# Great Neapolitan Earthquake of 1857/Part I. Ch. XVII

CHAPTER XVII.
VALUES OF THE COEFFICIENT $\scriptstyle {\mathrm {L} }$.
Before concluding this section, it remains to assign the values of the coefficient $\mathrm {L}$ for practical use.
It consists of two factors: the tenacity or resistance to rupture, by a force suddenly applied; and the specific gravity of the mass fractured off, by direct pull from an unit of section.
When a direct force, producing fracture by extension, is gradually applied to any prism, whose length and section are both unity, the work necessary to produce the rupture is
| $\mathrm {W} ={\frac {1}{2}}\mathrm {P} l$ | ${\mathfrak {A.}}$ |
$\mathrm {P}$
$l$
$\mathrm {P}$
$2\mathrm {W} =\mathrm {P} l$
${\frac {\mathrm {P} }{2}}=$
This force we suppose applied by the weight of a prism of the material fractured, whose base is the unit of section fractured; or being the specific gravity
| $\mathrm {L} =L\delta ={\frac {\mathrm {P} }{2}}$ | ${\mathfrak {B.}}$ |
​When the question relates to the fracture of a homogeneous body—such as a column shaft, of one block of stone for example then the force $\mathrm {P}$ to be taken, is that which applies to the material, and $\delta$ its sp. gr. But when the fracture occurs in walls of whatever sort, it takes place by the giving way, by loss of adhesion (generally), or sometimes of its own cohesion, of the mortar or other cement, as being the weakest part of the heterogeneous mass: in which case, $\mathrm {P}$ is to be taken for the rupturing force of either the adhesion or cohesion (as the case may be) of the mortar or cement, and $\delta$ the specific gravity due to the whole mass of masonry.
Fracture seldom or never occurs through the solid stone, in masonry, but always at the mortar joints, and generally by their loss of adhesion, to the stone at the faces of the joint. It rarely occurs through the brick, in brickwork, and only when the cohesion of the brick itself, is less than that of the cement.
To enable these equations to be applied generally, in earthquake countries, I have arranged the two following tables, I. and II., which embrace almost all the reliable information we as yet have, applicable to the matter, and from which, the value of $\mathrm {L}$ may be deduced, for a great variety of cases.
Many of the numbers, for want of better experimental data, can only be viewed as approximative.
​The use of the coefficient $\mathrm {L}$, in Eq. XXI. et seq., considers the value of $l$ (Eq. ${\mathfrak {A}}$) evanescent, so that the prism at the moment of fracture has not risen through an appreciable angle, at the surface of fracture, and from the extremely small extensibility of mortar, stones, &c., this is sufficiently true to nature.
Table I.
Factors for the coefficient $\scriptstyle {\mathrm {L} }$.
It appears, from the few experiments that have been made, that the resistance of stones, &c., to tension, varies from ${\tfrac {1}{8}}$th to ${\tfrac {1}{10}}$th the resistance of the same material to ​compression. The third column is here calculated on the mean of such data. It cannot be viewed as more than an approximation, except in the cases of mortars, which are from actual experiment, as given by Gauthey ('Sur la Construction des Ponts'), and by Rondelet ('L'Art de Bâtir').
Table II.
Of the specific gravities, cohesion, and mutual adhesion, of various building materials. Factors for the coefficient $\scriptstyle {\mathrm {L} }$.
​In the preceding table, the cements had, in all cases, six months to indurate, and the mortars (except in the second case) from six months' to seventeen months' induration.
Examples of very old and good, lime and sand mortar, may be found occasionally in good brickwork—such as that of the Roman amphitheatre at Pozzuoli, for example; or in rubble masonry, where the bond of the stone with lime mortar is peculiarly strong, as with the oolitic building stones, and limestones generally, and with a few sandstones and porous traps, in which the adhesion, of the indurated mortar, becomes fully equal to its cohesion, and both rise above 50 lbs. to the square inch, for forces suddenly applied.
In determining the mean specific gravity, of brickwork and rubble masonry, the proportion of mortar, to the brick or stone in a given volume, may be taken at from ${\tfrac {1}{12}}$th to ${\tfrac {1}{7}}$th, according to the goodness of the work.
Table III.
Deduced values, under different conditions, for the coefficient $\scriptstyle {\mathrm {L} }$.
​These values of $\mathrm {L}$, are all, for the mortar when yielding in cohesion. When observed to yield in adhesion, the coefficient in each case becomes 0.1,$\mathrm {L}$ for brickwork and 0.083,$\mathrm {L}$ for limestone. The values given, are also all for ancient and fully indurated (except 5 and 9) mortar; where the latter is under twenty-five years laid, the value of $\mathrm {L}$ should be taken (quam prox.) at 3/5 $\mathrm {L}$ in the table.
Proceeding now to