Analysis of the versions of the explosions.
«We live in conditions of the threatening spread of terrorism.
And this means — it is necessary to unite all the forces of society and the state to repel the internal enemy. This enemy has no conscience, pity and honor.
No face, nationality and faith.»
(from the televised address of the President of Russia B.N. Yeltsin, September 14, 1999)
A domestic explosion or a terrorist act?
The actions of law enforcement agencies, civil defense and emergency response units, city utilities, medical institutions, city authorities and even federal authorities depend on a quick and correct answer to this question.
Explosion Resistance Scientific and Technical Center specialists answer this question using precise mathematical calculations
Early in the morning, at 5:02 a.m., an eight-story building on Kashirskoye Shosse was blown up and completely destroyed. The specialists who quickly arrived at the scene have no doubt that this was a terrorist attack.
[Izvestia, September 14, 1999].
On September 8, 1999, at 11:58 p.m. in the residential area of Pechatniki, as a result of a powerful explosion, two entrances of a nine-story apartment building on Guryanova Street were blown up and completely destroyed.
[ITAR-TASS, September 9, 1999].
Explosion in Volgodonsk on September 16, 1999
As a result of the explosion carried out by terrorists in the courtyard of a residential building in the city of Volgodonsk, a crater with a radius (R) of 6.5 m and a visible depth (D) of 3.5 m was formed. A residential panel building located 15 m from the center of the crater was heavily damaged. All the wall panels of the multi-story building facing the center of the explosion were destroyed and fell off from the rest of the building, but the building itself remained standing.
There are two versions of the cause of the explosion.
The first of them suggests that the explosive was planted in a sewer manhole, which was located under the GAZ-54 vehicle. According to the second version, the explosive could have been detonated in the vehicle itself.
Version of an explosion in a sewer manhole
Let's assume that the explosive was planted in a sewer manhole.
Let's make a preliminary assessment of the value of the explosion index n:
where B is the radius of the funnel; h is the line of least resistance (depth of charge placement); H is the depth of the funnel.
The visible depth of the funnel H is always less than h — the line of least resistance, which can be 3-10°o greater than H. Using the data in Table II (from the book by V.N. Rodionov et al. «The Mechanical Effect of an Underground Explosion». Moscow: Nedra, 1971), we see that the closest in value of n is an explosion produced in clay soil. In this explosion, a 0.115 t explosive charge was detonated at a depth of h = 2.65 m, resulting in the formation of a crater with a radius of B = 4.8 m. Applying the law of geometric similarity, we find the value of h during the explosion in Volgodonsk:
We finally estimate the value of the explosion indicator in Volgodonsk: 
«The total number of victims of the explosion in the city of Volgodonsk in the Rostov region has reached 310 people. According to the regional North Caucasus Center for Civil Defense and Emergencies, 17 people died, including two children…»
(«Rossiyskaya Gazeta», September 17, 1999)
Therefore, in our version, the explosion index n = 1.81. This makes it easy to establish the explosion energy in TNT equivalent, assuming that the explosion occurred in a well:
Let us estimate the values of the parameters of the air shock wave (ASW) formed when explosion products break through a soil dome rising at high speed. Unfortunately, in the extensive literature on blasts in soils there is no data at all on the parameters of the ASW during breakthroughs of gases located under a rising dome.
Since the area of the inner surface of the dome constantly increases during the rise due to the upward bend, tensile stresses arise in the soil and when the soil is deformed by about 2-3%, a network of cracks develops in it. In clay plastic soils this happens later, and in alluvium — earlier. Through these cracks, the products of the explosion break through into the atmosphere.
At the moment of the gas breakthrough, the speed of the dome's rise begins to decrease. At the value of the maximum speed of the dome's rise, when its acceleration is zero, the pressure inside the cavity of the resulting funnel and the weight of the dome are balanced, and this makes it possible to determine the pressure of the explosion products at a given moment.
Let's find the volume of the funnel under the dome, equal, as is commonly believed, to the volume of an inverted cone with a height equal to h and a base equal to the area of the horizontal section of the funnel at the level of the daylight surface.
Volgodonsk is located in the steppe zone. The region is dominated by slightly and moderately water-saturated loams and clays with a bulk density of y = 1.8 tf/m^3. Based on this, we determine the weight of the soil in the dome, equal to the product of the bulk density of the soil and the volume of soil in the funnel, lifted by the dome:
Let's find the area of the horizontal section of the funnel at the level of the daylight surface:
The value of excess pressure that balances the weight of the soil dome is determined by the formula:
At the moment of gas breakthrough through the thickness of the dome, a spontaneous rupture disintegrates due to differences in pressure and temperature in the explosion products and in the atmosphere.
Solving the problem of the decay of an arbitrary rupture, we obtain for the considered version of the explosion the value of excess pressure on the outer surface of the dome equal to 15 kPa. At a distance of 15 m to the wall of the nearest residential building, the excess pressure will be about 6.5 kPa. With such a value of excess pressure affecting the building, in addition to the destruction of glass, there could not be other more serious destruction. Therefore, we can already say that the version of the explosion in the well is out of the question.
Let us estimate the amount of energy contained under the dome at the moment of the breakthrough of the explosion products into the atmosphere. At this moment, as shown by the estimates made, taking into account the fact that during the development of the dome, additional space in the form of a spherical segment was formed under it, the volume of gases is approximately 1.5 times greater than the volume of the previously calculated soil cone and is about 240 m ^ 3.
The amount of energy contained in the explosion products is found from the expression 
where p0 is the atmospheric pressure, k is the adiabatic index.
In TNT equivalent, given that 1 kg of TNT is equivalent to 4240 kJ, this is 23.15 kg.
Let's estimate the share of energy accounted for by the explosion products at the moment they break through the dome: 
With the funnel parameters most often used in ejection explosions (n = 1-2), this share in a wide range of charge weights from 0.1 to 981 te is within 4-11%. It is this share of energy that is spent on the formation of the shock wave.
It is of interest what was the maximum speed of the dome rise vm:
where E is the charge weight, kgf (explosion energy in TNT equivalent); h is the charge placement depth, m
The impact of clods of soil flying at a speed of 69.5 m/s on the building could have caused great destruction, but they could not have caused the destruction that actually took place.
Taking into account all of the above, we come to the conclusion that the version of the explosion in the sewer manhole is incorrect.
Version of the explosion in the GAZ-54 vehicle
According to the second version of the explosion in Volgodonsk, the explosive substance (EV) was planted in the GAZ-54 vehicle.
In case of a ground explosion of a charge of explosive on the ground surface according to the method of the V.V. Kuibyshev VIA, the radius of the funnel is B=0.84*C^1/3. Hence, the weight of the charge (the total energy of the explosion in TNT equivalent) for the known value B=6.5 m is calculated by the formula:
In a ground explosion, 15-18% of the charge energy is spent on forming a funnel and loosening the soil. Consequently, only 85-82% is used to create a shock wave. Thus, Suv = 0.82 x 463 = 380 kgf.
The given distances for a ground explosion are found from the dependence
where R is the distance from the charge center. The distances from the explosion center to different floors of the residential building were different, as were the angles at which the shock wave front met the wall surface at the level of different floors. Therefore, the problem of oblique reflection of the shock wave arose.
Excess pressures at the wave front were calculated using the formula of M. A. Sadovsky, and the values of excess pressure reflected at different angles of meeting were calculated using the Friedrichs and Courant algorithm. According to the calculation, the values were 302 kPa on the first floor and 35 kPa on the sixteenth floor, and the values were 1147 and 56 kPa, respectively.
The duration of the positive phase of the shock wave was estimated using the formula
Thus, at a distance of 15 m from the center of the explosion on the first floor, the wave action time was 17.55 ms (effective action time 8.95 ms), and at a distance of 43 m at the level of the sixteenth floor — 39.6 ms (effective time 29.8 ms).
We compare these values with the periods of natural vibrations of the slabs — wall panels Tpl and the building itself as a whole Tзд. The period of natural vibrations of a reinforced concrete slab with dimensions of 0.15×2.7x4m, taking into account the weakening by window openings, is about 60 ms. Thus, the load on the panels should be calculated as impulsive. The period of natural vibrations of a multi-story building as a whole is about 3000 ms, therefore, its stability is also determined by the impulsive load.
The equivalent static load under the influence of an impulsive load is determined as the product of the specific impulse and the frequency of natural vibrations:
Since the specific impulse acting on each floor on the panel and on the entire building is the same, the equivalent static load transmitted to the panels and to the entire building will be inversely proportional to the periods of their natural oscillations, i.e. the load on the panels will be 50 times greater (3000/60) than on the entire building. This will lead to the wall panels collapsing, thereby saving the entire building from possible overturning.
In reality, this is what happened: the wall panels from the first to the last floor collapsed, and the entire building remained in place. (In this case, the wall panels played the role of a protective collapsing screen in relation to the entire building).
The calculations show that the version of an explosive explosion in a car is more plausible.
Explosion in house 6, building 3 on Kashirskoe shosse on 13.09.99
«The power of the explosion in house 6, bldg. 3 on Kashirskoe shosse was such that only debris remained of the house. The explosive charge was from 200 to 250 kgf of TNT. At the same time, the nature of the destruction of neighboring houses (in most of them only windows were broken) allows us to state unequivocally that the explosion was not volumetric, therefore, the charge was planted in the basement of the building…» (From an interview with the head of the Scientific and Technical Center «Explosion Resistance», Academician A. Mishuev, to Russian media)
The blown-up eight-story brick building was located in a block bordered to the north by Khlebozavodsky Proezd and to the east by Kashirskoye Shosse (see the layout diagram in Fig. 1). The block is built up with 8- and 9-story brick buildings, with the exception of 2-story buildings: bldg. 6, bldg. 4 (kindergarten) and bldg. 8, bldg. 5, as well as one five-story building of school No. 543 (bldg. 6, bldg. 2). The foundations of the 2-story buildings are located on a terrace rising above the rest of the block by approximately 1.8-1.9 m.
The block is greened with tall trees up to the fourth or fifth floor. In the middle of the block there is a five-story building — 6, bldg. 2 (school #543). The exploded building on Kashirskoe Shosse, 6, bldg. 3 occupies a position in the block slightly shifted to the west from the center of the block.
General analysis of the state of the block and the explosion
Across the entire area of the block, in all directions, the ground surface is covered with a large number of brick fragments and concrete structures. The highest density of fragmentation is observed in the direction of bldg. 6, bldg. 4 and in the gap between buildings bldg. 8, bldg. 4 and bldg. 8, bldg. 3. This gives grounds to assume that the center of the explosion was in the southeastern part of building bldg. 6, bldg. 3. All the walls of the neighboring buildings facing the center of the explosion are covered with finely dispersed orange brick dust. The same dust settled on the leaves of the trees and stuck to their trunks, especially on the side of the center of the explosion. Such dustiness is typical after explosions of explosives (HE) that have a high brisant (crushing) effect, and is practically absent in explosions of gas-air mixtures, which, due to the low pressures developing during internal explosions (10-15 kPa), do not have a brisant effect. For example, during ground deflagration and detonation explosions of gas-vapor-air mixtures (GVAM), an explosion funnel is not formed.
On this basis, an unambiguous conclusion was made: the explosion that occurred was a sabotage explosion of explosives.
The blast wave has a spherical shape, and in the case of a ground explosion — hemispherical. Considering that the destruction of the glazing was mainly on the first floors of the surrounding houses, we conclude that the center of the explosion was at the level of the first floor of the building 6, bldg. 3 or in its basement. In this case, all distances from the center of the explosion have the smallest values at the ground level, and the pressure in the air shock wave (ASW) — the highest values.
Explosion energy assessment
The explosion occurred inside the building, so it could be expected in advance that the air shock wave outside the exploded building would account for a small portion of the explosion energy. The most reliable assessment could be made based on the destruction of the glazing on the wall of building no. 6, K.I, facing the center of the explosion. Moreover, there are no buildings in the path of the VSHW propagation in this direction. The destruction of the glazing on this wall corresponds to the impact of a reflected wave with an intensity of about 10 kPa. Consequently, there were about 5 kPa in the transmitted wave. The distance from the center of the explosion to the wall in question is 84 m. These data allow us to estimate the value of the energy in TNT equivalent that fell to the share of the shock wave.
To determine the reduced distance R for a ground explosion, we use the formula
From here, the energy of shock wave formation is:
At a pressure at the front of the passing wave of 5 kPa, the reduced distance is R = 19.55 m/kgf ^ 1/3. Thus, Suv = 39.66 = 40 kgf.
We will estimate the total energy of the explosion using the formula given in the “Uniform Safety Rules for Blasting Operations”:
According to the «Unified Safety Rules for Blasting Operations», in the event of complete destruction of glazing, partial damage to frames, doors, damage to plaster and internal light partitions, the coefficient K for a charge buried to its height has a value of K = 2-4. If the charge is located on the ground surface, the specified coefficient has a value of K = 8-10. In this case, it is advisable to take the value K = 6. Then the total explosion energy:
Thus, about 20% of the total explosion energy was spent on the formation of the external shock wave, and 80% of the total explosion energy was spent on the destruction of the building.
On the nature of the explosive substance (HE) used
According to the latest data, the explosion in this case involved a mixture of ammonium nitrate (20%) and TNT (80%) mixed with granulated sugar. The use of granulated sugar is important not only for masking the explosive. During an explosion of explosives, granulated sugar serves as a kind of tamping (stopper), ensuring the complete reaction of the entire mass of explosives. In addition, granulated sugar, converted into powder during the explosion of explosives and coming into contact with atmospheric oxygen, explodes itself. This dust explosion of finely dispersed organic combustible material enhances the throwing (scattering) effect of the explosion.
This combination of explosives and combustible material was apparently worked out in advance. Other combinations can be expected, for example, with coal dust, flour, aluminum powder, etc.
The influence of mutual shielding of neighboring buildings and green spaces on the nature of destruction
At the time of the examination of the consequences of the explosion, it was established that the height of the piled-up remains of the exploded building in its northern part reached the height of the fourth floor of the neighboring nine-story building (building 4, building 3), and in the southern part — the height of the second floor. This indicates that the charge was located in the southeastern part of the building, building 6, building 3. Over the entire area of the block, the effect of the shock wave was somewhat weakened by the crowns of large trees.
Table 1 shows the parameters of the shock wave for a ground explosion of explosives (TNT) weighing 40 kg in an open area, at distances tied to individual corners of buildings located in the block exposed to the shock wave (see Fig. 1).
In Table 1, point «O» is the center of the explosion. The distances from the center of the explosion to the points indicated in Fig. 1 are determined using the Atlas of Moscow. The excess pressures Drd, at the shock wave front are calculated using the formula of M.A. Sadovsky, written for ease of calculation in Horner's form:
where R is the reduced distance, m/kgf^1/3. The values of the velocity pressure qф at the wave front are calculated using the formula
The reflected pressure was determined using the expression
The values of the air shock wave parameters given in Table 1 are of an estimated nature. They are valid for flat open terrain. The actual distribution of the air shock wave parameters was strongly influenced by the mutual shielding of buildings and large trees with unfallen leaves growing on the site. Nevertheless, the data in Table 1 allow one to orient oneself in the order of magnitude and are therefore useful in analyzing the nature of destruction and identifying anomalies.
Since the TNT equivalent for a shock wave is 40 kgf, the action time of the positive phase of the air shock wave T^ is very short and is, for example, 7.8 and 18.5 ms, respectively, at distances of 18 and 100 m from the center of the explosion.
This means that the destruction of buildings with a natural oscillation period of more than 1000 ms (significantly longer than the duration of the positive phase of the wave) will depend on the magnitude of the impulse in the shock wave, and the destruction of glass with a small oscillation period will depend on the magnitude of the excess pressure in the shock wave.
Specific impulse of a shock wave, determined by the dependence
is 325 and 58 Pa at distances of 18 and 100 m, respectively. This specific impulse is small and cannot damage the building.
Buildings located next to the exploded building, bldg. 6, bldg. 3, received damage mainly to the glazing, and only at close distances (bldg. 8, bldg. 4) were the internal partitions damaged by the shock wave that had leaked in. The glazing of windows that were not in the geometric shadow of the buildings in front were destroyed by the combined action of flying brick fragments and excess shock wave pressure. Individual buildings that had walls perpendicular to the direction of the shock wave movement were affected by the reflected wave (L-shaped walls in the plan of bldg. 4, bldg. 3; bldg. 4, Bldg. I and bldg. 6, bldg. 1).
On the nature of the destruction
Research conducted at MGSU on the patterns of glazing failure showed that they are probabilistic in nature and subject to the Weibull distribution law. This is due to the fact that glass made from silicate sand from different quarries has different strength data. During the technological process of manufacturing, transportation and installation, glass gets various microcracks that weaken it.
The nature of the failure depends on the thickness of the glass, the shape and size of the sheet, the method of fastening the glass (on clamps, glazing beads, putty), and also on the rows of glazing. Double and triple glazing are much stronger than single glazing.
To significantly increase the stability of glazing in houses, it may be recommended to use frames with small cells. Such a measure is effective in residential buildings and in industries where explosive mixtures are not formed.
A survey of the consequences of the terrorist attack on Kashirskoe Shosse showed that when exposed to a short air shock wave moving along the surface of the glass, the resistance of the glass to destruction increases. (This is shown in the example of the glazing of the facade of school No. 543, where on the upper floors it remained intact when exposed to an oncoming wave with an intensity of 15-7 kPa.)
The survey showed that the explosion in the building on Kashirskoe Shosse, Bldg. 6, Bldg. 3 is a consequence of a terrorist act.
The explosion center was located on the first floor in the southeastern part of the building, which led to the asymmetrical nature of the destruction.
The main and massive nature of the destruction in neighboring buildings is the destruction of glazing, which is extremely dangerous in winter, leading to the defrosting of water heating systems and their failure for a long time.
The destruction of glazing is observed mainly on the lower floors and on the walls facing the center of the explosion, which were also exposed to the flow of brick debris.
Glass in frames with smaller cells (opening side transoms, windows of school No. 543) in many cases remained intact.
The total energy of the explosion in TNT equivalent is estimated at approximately 200 kg, of which about 40 kg was converted into shock wave energy. The explosive substance, according to a number of features, was a mixture of ammonium nitrate and TNT mixed with granulated sugar. Granulated sugar increases the completeness of the energy release of TNT and enhances the propelling effect of the explosion.