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

by Robert Mallet

CHAPTER VII.

DIRECTION OF FRACTURES IN RECTANGULAR BUILDINGS BY SUBABNORMAL SHOCK.

---

We now proceed to the fourth of our classes of waves, namely, the subabnormal, or the wave whose direction of transit, is diagonally in both an horizontal and a vertical plane, passing through the building; a wave which is at once abnormal and emergent.

On referring to Figs. 43 and 44 the general character of the dislocation produced by such will be evident.

The wave emergent in the direction $a$ to $b$ dislodges the portions of the quoin $c$, which it first reaches by inertia, during the time of the first semi-vibration, and those of the diagonally opposite quoin, are thrown forward and often projected out of place also, by inertia in the time of the second half vibration. As explained for the abnormal corresponding to the subabnormal sought; and the path of the latter will be found in a vertical plane passing through

$a'b'$. Join $op$, which is in the same vertical plane, and also in the plane of the fissures or fractures, $mp$, $qp$, and through the point of the quoin $e$ intersecting $a'b'$, draw $ab$ perpendicular to $op$; $ab$ is then the path of the subabnormal wave, emergent in the direction $a$ to $b$. This is tantamount to finding the resultant of all the parallel forces that resisted fracture, and of course assumes, that the masonry fractures equally readily everywhere.

This is practically sufficiently near the fact, except, perhaps, when the horizontal obliquity of the wavepath, or abnormal angle is very great; in that case one wall is broken by direct pull nearly, and the other nearly transversely, which may give rise to an unbalanced couple at $u$, in a line parallel to $qm$; and in that case the wedge-shaped mass, in place of simply sliding down or turning over, in a plane passing vertically through $a'b'$, and falling to pieces, at the base of the quoin, will have a small amount of rotation, either to the right or left of that plane; and the centre of gravity of the mass of debris, will be found correspondingly posited, to the right or left of the base of the quoin. No case has been observed by the author (amongst very many of the class), in which from this cause a perturbation was produced, that could render the determination of the wave-path uncertain in the horizontal element, by more than 2° or 3°.

The arc corresponding to the sine of $teo$, or of $seo$, viz. $\theta$ and $\theta '$ of a former case, gives the abnormal angle or bearing, of the horizontal element of the wave-path, if the building be cardinal: or this, + or − the azimuth of the walls, gives it if ordinal, and the angle $oeu=\theta ''$ is equal the angle of emergence with the horizon, to which it is alternate.

The whole of the preceding may be readily done trigonometrically, and by an observer accustomed to such operations that method will be found more advantageous, as greatly economizing time on the ground, and enabling the results to be worked out at leisure.

Let the dark lines $me$, $qe$ be the level top of the adjacent walls (Fig. 48 bis), $ef$ the quoin, $e$ the solid angle at top, $n$, $w$, $p$, the points of fracture in those lines.

As $ef$ is plumb, $mef$, and $qef$, are each 90°. Let $qem$, any angle be given, and also the distances, $ne$, $we$, $pe$, the two former proportionate to $f$ and $f'$, the component forces that produced the fractures at $n$ and $w$; $f''$ the vertical component corresponding to these; $u$ the intersection of the polar (or direction of emergence of the wave), with the plane passing through $n$, $w$ and $p$.

The angles made by $op$ and $ab$ = 90°.

Calling the angle		$qem$	=	$\phi$
"	"	$neo$	=	$\theta$
"	"	$weo$	=	$\theta '$
"	"	$ueo$	=	$\theta ''$ = $epo$ = angle of emergence
"	"	$wne$	=	$\sigma$
"	"	$nwe$	=	$\tau$
"	"	$\sigma +\tau$	=	$\rho$ = 180° − $\phi$
The arc of		$\phi -\phi '$	=	$\psi$
The line		$ne$	=	$\mathrm {L} =f$
"		$we$	=	$\mathrm {M} =f'$
"		$pe$	=	$\mathrm {N}$
"		$no$	=	$z=wo$
"		$oe$	=	$y$
"		$re$	=	$z={\frac {\mathrm {R} }{2}}$

R being the common resultant of $f$, $f'$, $f''$, in $ab$.

Then $\mathrm {L} +\mathrm {M} :\mathrm {L} -\mathrm {M} ::\tan {\frac {1}{2}}\phi :\tan {\frac {1}{2}}(\theta -\theta ')$

$\tan {\frac {1}{2}}(\theta -\theta ')={\frac {\tan {\frac {1}{2}}\phi (\mathrm {L} -\mathrm {M} )}{\mathrm {L} +\mathrm {M} }}$

$\theta ={\frac {\phi -\psi }{2}}$

$\theta '={\frac {\phi +\psi }{2}}$

But $\sin \theta :\mathrm {L} ::\sin \rho :2\mathrm {Y}$

as the diagonals $se$ and $wn$ mutually bisect
and

$2y={\frac {L\sin \rho }{\sin \theta }}$

but as ${\text{L}}+{\text{M}}:{\text{L}}-{\text{M}}::\tan {\frac {1}{2}}\rho :\tan {\frac {1}{2}}(\sigma -\tau )$

$\tan {\frac {1}{2}}(\sigma -\tau )={\frac {({\text{L}}-{\text{M}})\tan {\frac {1}{2}}\rho }{{\text{L}}+{\text{M}}}}$,

and $\angle$$\tau ={\frac {1}{2}}\rho$, + arc corresponding to, $\tan {\frac {1}{2}}(\sigma -\tau )$
$\angle$$\sigma ={\frac {1}{2}}\rho$ – the same arc.

Again, $\sin \tau :{\text{L}}::\sin \phi :2x$

$2x={\frac {{\text{L}}\sin \phi }{\sin \tau }}$ = the distance from one fracture to the other diagonally opposite

$2y$ = the resultant of $f$ and $f'$

${\text{N}}:y::1:\tan \theta ''$

$\tan \theta ={\frac {y}{\text{N}}}$

which gives the angle of emergence, or that made by the polar $a,~b,$ of the subabnormal wave, with the horizon.

${\text{Cos}}\theta '':y::1:z$

$z={\frac {y}{\cos \theta ''}}$

$2\times {\frac {y}{\cos \theta ''}}$ = R, the common resultant in the polar $ab,$ and, $2y\tan \theta ''=f''$ the vertical component.

An extremely easy method may be practised of finding the path of a subabnormal wave by an observer in the field.

Referring to Fig. 44. Let a line be stretched across the top of the walls (or anywhere below that, but horizontally), from the exterior or interior angle of fracture, on one wall, to that on the other, $w$ to $n$, and the length be divided in the proportion of $ew$ to $en$; if from the dividing point $'p$, a plumb-line be dropped, it will lie in the vertical plane in which the path of the wave is situate. Let now another line be stretched, or a light straight edge of wood be held, between the points $'p$ and $p$ (corresponding to the line $op$, of Fig. 46); lastly, stretch a line from the point $'p$, so that it shall be square to the line $'pp$, and holding it in the hand, "sight it," to coincide visually with the plumb-line: this line or string will then be, in the path of the subabnormal wave, and its azimuth and inclination, may each be at once got, by compass and clinometer, or by two measurements, without the latter instrument. This method admits of quite sufficient accuracy, if the fractured-out pyramid be not too large, but such, that either a straight edge (a straight rafter or joist will answer, of which plenty may generally be found loose about) or a stout cord can be stretched tight across, from $w$ to $n$. The direction can be thus obtained within a degree or two at most.

Fig. 44.

It is obvious that if the path of this wave, be referred to its component path, in either of the two walls, by a plane, normal to a vertical plane, and both passing through the wave-path, then the former plane, will cut the surfaces of the walls, in directions perpendicular to the fracture in each respectively, as in $a'$ $b'$ and $a''b''$ (Fig. 44), which coincides with what was stated before, as to the general. fact, that the lines of fissures from subnormal waves (i. e. those emergent in the

Photo Pl. 50		Photo Pl. 49
	⁠
Vincent Brooks, lith. London
The Cathedral Paterno.	⁠	Auletta.

plane of the wall) are perpendicular to the path of emergence.

The position of the point $p$, or the distance down the quoin, at which the two adjacent fractures meet, apart from any variations caused by differences of masonry, &c., depends both upon the velocity, and the angle of emergence of this wave.

For referring to Fig. 47, if $a^{2}b^{2}$, be the path of emergence, referred to one of the walls, and $w^{2}j^{2}$, be the line of fracture therein corresponding, then as the mass thrown out is greater, as the velocity is so, and as the angle of fracture with the quoin is constant, while the emergence is so, it follows that if the velocity be reduced, the fracture will be somewhere as at $xy$, still parallel to $w^{2}p^{2}$, but higher up, and vice versa, and so for any other emergence: but if the velocity being the same, the angle of emergence vary, then, in order that the line of fracture may still continue perpendicular thereto, the point $p$ must ascend or descend along the quoin as with the path $ab$, producing the line of fracture $wp$, which, when it makes a very acute angle with the quoin, and therefore the angle of emergence small, and the velocity also great, may even follow back along the wall as to $xz$, so that the point $p$, would fall below the base of the wall, if the lines of the fractures were produced.

In the Photog. No. 49, which gives a very good illustration of this class of fracture, as observed at Auletta, this was actually the case. In this view, one of the other large fissures, corresponding to $n$ and $k$ (Fig. 45) may be remarked, as also in Photog. No. 50, at Paterno church.

In the Photog. No. 51 (Coll. Roy. Soc.), also in a street in the town of Polla, a mass, thrown by an extremely oblique subabnormal wave, will be observed to the left hand, in which the effect of this obliquity, and of the miserable class of "nobbly" rubble masonry, and of a floor within, have perturbed the phenomena, as respects the wall, seen most nearly in the plane of the picture. The direction of shock, was in this instance, nearly along the line of the street, towards the spectator, and a little from the left, towards the right; and in further evidence of the general direction here, it should be remarked that the whole side of the street to the right hand where the fronts of the houses are nearly in the line of shock, and supported as well as held in by the floors, &c., against the small transverse component of shock, all remains standing, though propped here and there; while at the far end of the street, the fronts of the houses in the street running obliquely across that going from the spectator are all down.