of stout planks 
 18 or 20 inches in width, are laid one at 
 a time beginning at one end of the cen- 
 tring. The bricklayer whose duty it is 
 to key-in the arch stands with his head 
 and shoulders between the brickwork 
 A, A, and starting at the end of the last 
 piece of completed arch places the first 
 cross lagging, and keys in the arch over 
 it ; then a second, and in like manner 
 keys in the arch over it, and thus re- 
 treats along the entire opening until the 
 whole length of arch is keyed in. 
 
 Among the varieties of patent centres 
 that planned by Mr. Frazer, affords a 
 most excellent specimen, and both from 
 its strength, economy, ease of shifting 
 and the small amount of space it occu- 
 pies in the tunnel, has met with much 
 approval from the engineering profession 
 in England. This centre consists of but 
 three ribs each differing from the other 
 two in design as shown in Figs. 9, 10 
 and 11, of which 9 is the leading, 10 the 
 middle and 1 1 the back rib. Each rib is 
 
98 
 
 FIG. 9. 
 
 FIG. 8. 
 
99 
 
 constructed of four pieces of timber four 
 and one half in. thick by 16 inches wide, 
 scarfed together as shown in the draw- 
 ings. In centres of the ordinary con- 
 struction, the ribs when the laggings 
 are laid upon them are all of precisely 
 the same size, and of the same span and 
 rise as the soffit of the intended arch. 
 In Mr. Frazer's plan, however, all the 
 ribs differ in the length of their radii ; 
 the Radius of the outer curve of the lead- 
 ing rib (Fig. 9) being greater ; that of 
 the middle 3 inches less than, and that of 
 the back rib yet smaller than the radius 
 of the soffit ; so that the middle centre 
 is the only one of the three which acts in 
 the same way as the ordinary centre 
 frame, that is to say with the laggings 
 and arch resting immediately upon the 
 rib, and is consequently with the lag- 
 gings on it of the same rise and span as 
 ahe arch. 
 
 The leading rib has for its outlet edge 
 a radius 12^ inches larger than that of 
 the arch soffit, and for its inner edge one 
 3^ inches less than the same radius (thus 
 
100 
 
 making the 16 in. thickness) and is 
 plated on * both the inner and outer sur- 
 face with half inch iron plates bolted 
 quite through. The plate on the inner 
 surface is six inches broad and projects 
 2 inches over that side of the rib which 
 is turned towards the middle rib, thus 
 forming a flange on which the laggings 
 rest (see Fig. 9). When this rib then 
 is in place, it must be its whole thickness 
 in advance of the end of the intended 
 arch, and as it stands 12^ inches above 
 the soffit will cover 12^ inches of the 
 toothing ends of the brickwork, thus 
 forming a sort of mould to guide the 
 toothing. 
 
 The middle rib (Fig. 10) is also covered 
 on the under surface with half inch plate 
 iron in one piece and bolted through as 
 shown in figure, thus giving the rib the 
 strength it would have if supported by 
 the usual struts and braces. The lag- 
 gings rest immediately upon the upper 
 surface of the rib, and therefore the 
 radius of this side must be the same as 
 that of the arch soffit, less three inches to 
 allow for the thickness of the laggings. 
 
Fift. 10. 
 
 FIG. 11. 
 
* 102 
 
 The back rib (Fig. 11) is covered on 
 the under surface with a coating of half 
 inch plate iron an one piece, which is 
 bolted through as in the case of the mid- 
 dle rib. Between each bolt a hole is 
 made quite through the rib and its 
 plating, and in it is placed the stem of 
 a bearing iron. There are as many of 
 these irons as there are laggings, the ob- 
 ject of using them being to support the 
 laggings which it will be observed do 
 not rest on the rib but on the projecting 
 irons. The amount of projection is regu- 
 lated by means of adjusting screws, by 
 screwing which the laggings may be 
 raised to the required level, or by un- 
 screwing lowered one by one from the 
 arch when completed. These last two 
 ribs are permanently attached to trestling 
 by brackets, straps and bolts, and the 
 trestling in turn mounted on iron rollers 
 which run on half timbers laid longitud- 
 inally as a kind of tramway. They are 
 also steadied at the crown by long iron 
 hooks attached to one rib and fitting into 
 eyes in the other. 
 
103 
 
 The leading rib is supported upon 
 slack blocks placed on top the brick- 
 work of the side walls and by the prop 
 A. This prop, to allow for any inequali- 
 ties of the invert on which it rests, is 
 mounted at the lower end on a screw by 
 which it may be raised or lowered. 
 
 In setting this patent centre, the lead- 
 ing rib is first brought forward into place 
 and wedged up on the edge of the brick- 
 work to its desired level, and the prop A 
 screwed up tight under the heel. The 
 trestles bearing the middle and back 
 ribs are then rolled forward till the mid- 
 dle rib is at the proper distance from the 
 leading one. Three pairs of wedges are 
 then placed between the bottom piece of 
 the trestles and the tramway, and the 
 trestles thus wedged up until the top of 
 the middle rib is on a level with the 
 flange of the leading one, thus giving two 
 level bearings for the laggings. The 
 bearing irons of the back rib are then 
 pushed out by the adjusting screws until 
 the top of each of them is also on a level 
 with the flange of the leading rib. The 
 
104 
 
 three bearings then, of each lagging, 
 when the ribs are thus arranged is first 
 upon the flange of the leading rib, then 
 upon the middle rib itself, and finally 
 upon the bearing irons of the back rib. 
 When this centre is to be again moved 
 forward on the completion of this length 
 of arch, a fourth rib called the "jack 
 rib " is first fixed under the laggings in 
 the rear of the back rib, this last named 
 rib consists simply of a band of iron 1 
 inch thick by 2 \ wide, bent into the shape 
 of the arch. Opposite every alternate 
 joint of the laggings a screw passes 
 through the rib, and is furnished on its 
 outer end with a square head similar to 
 that of the bearing plates of the back 
 rib, and on its inner or lower end is a 
 loop so that it may be easily turned with 
 a lever. The object of placing these 
 screws opposite each alternate joint is 
 that by this arrangement only half as 
 many screws are needed as there are 
 laggings. The jack rib is itself support- 
 ed at each end by an iron bar 2 feet 
 long driven temporarily into the wall. 
 
105 
 
 As soon as this latter rib is adjusted 
 to take the ends of the laggings, the 
 wedges are driven from under the tres- 
 tles and its rollers thus brought down 
 upon the tramway prepared for them. 
 When thus lowered, it is evident that 
 the two ribs (middle and back) will be 
 so much below the leading rib which is 
 left standing that they will easily pass 
 under it. The trestle and its ribs is then 
 moved forward until the back rib is 
 within 8 inches of the ends of the lag- 
 gings, when it is wedged up as before. 
 The bearing screws are then screwed up 
 tight against the laggings, giving these 
 latter the same support hitherto obtained 
 from the leading rib, which now stands 
 between the middle and back rib. The 
 wedges under the ends of the leading 
 rib (see Fig. 9) are then removed and 
 the rib carried forward over the top of 
 the middle rib and adjusted, as previously 
 described, on the top of the newly built 
 side walls. The laggings are then drawn 
 forward one or two at a time as they are 
 needed, beginning at the springing line. 
 
106 
 
 The great advantage which these 
 patent centres appear to possess over 
 those of the ordinary construction, is the 
 total absence of all struts, ties and braces, 
 thus leaving a fine open space for the 
 scaffolding and materials of the masons. 
 The amount of repairs also is very trivial, 
 as they are not so liable to be injured by 
 flying rocks. In point of economy, 
 though the first cost of patent centres is 
 much greater than that of the segment 
 or queen post centres, the amount expend- 
 ed in repairing the latter soon makes up 
 the difference. In point of strength, it 
 must be acknowledged that, when work- 
 ing through heavy earth, the patent cen- 
 tre of three ribs is by no means so reli- 
 able as the all-wood centre of five ribs 
 and two sets of laggings, used as above 
 described. And this is certainly a seri- 
 ous objection in that, it is impossible to 
 tell beforehand at what moment, owing 
 to a fault or to the displacement of the 
 local beds, the character of the earth may 
 change completely from a light soil to 
 one of great heaviness. 
 
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