III.
ON THE NATURE OF THE ARCH.
779.
WHAT IS AN ARCH?
The arch is nothing else than a force originated by two weaknesses, for the arch in buildings is composed of two segments of a circle, each of which being very weak in itself tends to fall; but as each opposes this tendency in the other, the two weaknesses combine to form one strength.
OF THE KIND OF PRESSURE IN ARCHES.
As the arch is a composite force it remains in equilibrium because the thrust is equal from both sides; and if one of the segments weighs more than the other the stability is lost, because the greater pressure will outweigh the lesser.
OF DISTRIBUTING THE PRESSURE ABOVE AN ARCH.
Next to giving the segments of the circle equal weight it is necessary to load them equally, or you will fall into the same defect as before.
WHERE AN ARCH BREAKS.
An arch breaks at the part which lies below half way from the centre.
SECOND RUPTURE OF THE ARCH.
If the excess of weight be placed in the middle of the arch at the point _a_, that weight tends to fall towards _b_, and the arch breaks at 2/3 of its height at _c e_; and _g e_ is as many times stronger than _e a_, as _m o_ goes into _m n_.
ON ANOTHER CAUSE OF RUIN.
The arch will likewise give way under a transversal thrust, for when the charge is not thrown directly on the foot of the arch, the arch lasts but a short time.
780.
ON THE STRENGTH OF THE ARCH.
The way to give stability to the arch is to fill the spandrils with good masonry up to the level of its summit.
ON THE LOADING OF ROUND ARCHES.
ON THE PROPER MANNER OF LOADING THE POINTED ARCH.
ON THE EVIL EFFECTS OF LOADING THE POINTED ARCH DIRECTLY ABOVE ITS CROWN.
ON THE DAMAGE DONE TO THE POINTED ARCH BY THROWING THE PRESSURE ON THE FLANKS.
An arch of small curve is safe in itself, but if it be heavily charged, it is necessary to strengthen the flanks well. An arch of a very large curve is weak in itself, and stronger if it be charged, and will do little harm to its abutments, and its places of giving way are _o p_.
[Footnote: Inside the large figure on the righi is the note: _Da pesare la forza dell' archo_.]
781.
ON THE REMEDY FOR EARTHQUAKES.
The arch which throws its pressure perpendicularly on the abutments will fulfil its function whatever be its direction, upside down, sideways or upright.
The arch will not break if the chord of the outer arch does not touch the inner arch. This is manifest by experience, because whenever the chord _a o n_ of the outer arch _n r a_ approaches the inner arch _x b y_ the arch will be weak, and it will be weaker in proportion as the inner arch passes beyond that chord. When an arch is loaded only on one side the thrust will press on the top of the other side and be transmitted to the spring of the arch on that side; and it will break at a point half way between its two extremes, where it is farthest from the chord.
782.
A continuous body which has been forcibly bent into an arch, thrusts in the direction of the straight line, which it tends to recover.
783.
In an arch judiciously weighted the thrust is oblique, so that the triangle _c n b_ has no weight upon it.
784.
I here ask what weight will be needed to counterpoise and resist the tendency of each of these arches to give way?
[Footnote: The two lower sketches are taken from the MS. S. K. M. III, 10a; they have there no explanatory text.]
785.
ON THE STRENGTH OF THE ARCH IN ARCHITECTURE.
The stability of the arch built by an architect resides in the tie and in the flanks.
ON THE POSITION OF THE TIE IN THE ABOVE NAMED ARCH.
The position of the tie is of the same importance at the beginning of the arch and at the top of the perpendicular pier on which it rests. This is proved by the 2nd "of supports" which says: that part of a support has least resistance which is farthest from its solid attachment; hence, as the top of the pier is farthest from the middle of its true foundation and the same being the case at the opposite extremities of the arch which are the points farthest from the middle, which is really its [upper] attachment, we have concluded that the tie _a b_ requires to be in such a position as that its opposite ends are between the four above-mentioned extremes.
The adversary says that this arch must be more than half a circle, and that then it will not need a tie, because then the ends will not thrust outwards but inwards, as is seen in the excess at _a c_, _b d_. To this it must be answered that this would be a very poor device, for three reasons. The first refers to the strength of the arch, since it is proved that the circular parallel being composed of two semicircles will only break where these semicircles cross each other, as is seen in the figure _n m;_ besides this it follows that there is a wider space between the extremes of the semicircle than between the plane of the walls; the third reason is that the weight placed to counterbalance the strength of the arch diminishes in proportion as the piers of the arch are wider than the space between the piers. Fourthly in proportion as the parts at _c a b d_ turn outwards, the piers are weaker to support the arch above them. The 5th is that all the material and weight of the arch which are in excess of the semicircle are useless and indeed mischievous; and here it is to be noted that the weight placed above the arch will be more likely to break the arch at _a b_, where the curve of the excess begins that is added to the semicircle, than if the pier were straight up to its junction with the semicircle [spring of the arch].
AN ARCH LOADED OVER THE CROWN WILL GIVE WAY AT THE LEFT HAND AND RIGHT HAND QUARTERS.
This is proved by the 7th of this which says: The opposite ends of the support are equally pressed upon by the weight suspended to them; hence the weight shown at _f_ is felt at _b c_, that is half at each extremity; and by the third which says: in a support of equal strength [throughout] that portion will give way soonest which is farthest from its attachment; whence it follows that _d_ being equally distant from _f, e_ .....
If the centering of the arch does not settle as the arch settles, the mortar, as it dries, will shrink and detach itself from the bricks between which it was laid to keep them together; and as it thus leaves them disjoined the vault will remain loosely built, and the rains will soon destroy it.
786.
ON THE STRENGTH AND NATURE OF ARCHES, AND WHERE THEY ARE STRONG OR WEAK; AND THE SAME AS TO COLUMNS.
That part of the arch which is nearer to the horizontal offers least resistance to the weight placed on it.
When the triangle _a z n_, by settling, drives backwards the 2/3 of each 1/2 circle that is _a s_ and in the same way _z m_, the reason is that _a_ is perpendicularly over _b_ and so likewise _z_ is above _f_.
Either half of an arch, if overweighted, will break at 2/3 of its height, the point which corresponds to the perpendicular line above the middle of its bases, as is seen at _a b_; and this happens because the weight tends to fall past the point _r_.--And if, against its nature it should tend to fall towards the point _s_ the arch _n s_ would break precisely in its middle. If the arch _n s_ were of a single piece of timber, if the weight placed at _n_ should tend to fall in the line _n m_, the arch would break in the middle of the arch _e m_, otherwise it will break at one third from the top at the point a because from _a_ to _n_ the arch is nearer to the horizontal than from _a_ to _o_ and from _o_ to _s_, in proportion as _p t_ is greater than _t n_, _a o_ will be stronger than _a n_ and likewise in proportion as _s o_ is stronger than _o a_, _r p_ will be greater than _p t_.
The arch which is doubled to four times of its thickness will bear four times the weight that the single arch could carry, and more in proportion as the diameter of its thickness goes a smaller number of times into its length. That is to say that if the thickness of the single arch goes ten times into its length, the thickness of the doubled arch will go five times into its length. Hence as the thickness of the double arch goes only half as many times into its length as that of the single arch does, it is reasonable that it should carry half as much more weight as it would have to carry if it were in direct proportion to the single arch. Hence as this double arch has 4 times the thickness of the single arch, it would seem that it ought to bear 4 times the weight; but by the above rule it is shown that it will bear exactly 8 times as much.
THAT PIER, WHICH is CHARGED MOST UNEQUALLY, WILL SOONEST GIVE WAY.
The column _c b_, being charged with an equal weight, [on each side] will be most durable, and the other two outward columns require on the part outside of their centre as much pressure as there is inside of their centre, that is, from the centre of the column, towards the middle of the arch.
Arches which depend on chains for their support will not be very durable.
THAT ARCH WILL BE OF LONGER DURATION WHICH HAS A GOOD ABUTMENT OPPOSED TO ITS THRUST.
The arch itself tends to fall. If the arch be 30 braccia and the interval between the walls which carry it be 20, we know that 30 cannot pass through the 20 unless 20 becomes likewise 30. Hence the arch being crushed by the excess of weight, and the walls offering insufficient resistance, part, and afford room between them, for the fall of the arch.
But if you do not wish to strengthen the arch with an iron tie you must give it such abutments as can resist the thrust; and you can do this thus: fill up the spandrels _m n_ with stones, and direct the lines of the joints between them to the centre of the circle of the arch, and the reason why this makes the arch durable is this. We know very well that if the arch is loaded with an excess of weight above its quarter as _a b_, the wall _f g_ will be thrust outwards because the arch would yield in that direction; if the other quarter _b c_ were loaded, the wall _f g_ would be thrust inwards, if it were not for the line of stones _x y_ which resists this.
787.
PLAN.
Here it is shown how the arches made in the side of the octagon thrust the piers of the angles outwards, as is shown by the line _h c_ and by the line _t d_ which thrust out the pier _m_; that is they tend to force it away from the centre of such an octagon.
788.
An Experiment to show that a weight placed on an arch does not discharge itself entirely on its columns; on the contrary the greater the weight placed on the arches, the less the arch transmits the weight to the columns. The experiment is the following. Let a man be placed on a steel yard in the middle of the shaft of a well, then let him spread out his hands and feet between the walls of the well, and you will see him weigh much less on the steel yard; give him a weight on the shoulders, you will see by experiment, that the greater the weight you give him the greater effort he will make in spreading his arms and legs, and in pressing against the wall and the less weight will be thrown on the steel yard.