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

by Robert Mallet

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

$\mathrm {W} ={\frac {1}{2}}\mathrm {P} l$

${\mathfrak {A.}}$

$\mathrm {P}$ being the static load gradually applied, and $l$ the amount of extension of the body on the unit of length at the limit of rupture. But if $\mathrm {P}$ be applied at once (suddenly), then $2\mathrm {W} =\mathrm {P} l$, the accumulated work, is twice that necessary for fracture, or ${\frac {\mathrm {P} }{2}}=$ the force, whose tension suddenly applied, as by an earthquake shock, shall rupture the prism.

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 most important numbers by far, are those relating to the adhesion and cohesion, of the varieties of common mortar; and, fortunately, these have been ascertained by Boistard, Gauthey, Treussart, and Colonel Totten, with considerable accuracy.

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 $\scriptstyle {\mathrm {L} }$.

Material	1. Weight in pounds per cub. foot Sp. gr.	2. Pounds per square inch Resistance to Pressure.	3. Pounds per square inch Resistance to Tension.	3. Authority for 1 and 2.
Limestone, Caserta, Naples	170	8173	908	Rondelet
Upper Limestone, Geneva	169	4917	546	Gauthey
Jurassic Limestone, Givry	148	4232	496	..
Cretaceous Limestone (Compeigne)	154	3007	334	Rondelet
Lava, Hard Vesuvian	166	8735	972	..
Lava, Soft Vesuvian	107	2209	246	..
Lava, Piperno (Pozzuoli)	162	8140	905	..
Travertino, Old Roman	147	2297	255	..
Travertino, Pæstum	141	3102	345	..
Peperino, Roman	123	3135	347	..
Tufa, Old Roman	78	797	89	..
Tufa, Naples	82	718	80	..
Hard brick	98	1851	206	..
Soft ill-burnt brick	91	1200	133	..
Mortar, lime, and sand, unground	102	423	47	..
Ditto, ditto, ground	119	577	64	..
Mortar, Pozzolano, of Rome and Naples, unground	92	503	56	..
Ditto, ditto, ground	105	732	81	..
Mortar, Old Roman (Campagna)	97	1047	105	..
Mortar, Old French (Bastile)	94	753	84	..
Plaster of Paris (mean)	..	500	55	Laisne

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 $\scriptstyle {\mathrm {L} }$.

Material	1. Weight in pounds per cub. foot Specific gravity	2. Resistance to Tension lbs. per square inch.	3. Adherent Resistance lbs. per square inch.	3. Authority.
Granite	164	1200?	..	T.
Granite to Portland cement	..	..	97	W.
Granite to Parker's cement	..	..	22	W.
Silurian slate	170	2300?	..	T.
Oolite (Portland)	132	270	..	W.
Oolite to Portland cement	..	..	146	W.
Oolite to Parker's cement	..	..	42	W.
Sandstone, coal measure	147	234 to 250	..	T.
Millstone grit and Portland cement	..	..	76	W.
Sandstone (Whitby) and Portland cement	..	..	57	W.
Kentish rag and Parker's cement	..	..	29	W.
Brick, best English	135	200 to 230	..	..
Brick, inferior	97	40 to 80	..	B.
English brick in Portland cement	107	..	..	W.
Portland cement	127	400	..	W.
Parker's cement	120	300	..	W.
Mortar (sand 3, lime 1)	100 to 119	11 to 20	9.88	B.G.
Green and fresh	..	2	..	T. T.
Mortar, ground lime and tiles	100 to 120	40 to 80	5.26	B. G.
Hydraulic mortar				T. T.
Jurassic limestone to mortar	..	..	3.80	R.
Brick and tile to mortar	..	..	8.27	R.

Authorities.—T., Tredgold. W., White, 'Trans. Inst. C. E.' B., Barlow. B.G., Boistard and Ganthey, 'Const. des Ponts.' T. T., Treussart and Totten. R., Rondelet.

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 $\scriptstyle {\mathrm {L} }$.

No.	Conditions of Fracture.	Value of $\scriptstyle {\mathrm {L} }$
1	Apennine limestone, broken through the stone	225
2	Cretaceous limestone, ditto ditto	154
3	Apennine limestone rubble masonry, of best quality, broken through the joints	52
4	Apennine limestone rubble masonry, of inferior quality, broken through the joints	30
5	Apennine limestone, rubble masonry of best quality, mortar not indurated	3.9
6	Argillaceous rubble masonry of the Murgia (Apennine marl rocks), best quality, with indurated mortar	55
7	Best Italian or Roman brickwork in mortar	63
8	Inferior brickwork in mortar	30
9	Brickwork, the mortar not yet indurated	2.5
10	Rubble masonry of tufa and mortar, good, with mortar indurated	87
11	Rubble masonry of Travertino, or Peperino, and mortar indurated	51

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