JPRS ID: 9123 TRANSLATIONS STRUCTURAL DESIGN OF CIVIL DEFENSE SHELTERS BY M.D. BODANSKIY, L.M. GORSHKOV ET AL

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JPRS L/9123
3 June 1980
Translatiar~
STRUCTURAL DESI~N OF CIVIL DEFENSE SHELTERS
By
M.D. Bodanskiy, L.M. Gorshkov et al.
r
' ,y!' ,
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JPRS L/91.23
_ 3 June 1980
STRUCTURAL DESIGN OF CIVIL DEFENSE SHELTERS
Moscow RASCHET KONSTRUKTSIY UBEZHI5HCH i~ Russian 1974 signed to ~
press 14 Feb 74 pp 1-208
Book by M.D. ~3odanskiy, L.M. Gorshkov et al., Stroyizdat Pub-
lishers, 24,000 copies _
CONTENTS
Foreword 2
CHAPTER I. CIVIL DEFENSE SHELTERS
l. Classification of Shelters 4 ~
2. Planning Solutions and Struc~~sral Designs of
Shelters 10
CHAPTER II. DYNAMIC LOADS ON THE STRUCTURAL ELEMENTS OF CIVIL
DEFENSE SHELTERS
1. Parameters of the Air Shock Wave 29
2. Loads on the Structure from an Air Shock Wave
of a Nuclear Blast 37
3. Loans an the Structures from Gas-Air Mixr.ure Blasts 44
Giir~PTER III. CALCULATION OF REINFORCED CONCRETE FLOORS, CEILINGS AND
WALLS ON THE EFFECT OF AIR SHOCK WAVES
1. General Principles 56
2. Calculation of the Structural Elements in the
Elastic Stage 57
3. Dynamic Strength of Reinforcing Steel and Concrete. 63
4. Characteristic Limited States of Bent Structural
Elements 68
5. Calculation of a Hinge-Supported Beam in the
Plastic Stage 71
6. Calculation of a Beam with Clamped Ends in the
Plastic Stage 86 _
- a - [II- USSR - ~'OUO]
[III - USSR - 4 FOUO]
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7. Calculation of a Beam With One Clamped End and the
Other Hinged End in the Plastic Stage 96
8. Calculation of Continuous Beams lb6
9. Calculation of Rectangular Slabs 109
10. Cxample Calculations 117
CHAPTER IV. CALCULATION OF THE STRUCTURAL DESIGNS OF CEILINGS AND
FLOORS UNDER THE EFFECT OF COMPRESSION WAVES IN THE GROUND
' 1. Basic Prerequisites 128
2. Loading Wave Propagation 131
3. Process of Lowering the Pressure (Unloading)....... 135
- 4. Compression Wave Reflection From a Stationary
Barrier 143
- 5. Interaction of Compression ~daves in the Ground With
Fles:~ble Structures 146
CHAPTER V. CALCULA'"ION OF ROCK WALLS AND COLUMNS FOR THE EFFECT
OF SHOCK WAV~ LOADS
1. Loads 152
2. Characteristic of the Limiting States 153
3. Calculation of Outside Stone Walls 156
4. Calculation of Columns and Inside Walls............ 163
CHAPTER VI. CALCULATION OF THE ENCLOSING STRUCTURES FOR THE THERMAL
EFFECTS OF MASS FIRES
L. Calculation Thermal Effects 171
2. Protection of the Enclosing Structures of Shelters
From Heating 175
3. Designing Enclosing Structures of Shelters fox
Heating During Fires 176
4. Effect of Heating on th~ Bearing Capacity of the
Ceilings and Floors and Seal of the Structure...... 184
S. Example Calculations 186
BIBLIOGRAPHY 190
- b - ~
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PUBLICATION DATA
English title :~STRUCTURAL DESIGN OF CIVIL DEFENSE
SHELTERS
Russian title ; RASCHET KONSTRUKTSIY UBEZHISHCH
Author (s) ; M. D. Bodanskiy, L. M. Gorshkov, et al.
Ed~Ltor (s) ,
- Publishing House ; Stroyizdat
Placc. of Publication ; Moscow
Date of Publication , 1974
Signed to press . 14 Feb 74
Copies , 24,000 -
COPYRIGHT , Stroyizdat, 1~74
- c -
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UDC 699.852.001.24
STRUCTURAL llESIGN OF CIVIL DEFENSE SHELTERS
Moscow RASCHET KONSTRUKTSIY UBEZHISHCH ir Russian 1974 pp 1-208
[Article by M. D. Bodanskiy, L. M. Gorshkov, V. I. Morozov, B, S. Rastorguyev] _
[Text] A discussion is presented of ttie calculation of the sugporting struc-
tures of civil defense shelters against the effect of the shock waves from
nuclear blasts, the explosion of gas-air mixtures and heating during fires.
A study is ma.de of the methods of dynamic calculation of flexible reinforced
concrete structures in the elastic and plastic stages under various support-
ing conditions; the extracentric compressed elements brick walls consider-
ing the opening of horizontal cracks; inside columns and walls considering
their ,joint operation with the ceilings and floors and with the dirt founda-
tion.
There are 12 tables, 69 illustrations and 74 references.
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rva vrrl~i~w uor. v..L~ -
FOREWORD -
One of the primary goals of civil defense is the timely design and construc-
tion of structures to protect the population against weapons of mass de-
- struction.
Among such structures a special role is playsd by the civil defense shelters,
- a special role is played by.the civil defense shel~ers, the construction of ~
which requirea defined material expenditures and time. The structural de-
signa of the shelters and their internal equipment must be calculated and
designed so as to protect people from the destructive factors of s nuclear
blaet in the zone of total destruction of buildings, against high tempera- -
tures and smoke in the case of mass fires and also against the effect of
chemical and bacteriological weapons.
The probleme of calculating the structur.al designs of shelters for dynamic
loads occurring under the effect of ~ nuclear blast ahock wave have been
discussed inadequately in the literature and are little known to construc-
tion engineers, planners and designers. At the same time, the methods and
the calculation of protection against the radiation of radioactive fallout
and primary radiation of a nuclear blast are the sub~ect of a broad litera- -
ture [lk, 17, 21, 28, 38, 39, 46]; therefore they will not be investigated
here.
The primary goal of calculating the structural elFUnents of shelters loaded
from a shock wave, the magnitudes of which are tens and hundreds of times
more than the loads normalized in industrial and civil construction, is the
establishment of the dimensions (cross sectinn), guaranteeing the same pre-
sence of people in the shelter under the effect of modern methods of mass
destruction. A characteristic feature of the calculation is the specific
situation which is rare in the ord~.iiary designing of industrial and civil
structures: the structural element~ ~zf shelters are designed for shock
wave load, which will occur once or t~ice daring its entire service life.
Thia makes it possible to approach the selection of the methods of calcula-
ting the structural designs with less rigid requirements, the basic one of
wh'ich consisted in the f act that the structural element must wi.thstand the
load without collapaing [48]. Here it is possible to permit significan re-
aidual deformations .in th~ shelter structures accompanied by the opening of
cracks and large deflec~ions.
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The calculation of structures considering the effect of the changes in their
geometric diagrams in the process of deformation and the consideration of
the working of the material of the structure beyond the elastic strings that
will perm~~ lowering of the expendi.*.ures on the construction and the provi-
sion of prutection for a larger num~er of people.
Iiased on the mentioned prerequisitea, in this paper the specific nature of
the design of ahelters is reflected, the procedure ia given for determining
th~ dynamic loads from the explosian of gas-air mixtures and a nuclear
blast. A discussion is presented of the methods of calculating the basic
bearing structures (elements of reinf orced con~rete floors and ceilings,
- columns, walls and foundation~) on the effect of the dynamic loads and also
on the heating during fires. These methods can find application also for
_ calculation of the structural designs of civil and industrial structures for -
the eff ect of dynamic loads in emergencies. ,
Chapter I was written by L. M. Gorshkov, Chapter II was written by V. I.
~torazov; Chapters III and IV were written by B. S. Rastorguyev with the
participation of V. I. Morozov, Chapter V was written jointly by V. I.
Morozov and R. S. Rastorguyev, Chapter VI and the characteristics of the
mass fires in Chapter I were written by M. D. Bodanskiy. -
The authors hope tha.t the book will be usef ul to a broad class of scientific
workers and engineers enga~ed in designing shelters and specialists working
~.n the field of dynamic structural design calculations.
The authors express deep appreciation to doctor of technical sciences, pro-
feasor ~I. Popov for advice and suggestions which were taken into account
when preparing the manuscript for publication; the authors also express
their heartfelt thanks to candidates of technical sciences V. I. Ganushkin
and A. I. Kostin for their work in reviewing the manuacript.
3
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CHAPTER I. CIVIL DEFENSE SHELTERS -
1. Classif ication of Shelters
Civi.l defense shelters are designed to protect the population against -
means of mass in~ury. The requirements on such st~:~~ctures are defined
beginning with evaluation of the destructive factors of a nuclear blast
and also evaluation of the in~urious erfect of poisons and bacterial means
[1, 2, 23]. For the bearing sttuctures of the shelters, the requirements
arising from the destructive factors of a nuclear blast are defini.ng. _
General Characteristic of the Destructive Fa:tor:~ in a Nuclear Blast
_ A nuclear blast in a populated area causes destruction, fires and radio~-
active casualties. The nature of the destruction of the buildings and
structures by the shock wave of a nuclear blast and in3uries to people is
presented in Table 1. When compiling the table it is considered tYiat a -
shock wa~*e also aff ects people indirectly in~uries from fragments of
destroyed buildings and structures, pieces of g:lass. The in~uries from
- a shock wave are observed at pressures of 0.03 kg/cm2 and more.
The mass destruction of above-ground buildings and structures in citiee
- can be accompanied by blocking of the streets with pieces of walls, metal,
reinforced concrete and wooden beams, brick and other materials which leads
to the formation of one-sided or two-sided obst:ruction of the streets.
- With a shock wave pressure of more than 1.2 kg/c:m2, as a rule, solid
obstructions are formed over the entire built-up area. The characteris-
_ tics of the obstructions (composition, height, d:istance the main part of
the fragments are scattered) are presented in ref'erences [14, 64].
' An unavoidable cv.-.3equence of nuclear blast is the fires caused both by
the direct effect of the light emiesion on open contbustible structural
elements and materials and by the destruction of fu:rnaces, industrial
heating units, damage to the gas ma~ns, electrical networks, and so on.
The conditions for spreading of a fire in buildings s~fter damage of the
bearing structures by the shock wave and destruction of windows and doors
will be ideal [21]. The control of fires is complicated. if the water
lines are out of order.
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Tahle 1
U~~structive Eff ect of the Shock Wave of a Nuclear Blast [14J -
- Killing radius
Pressure at the for a ground
Nature of destruction shock wave blast of
front in kg/cm2 1 million tons
in km -
Destruction of glazing 0.03 31
Destruction of the roofs of buildings, 0.18-0.1 7-11.5 -
partitions, ceilings and floors on
woaden beams
Destruction of wooden buildings 0.2 -0.14 7-10
Contusions and slight injuries to 0.4 -0.2 4.5-7
peop~e
Destruction of stone buildings 0.45-0.35 4.5-5
rtedium serious injurie~ of people 0.5-0.4 4-4.5
_ 5erious injuries of people 1-0.5 2.8-4
~estruction of industrial-type buiidings 1-0.8 2~g_3.2 -
Destruction of strip foun3a~ions of 4 1.5
_ residential buildings
Destruc*_ion of underground reinforced 15-12 0.8-1
concrete pipe 1.5 meters in diameter
with a wall thickness of 0.2 meters
During World War II, mass fires were one of the primary causes of the
destruction of buildings and loss of Iife. It has been documented that
out of the total amount of destruction in large populated areas subjected
to air attacks, up to 80% was caused by fires [69, 73]. After an air
burst of a 20 megaton nuclear bomb as a result of the combined effect of
a ahock wave and light emission the mass fires would occur at distances
of up tn 30 lun from the blast epicenter [49] .
Lepending on the density of building and atmospheric conditions, various
~ypes oi' fires can occur in the city. The individual fires, so~id fires,
storm fires and versions of them cyclone fires, fires in obstructions
a~-e distinpuished.
It is known [21] that the probability of the spread of f ires with less
thar~ 2G% density of building is very low. This is explained by signifi-
canr snac{.i~~; between the bu~ldings which prevents the combination and
a_;tuai :,f,:~:;.t ~f the temperature f~.el.ds. :of: adjacent individual fires.
tinder these conditions even when all of the buildings are burning simul-
taneously in an area with the indicated building density each fire
develops independently. Such ma.ss fires are called individual. With an
increase 3n the building density and corresponding decrease in the spacin$ _
between buildings, individual fires begin to have a mutual effect by the
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radiation f ields. These are solid fires. If the buildings are closer
*_ogether, the thermal fields of individual fires merge in general (storm
fires) ~ -
, The first of these fires came to be called solid f 3res as a result of the ~
fact that over the entire area encompaesed by the fire, as a result of the _
interaction of the radiation fields of the burning objects, approximately
the same temperature regime is established, and the second groun of f ires
are so-called because in their presence merging of the flame of all of the _
burning ob~ects in the investigated area is possible, and the wind veloci-
ties reach storm values. '
In some cases during the occurrence of a storm fire in a section, the dimen- -
sions of which exceed lxl 1_an, its ~rowth into a fire storm or a fire _
cyclone, as it is called, is possible.
Th~ f ire cyclone is characterized by extremely intense burning of various
combustible m~terials with the occurrence of powerful centripetal air
flows and also the formation of a column of heated air and combustion
products rising to an altitude of up to 4000 meters. The f ire cyclone
usually does not go beyond the limits of the region where it occurred.
The velocity of the air flows directed toward the center of Che cyclone
fire reaches 35 m/sec or more, and the air temperature in the cyclone range
approaches 1000�C. Four to 5 hours after the occurrence of a cyclone fire
the region where it occurred is converted to a heap of incandescent struc- _
Cures and destroyed elements of buildings.
In addition to the enumerated types of mass fires which must be expected
in an area where the pressure at the shock wave front does not excee~
0.2 to 0.5 kg/cm2, ,at ~ the li.mits of the zor~e with greater pressures fires
in the obstruction are possible which are distinguished from the mass
fires of other types by the great duration and relatively low air tempera-
tures. The classification of the ~ass fires with respect to expected
atmospheric conditions developing within the built-up areas of cities is _
_ presented in Table 2.
Table 2
- Classification of Mass Fires by the Outside Air Parame!:ers
t~iax:tmum air Probable maximum Duration
Fire Type of temperature in concentrations in ~ of the fire
cate or fire �C CO CO 0 in hours -
I Storm 600~800 1.2 4.8 12.5 to 4
II Solid 500-600 0.5 2.4 16.5 6-8
III Individual to 200 0.3 1,4 18.5 6-8
IV In obstructions to 40 0.2 0.8 20 , to 24
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I.n order to discover the parts ~f the bt~ilt-up areas of cities in danger
from the point of vie~: of the occurrence of one type of mass fire or
another, i~ is possible in the first approximation to use the simplest -
proce3ure for estimating fire hazard ir~ reference [14] .
~ Here it is necessary to consider that the territory of categcry ITI of
fire hazard, the dimensions of which are less than 20Q ~00 meters9 located
wi*_hin the bounda;-ies of the calculated section of category I(II), is
al.l evaluated by the category of the last-mentioned section, and for
~ dimensions of greater than 200x200 meters with respect to category I(II)
only the 100-meter strip around its perimeter is evaluated. This correc-
tion takes into account the calculated spread of the increased tempera-
_ tures and concentrations of the products of combustion to a distance of up
to 100 meters from the burning buildings located in the sections of higher
category oF their hazard by co~parison with the investigated. territory.
People can receive burns from the light emission during a nuclear blast.
In the casz of a ground l~urst of 1 million tons in clear weather the
exposed parts of the skin will receive f irst degree burns at a distance of
9-12 km from the radiation sourcey second degree burns at a distance of
7-9 km, and third degree burns at a distance of 5-7 km. People in build-
- ings can escape burns from light emission, but suffer from fragments of
the destroyed buildings and also from other injurious factors of the
r.uclear blast: penetrating radiation and radioactive contamination.
_ Along with the shock wa~e and light emission the penetrating radiation is
ene of the decisive factors when determining the requirements on shelters.
Penetrat:ing radiation Kahich acts up to 15 seconds aFter the time of the
blast consists basically of gamma radiation and a neutron flux. The
- gaimna radiation is the most dangerous in view of its high penetrating
capacity, large radius of effect and capacity to disperse in the air. In
the case of ~ 1-million ton ground burst the gamma radiation dosage will
be as follows [14):
At a distance of 4 tan (pressure 0.5 kg/cm2) 4 roentgens
At a distance of 2.8 km (pressure 1 kg/cm2) gp
At a distance of 2.26 km (pressure of 1.5 kg/cm2) 1000 " -
At a distance of 2.km (pressure of 2 kg/cmz) 4000 "
At a distance of 1.78 km (pressure o= 2.5 kg/cm2) 10000 "
As a r.esult of fallout of the radioactive particles on the ground from the
cloud of a nucJr.ar blast over large territories, radioactive contamination
~:akes placc:, "I't~~tis, for e~ct~ntple., for a 1-million ton blast at the edge of ~
a large city on the windward side with a wind velocity of 24 1an/hour, the
part of the area of the city c~vered by the zone with a contamination
level at the outer boundary of 10 roentgens/hour, 1 hour after the blast -
will be 60 to 70% of the entire area of the city; with a level of contam-
ination of 100 roentgens/hour, 40-50%, and with a contamination level of
300 roentgens/hour, 30-40%.
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If the city is covered by a radioactive cloud from adjacent blasts, the
degree of radioactive contamination of the territory increases by several
times. The radioactive contamination of the area which is dangerous to
. people can occur at a distance of hundreds of kilometers from the center
_ of the nuclear blast. The gamma radiati~n from the radioactive fallout
- causing general external irradiation is of primary danger to man. The
radioactive particles can get into the organism through the respiratory
organs, together with water and food, cau~ing internal irradiation.
Taking the simplest measures of protection of the respiratory organs from
dust and observing sanitary-hygienic requirements when eating~ it is
possible ~o exclude or significantly de~rease the incidence of radioactive
materials in the organism. _
There are no effective measures of protection against external irradiation
except a shelter. Therefore, irradiation presents the primary danger, and
protection from it is the main problem in areas of radioactive contamina-
tion.
The protective properties of buildings and structures with respect to
_ gamma radiation are characterized by~ the radiation attenuation factor
which indicates how many ti.mes the irradiation dosage of man is diminished
~.n the building (structure) by comparison with the irradiation dosage in
- the open. The admissible dosage of the total single external irradiation,
considering its quickest af tereffect4, must not exceed 25 to 75 roentgens.
A more detailed characteristic of the injurious factors of a nuclear blast
can be found in references [10, 14, 21, 25~ 39, 49].
~ In the set of destructive factors connected with the effect of modern
means of destruction, it is necessary also to consider the so-called
secondary destructive factors which are a consequence of the effect of the
shock wave and light emission of the nuclear blast. These include
destruction of reservoirs and various process units with easily inflammable,
_ combustible, explosive and poisonous liquids and gases, as a result of
which destruction of the surrounding buildings and structures, intense
fires, and so on occur.
Requirements on Shelters
Considering the discussed peculiarities oF the eff ect of the destructive
factors of a nuclear blast, it is poss3,ble to formulate the following
. basic requirements on civil defense shelters.
In the cities which can become the target of a nuclear attack, it is
necessary to protect people from a11 of the destructive factors of a
nu~lear blast. The shelters in these cities must have emergency exits
far independent exit of people to the ground service in case of destruc- _
- tion of above-ground buildings and structures and the formation of
obstructions. The enclosing structures of shelters must have the required
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thermal resistances, preventing heating of the inside surfaces during
fires. In built-in shelters the ceiling must not be punctured by individual
falling .r~.gments when the buildings are destroyed.
In rural populated areas and small cities which are not the target of.
nuclear attack, basically protection from radioactive contamination muat
be provided. However, near the limits of cities which can become the
target of a nuclear attack, the shelters must also withstand the effect
ot r.l~e shock wave (of corresponding less intensity), and in individual
cases, the secondary destructive factors (for example, protection against
poisons which can spread in the direction of the prevailing wind far
beyond the city limits af ter ~he containers are destroyed in the city).
One of the most important requirements on civil defense shelters is the -
possibility of filling them with paople in a short time measured in
minutes. For shelters outside the cities which can become the target of
a nuclear attack, this time can be greater and will depend on the distance
to the ciry and the propagation rate of the radioactive cloud.
The civil defense structures must insure effective protection, that is,
essentially diminish the possible losses of population under wartime
conditions with the application of weapons of mass destruction; at the
same time the construction of these structures must not cause significant
additional expenditures. The basic increase in cost of the buildings and
structures adaptable as shelters is conn~cted with providing the required
strength of the c:tosl.ng structures designed for the impact of the shock
wav e .
Classification of She.lters
The civil defense shelters are divided into regular shelters and radiation-
proof shelters. The regular shelters are designed to protect people
against the damaging effects of a nuclear blast and also poisons and `
bacteri.al means. The radiation-proof shelters are designed to protect
peoPle against radioactive contamination (radioactive radiation, the
incldence of radioactive particles on the surface of human skin and also
chrough the respirat.ory organs).
Ti~.e shelters are classj.fied with respect to protective properties, loca-
tioi2 and erection time.
',d:~.~-3i ~~spect to protective praperties the regular shelters are divided
into �i.ve classes [23], depending on the calculated pressure of the air
_ sh~c': ~-.~ve of the nuclear b].ast. -
i:h~ -~ciiatiUn-~Proof shelters are divided into three groups with respect _
to degree of attenuation of radioactive radiation.
With respect to location the shelters can be built in (in the buried part
of buildings) or separately standing (located outside buildings).
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The separately standing shelters are divided into trench shelters and
underground shelters. The covering of the trench-type structures is
located, as a rule, at ground level. The structures of the underground
type are at significant depths and are protected by the natural thickness
of the ground.
The shelters are erected in advance, with the application of permanent
incombustible material, structural elements and internal equipment of
industrial manufacture.
When it is necessary to erect them in short time, local building materials ~
(including wood) and internal equipment manufactured by the population
and local industry are used to build the shelters.
The shelters (specially the regular shelters) are located near the places
where the people work and live in view of the limited time to take shelter
after the air alarm signal is given.
The most acceptable are the built-in shelters which can be filled with
people in the shortest tims. The distance from a separately standing
shelter to the exit f rom a building must not exceed the gathering radius,
the procedure for determination of which is presented in reference [14].
The separately standing shelters are located insofar as possible so that ~
one of the entrances will be removed from the nearest building by a
distance no less than the height of ti?is building and can serve as emer-
gency exit. If this solution is impossible, a special emergency exit is
provided.
2. Planning Solutions and Structural Designs of Shelters
Planning Solutions
The space-planning and structural designs of shelters are determined by
the requirements of protection against weapons of mass destruction and the
conditians of their operation and maintenance in peacetime in the national
_ economy. This combination of functions in one shelter is advantage~usly
economic and promotes more intense accumulation of the shelter pool.
During peacetime the shelt~rs can be used as domestic and production facil-
ities which do not require natural lighting, garages for cars, commercial
and public eating enterprises, warehouses and other facilities. The
space-planning solutions of such faciliti.es must correspond most completely
- to their purpose, insure the required sanitary-hygienic conditions for
the people working in them, and be economical and simple to the maximum
with respect to composition. The shelters must be designed considering
the erection of them by industrial methods coordinated with the structural
elements of the buildings and with the enclosing construction.
10 ~
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In the shelters (Fig 1, 2) provision is made for basic (facilities for
shelt~r:t~~ people, lQCk-type entries) and auxiliary facilities (f ilter-
ventilati~~r, chambers, sanitary facilities, diesel electric power plants).
The tieig'tit of the facilities is taken in accordance with the requirements
of their use in peacetime, but no less than 2,2 meters from the level of
the floor to the bottom of the protruding structures of the ceiling.
In a fac:ility for sheltering people provision must be made for places to
s:it cz� ~l.e dawn.
_ At one of the entr.ances to the shelter a lock chamber is built (single-
chamber or double-chamber depending on the capacity of the shelter),
the purpose of which is to prevent the danger of shock wave injury of the
people in the shelter when the doors are opened. Therefore the outside
and inside sealed protective doors of the lock chambers must have blocking
to exclude si.multaneous opening of them. Af ter the shelter is filled it
~.s expedient to use the lock chambers for people in the sitcing position.
The dimensions of the filter-ventilation chamber (FVK) and the facility
Lor the diesel electric power plant (DES) must be determined from the
condition of the minimum required space for the equipment.
In view of the great variety of versions of the use of the shelters in
peacetime it is expedient to standardize their space~planning solutions
wir.h Y:he c31~plication of standard spans and standard column network. Con-
sidering the significant calculated load on the enclosing structures of
the shelters, it is necessar.y to limie the size of the spans and the -
~ spacl.ng of. the columns. The ceiling span is usually taken equal to 6 m,
and the column spacing, 6, 4.5 or 3 meters.
The insicle longitudinal bearing walls are used comparatively rarely, for
in this case the operation and maintenance of the facilities in peacetime
are compJ.icated. However, in cases where this solution ts possible, it
must be widely used as a resul.t of the following aduantages: a decrease
in height of the facility, reducti.on of ttie types of prefabricated cover-
j.n~ ~lements from 3(slab, beam, column) to 1(slab), a decrease in depth
c~ the pit (replacement uf the column foundations with strip f oundations),
S~implicity of erection.
Fo~ ;~?ie].te-r.s located in the basements of one-story production buildings or
~o~� ::;e;-~arateJ.y standing shelr_ers, a column grid can be used di~t:inguished
from r},e standard one inasmuch as it is not connected with the structural
elentc~,~.r:= of the above-grcund part of ~hP building. Some decrease in the
~ol.u:r,_, F:. i';,r example, fram 6x6 to ~?x4 meterG, wi11 permit a reduction
~_{Z ti;e ~~,~r~s..~cnption o~ materials wlthout having a significant negative
~ffect on the possibilities for use of the facilities in peacetime.
- 11
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.I
5
~q 2
' ~ ~ G"r,
~
0 o a
. g�, a~ 3
~
2
, 6
3000~ � ~ � 3000
}==~-t---~"-�-" k
Figure 1. Planning a storage-shelter for 150-200 people in
' a basement.
1-- facility for sheltering people; 2-- lock chamber;
3-- filter-fan chamber; 4-- sanitary facility; 5-- elevator;
6 emergency exit
`Gi" 1 ~
I ~a-8 ~
I o000 ~
g~ e
9 0~00
~
9 � p 6 6 0 0
U-1lL~i' p o 0 6 N
0
N ' ~ 5
~
�D~ ~ , ~ � 1 ~
, ,
i ~ aeooe I
~ b b ,
Figure 2. Planning a high-capacity shelter in the basement of
a one-story industrial building with a column grid of 4x4 metera.
- 1-- prelock; 2-- lock chamber; 3-- lock; 4-- facility for
sheltering people; 5-- medical room; 6--- sanitary facility;
7-- water tank; 8--- housekeeping stoi::ige; 9-- produce storage;
10 ventilation chamber; 11 electric panel; 12 cooling
unit of the diesel electric power. plant (DES); 13 DES; 14 ~
- fuel and lubricant storage; 15 food waste storage; 16
drinking water; 17 benches and bunks ~
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Fig 2 shows the planning of a shelter in the basement of a one-story
building with a 24--meter span. In pea.cetime the basement i~ used as a
storaoc ~o~ the materials of an industrial enterprise. The shelter is
des~.gne:~ iui- 1000 people. There are three entrances in the shelter: two
with doors 120x200 cm and one with gates 180x240 cm. Loads which are
lowere~ from the rirat floor by a bridge crane are sent through the gates.
The loads are transported within the storage by electric cars. The
capacity of the shelter can be increased with placement of the people in
s~crage racilities and in the medical room wnich do not enter into
the standardized list of mandatory facillties.
If with respect to the conditions of operation and maintenance of the
. shelters in peacetime it is necessary co have a large-span facility with-
out insic~e walls and columns, then for separately standing shelters it is
expedient to use the arch enclosures (see Fig 3). In this case the
entrancss a.nd auYiliary facilities are placed at the ends which are made
- a:E f.lat structural elements.
For operation and maintenance in peacetime, an unloading and loading plat-
form, the loads to which are supplied by elevator, is provided in front
of the entra~ce to the storage.
When using the shelters in peacetime as transport structures (parking
= trucl~s, el~ctric cars) an inclined ramp is built for entrance of the trans-
po:'~ ,x;.ii~~, th~~ length of which reaches 35 to 40 meters. This compli-
cates fihe choice of the construction site under conditions of dense build-
ing up eF the territory. With a sma11 number of transport means instead
of an inclined ramp it is expedient to design a vertical 1ift.
Tt~e space-planning solutions of shelters in residential and public build-
ings also nave a iiumber of peculisrities connected with the insurance of
the required conditions of operation and maintenance of basement facili-
ti~s ar.Ad the buildings themselves during peacetime. The presence in the
-r.e~identia.l bi~ildings of freque~tiy occurring vertical stands of the
sewerage, tieacing, water and gas supply systems complicates the operation
of tlie base~ient facilities and sealing of the shelter~.
Thr.ee versions of the placement of the piges of the interna.l equipment
systems below floor level of the first floor of the residential buildings
a;:~. ;~ossibl.e; the first passage of the pipes through the basement in
stt:~..1 4~ubes (for the sewerage risers) or reinfor.ced concrete boxes;
seccnd ~.he construction of a special engineering floor for the pipes;
thir~l passage of the pipes in the cei.ling above the basement (in
r_~-:a_::~...~l.s . "~a,., the f]_oor~.
'~he fixsr version i.s preferable with respect to cost. By camparison with
it for. f~ve-story large-panel residential buildings the second version
is 5% more expensive, and the third is 6% more expensive. However, the -
second version has significant advantages: it insures a good ir.terior
of the baeement; it excludes the laying of the main lines through the
13
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~ _
FOR OFFICIAL USE ONLY
basement and the installation of sewerage risers~ it permits the laying of
the intrablock communications through the building; it provides for
convenience of operation of the internal equipment systeme of the build-
ing. In higher-rise buildings the construction of the engineering floor
is cheaper. 1'he third version is the worst with respect to all indexes.
' 65800
- ~
n
!
ii~
"
aaooao
3 4. t0 ~13 `6 "
O io~
~ 6 5 9 , ~ ~ '.18
N ' �[J ~ J ~ ~ i~ .J ~ ~ ~ ~ ~ I ~ Llil. ~s ? -
O ~ 1'~'
~~l
4
O ~
~
~ " 6000 ~ 6000 ___y
Figure 3. Arch type shelter:
- 1,2 entrances; 3-- prelock; 4,7 lock chamber; 5-- electric
panel; 6-- diesel electric power plant; 8-- facility for
sheltering people; 9-- sanitation facility; 10, 11 benches
and bunks; 12 filter-ventilation chamber; 13, 15 lock
chamber; 16 unloading area; 17 facility for entr.ance of
engineering communicationa; 18 freight elevator
In public buildings the internal equipment system ris~s are arranged more
compactly than in the residential buildings. Therefore it is possible to
run the pipes through the basement at defined locations, setting aside
amall areas for this if necessary insulated from the shelter facility by
a sealed-protective or sealed door.
Structural Designs of Shelters
The facilities adaptable for built-in shelters occupy the greater part of
the building, and therefore the volume of the enclosing structures designed
to take the shock wave impact is relatively small with respect to the
volume of the entire building. For the enclosing structures of shelters,
it ie possible to use standard reinforced concrete structural elements
which are widely used in industrial and civil construction. These include
14
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the II-20 series structural elements for production buildings with a
column ~-{cl of 6~;6 and 9x6 meters designed for loads from 500 to 2500 kg/m2,
TI-04 sez~^s for loads to 400 kg/m2, and so on.
The bearirig capacity of the ceiling of the shelter rer~uired by calculation
of the shock wa~ve impact can be attained by installing an auxiliary longi-
tudinal and transverse reinforcing in intervals specially left when
_ in~~.alt~.r.g Che ceiling (20-40 cm) between the prefabricated elementa,
increasing the working height of the ceiling by laying standard elements
of the monolithic concrete layer above (aee Fig 4). The amount of operating
reinforcing is determined from the condition of the working of the same
prefabricated monolithic cross section as monolithic. The joint working
of the prefabricated monolithic concrete in the slabs is provided for by
the forces of adhesion, and in the collar beams, projections of transverse
reinforcing can also be grovided.
TYie prefabricated monolithic structural elements are widely used in build-
ing shelters as a result of the follo*aing basic advantages:
Tne possibility of construction by industrial methods;
Increasing the spatial stability or" the structure as a result of insuring
the required rigidity of the corner structures by making them monolithic;
Simplicity of creation (joining of the assemblies without welding with
subsequent monolith3,r?g) of coutinuous prefabricated elements on intermediate -
supports and the possibilities of changing the rein�orcing in the supports; -
Improvement of the seal of the ~oints of the prefabricated elements as a
result of monolithing them; -
Decreasir.g the weight of the prefabricated elements when they are made
~n complete profile.
The pr~fabricatPd-monolithic structural elements car. be of three types;
Frou; prefabricated elements with monolithic seams and intervals between
elements;
FroJ ;:-efabr.j.cated elements of incomplete profile with respect to height
~aith relnforcing protruding to rhe outside supplemented with finish
. ~oncre~??~g on sites to the total calculated height;
~'~:c~:?: ba:,ic s~:pp~rting prefabricated elements and ~.ndividual sections
executed from monolithic reinforced concrete in the suspended form with
general monolithic and finish concreting of the prefabricated elaments to
the full profile.
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FOR OFFICIAL USE OrJLY
I 1-1
~r--~
~ 2 __~I ~
~o 00000 `
s s
4 ' ~ ~J ?
Figure 4. Structural design of the ceiling of structures with
the application of standard prefabricuted elements.
1-- II-03 aeriea slabs; 2-- monolithic reinforced concrete;
~ 3-- prefabricated collar beams (nonstandard); 4-- prefabri-
cated column (nons~andard)
In Table 3 the nomenclature for the first type structural elements is pre-
aented. This set of structures permits th2 design of sheltera (see Fig 5)
built into the one-story industrial buildinga. The monolithic concrpte
(15-20X of the total amount of concrete) is used for monolithing the
aeams, the construction of a reinforced coupler (5 cm) above the slab's,
in the framing above the wall panels, in the corner sections of the walls
and foundations, for the longitudinal beams built between the prefabricated
beams.
The number of types and sizes of the prefabricated elements ia reduced as -
a result of using concrete and steel of different types and different
percentage of reinforcing. The same form is used here. For 32 marka and
18 types and sizes of elementa only 10 types of forms are required.
The deficiencies of these structural designs inclule the following:
increased~�.consumption of concrete (1.46 m3 per m2) and steel (181 kg per
_ m2) as a result of separate working of the elements (by the beam system
with hinged supports), significant height of the ceiling (135 cm) as a
result of floor by floor coupling of the beams and. slabs, large volume of
welding operations on site ~oining the laid parts of the elements. The
coupling of the beams and slaba is possible within the height limits of
~he beam (Fig 6). However, the above-indicated deficiencies are also
characteristic of this version of construction.
In practice the fully prefabricated structural design (10y monolithic
concrete) is the construction of a shelter from T-beams (Fig 7) resting
on columns in mutually perpendicular directions, and from square flat
slabs laid on the webs of the beam. The continuous nature of the T-beams
and their rigid coupling to the columns are achieved by connecting the
pro~ections;of the reinforcing, installing additional reinforcing at the
~oints and monolithing with concrete. The ~oints of tha auxiliary rein-
forcing to the pro~ections are made dovetailed without welding. The
deficiency of this structural solution is the great mass of the T-beams~
_ reaching 15 tons, and the necessity for the application of a crane
(type SKG-50) rarely used in construction for installation of them. The
- advantages are in the higher degree of p3.ant readiness of the structural
16
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FOR OFFICI"~IL USE ONLY
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10
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10 1 / �4~
h
~ ~ Z Z h ~
.3 �j
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4
6 4 +':0 6'
� o ~
~ 4~
~ ~ ~ ~ ~
~ ' Foo eoos e S -~,s~
6000 6000 ~ 6QOD ~
~ ~
Figure 5. Structural diagram of a shelter made of prefabricated -
elemen,ts in a one-story industrial building.
' 1-- ribbed slabs; 2-- monolithic reinforced concrete unsplit
~ beam; 3-- gretabricated beams; 4-- prefabricated columns;
5-- prefabricated foundation; 6-- wall panel; 7-- monolithic
concrete; 8--- monolithic reinforced concrete ties
elements and the small number of types and size~ of prefabricated elements.
By comparison with the above-ir?vestigated solution the concrete consumption
is decreased by almost 1.5 times, and the steel consumption, by threefold.
2
I
I 3 *0,0
~ 4 I
Figure 6. Couplin,g of the beams and slabs at the level of
the top of the prefabricated ceiling
1-- wall parcel; 2-- ribbed slab; 3-- beaM; 4-- column
- Tt~e ful],y pr.efabricated structural e.lements with a~pan of more than ~
6-~eters have analogous def3.ciencies. For example, for a span of 9 m, the
co~ntiiiuous prefabricated slab (Fig 8) weighs 13 tons, and the steel
consump~tion is 140 kg/m2. The continuous na.ture is insured by welding
tYce large-diame[er reinforcing (40 mm) and subsequent monolithing of the
~oizt~.
The stractura.t designs of the first type (fully prefabricated) have ~a
17~.gh de~re~~ o~ prefabrication, but the complexity of providing for working
- with respect to the continuous scheme and the large mass of the elements
- are holding up their broad application.
19
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I-I -
. 1375 3250 f3751375 N
b
O
OO
~ a i i eoa (
~ a i, 3
i . r ' .
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1
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o 5 j,i Z T
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i il
~ ,
5 .
.fis~~== ~ v~
L -~C
5 II_II
, , -
_ Figure 7. Structural diagram of a built-in shelter with
ceiling made of continuous T-beams.
1-- T-beam; 2-- square flat alab; 3-- column; 4-- wall
panel; 5 monolithing sections
The prefabricated-monolithic structural elements with elements of incom-
- plete profile with respecC to height, especially the beamless ones, have
high technical-economic indexes. They consist (see Fig 9) of prefabri-
cated flat slabs (400x180 and ~20x18 0 cm, 8-10 cm thick), laid on pre-
fabricated column capitals (246 cm wide and 60 cm high) and a layer of
monolithic reinforced concrete, the thickness of which depends on the
degree of protection of the shelter. The capitals are executed in the
form of a hollow truncated tetrahedral pyramid installed on a steel mount-
ing table fastened at the upper level of the column. The reinforcing
projections from the column run through an opening in the capitals. In
the middle spans flat slabs are laid, and in the edge spans, slabs with
one geatle rib by which they rest on the wall panzls. All of the Yower
working reinforcing is in the slabs. The upper reinforcing above the
support in the form of welded gratings is laid in the monolithic layer of
the ceiling. Columns SOx50 cm are made in the foundation sl;~eves. The
column grid 4x4 meters permita the application o f this structural
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~
~
0
- ~
i
8800
~ 130
� 1
~ q
~
~ , -
4 1_I 100 3 S
o ~ 5 �
-
- - ~ i .
~`�a c v ~
~ o_ 'i
~o ~
2
F-igure 8. Continuous prefabricated slab with a 9-meter span.
1-- slab; 2-- support; 3-- welded joint of the working
reinforcing; 4-- loops of the projections into the
monolithing seam; 5-- monolithing zone
solution for shelters built into one-story buildings and standing
separately. The consumption of materials per m2 of basement will be
0.8 m3 of cancrete and 30 kg of steel.
o~~:~~:-:-�- ~~~1
~ ~ ~
e~+ t �,'r'-~3 1
~
~
"r
5 -
Figure 9. Prefabricated monolithic beamless shelter ceiling
assembly.
_ 1-- prefabricated flat slab; 2-- monolithic reinforced
~oncrete; 3-- fittings above the support; 4-- prefabricated
capitals; 5 mounting table
21
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11-11
i-�I 3
} ~ ~ ~i ~ ~ ~ 6 .
~ ~ . ~
I / / t~
i
~ ~ ~
i ~5~~
Z ~j
_
- /p-fQ 3' r
2 p , ? ~I ~ ~
3 -7.
2~' ~
ID ~ ~
3~-~-~.F
~ Figure 10. Assembly for joining wall panels to th% prefabricated
monolithic beam ceiling of a shelter.
1-- slab of the inverted I type; 2-- wall panel; 3-- mono-
lithic reinforc~d concrete
- The deficiencies of the prefabricated monolithic beaml~ss ceilings can
include the complexity c~f the form for manufacturir.~ the capitals and
significant weight of tnem with an increase in thp column grid to 6x6 m.
i
The beam ceiling (Fig li,) has simpler structural ~elements. It consists of
continuous beams l.aid un the columns and supported on the beams of the
prefabricated slabs of the inverted I type (dimensions 600x300 cm, 30 cm
- thick). After installing the support reinforcing of the beams and mono- -
~ 1 itl~ing to the calculated mark the slabs become continuous. All of the
5tres5ed span rcinforcing is placed in the prefabricated slabs. As a
result of the ~oining of the beam and the slab at the height limits of
the beam the height of the ceiling does not exceed 80 cm. The basement
walls are made of prefabricated panels (600 cm wide, height equal to the
height of the basement, thickness 30 cm). The panels are of rectangular
cross section (weighing up to 12 tons) with grooves for rigid connection
of the wall to the ceiling by installing additional reinforcing in them ;
' and monolithing the joint. The weight of the main ceiling elements (slabs,
beams, columns) is approximately the same and is 5 to 7 tons. The con-
sumption of materials per m2 of the basement is as follows: 0.8 m3 of
conerete, 55 kg of steel. The amount of monolithic (construction) con-
crete is about 32% of the total consumption.
The advantages of the investigated structural solution must include the ~
low structural height of the ceiling and the fact that all of the beams
are arranged in one direction arid protrude a total of 35 cm into the
basement. The.deficiency is the complex structural design of the wall
panels (as a result of the presence of grooves) and significant weight
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of theu~.greatly excecding the wei.ght of rhe ceiling elements, which is
disadvantageous from the point of view of selecting the crane equipment.
~ecreas ~ n~; Llie materlal consumption and weight oE the prefabricated ele-
ments i~;'th~~ most importanL goal oC improving the structural design of
ttie stielters. Tt~e application oE curviltnear Prefabricated monolithic
elements, i.n parricular, ttie arches of the c:urved outline, permits a_b-
n Lf Ir.ant decrease in thickness of the stielter ceiling (to 16 cm for a span
of s~U cm an~l rl:;e of 60 cm) . However, the frequent arrangement of the
bearing w~lls on which the arches are supported (or the correspouding
small column grid) leads to an increase in the concrete consumption and
limits ttie possibiliCy of using tlie facilities in peacetime. In addition,
the technological grocess equipment has still not been created for
cnechanizing the Forming of the curvilinear panels. The labor intensiveness
of installing the cylindrical shells is 1.5 to 2.5 times higher than the
s~andard flat structural elements,
An eFficient form of enclosing structures for shelters providing for the
most advantageous use oF the bearing capacity of concrete under compression
and tlie possibility of considering tiie plastic properties of the founda-
tion soil can be considered to be the arch construction (see Fig 3). The
curvature of the arch, the srructural design of the hinges and the shape
of Che embankment of a separately standing structure can be selected so
tllat the possibility of the occurrence of bend ing moments in the arch
elements is excluded.
The structural designs in the form ot sphere, the cupola and so on are
sometj.mes used abroad Eor ttie small-capacity shelters (Eamily shelters,
indivlciual shelters).
A successful structural design of shelters made of monolitliic reinforced
concrete is the beamless ceiling (see Fig 11).
lditti a column grid of 6x6 meters the slab thickness of the ceiling varies
f rom 2.5U to 450 mui depending on the load, the column cross section
f.r.om 500x500 to 100Qx1000 mm, the column capital size with respect to
height of 600 mm and the slope of the faces at 45�. For convenience of
cc~~uxeting i.t is expedient to design the walls flat (without pilasters)
wi.th continuous brackets next to the walls. In dry ground the foundations -
urdc:r the coJ.umns are columnar, and under the outside walls, strip. In
tl~e water-saturated soils ttie slab of a continuous foundation is designed
as an inverted beamless ceiling.
In c.L~: c:~;:e ~~f dry ground the shelter. walls can be made from prefabricated
~~oz~~~r~l.e ~;~oclules (see Fig 12) . In order to improve the wor'~.ing of the
wa1.1s uuder the joi.nt effect of vertical and horizontal loads, it is
necessary to place the beams along the structure, which insur.es uniform
loading of the walls from. the ceiling. The unwei~hted walls work on
bendiixg and requir.e reinforcement. The walls made of prefabricated con-
' crete blocks usually are reinforced by a monolithic reinforced concrete
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wall on the inside or the installation of monolithic columns at the break$
between the blocks which are laid in th~s case without tyiiig.
0 600
~ i 60 ~
I, 600600 i 600 600 II600
~I I i
~Op' 6000 6000 6000 6000 _ 300
Figure 11. Monolithic beamless ceiling of a shelter
The structural design of shelter walls made of prefabricated concrete
modules cannot be recognized as efficiex~t with r~spect to concrete consump-
tion, but it has become widespread as a result of the simplicity of
erection and ubiquitoua manufacture of these modules.
~ The foundations under the shelter walls are strip nade of standard pre-
fabricated reinforced concrete foundation modules or monoliths; under the
columns they are columned or atri~ped (with a column spacing of less.~
than 6 meters), prefabricated or more frequently monolithic. The standard
foundation modules, as a rule, have sufficient thickness from the condi-
tion of working under impact, but in some cases they require reinforcement ~
calculated for bending. Under the effect of a shock wave on the structure ,
the foundations undergo significant settling (to several centimeters). In
~ addition, the structure moves in the horizontal direction from the un-
" equalized loads acting on the front and rear.
a) . ~ b)
. '
~ o
0 3 U Z
3' f Z H
u
1 ' �4
500 o k00 100 -
. ~
5 5 ~
6000 ~ ,
Figure 12. Outside walls of shelters made of prefabricated
concrete modules.
- a-- weighted ceiling; b~, unweighted ceiling; 1-- concrete
module; 2-- prefabricated hollow slab; 3-- monolithic �
- reinforced concrete strip; 4-- monolithic reinforced concrete -
wall; 5 concrete preparation .
24 ~
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Thus, ~.he structure and its individual parts are involved in complex
movement in space under the effect of a shock wave, they are loaded and
unloFde? unsimultaneously.
In order to provide for general spatial stability of the shelters, it is
neceaear~ to take some structural measures: rigid building of the
columns and wall panels into the foundation; coupling of the wa11 and
ceiling elements and also all ~f the prefabricated reinforced concrete -
e~.ements to each other; reinforcing the corners and the ~oints of the
walls made of concrete modules; clamping of the ceiling slabs in the walls;
the conatruction of mon~olithic reinforced ~oncrete strips around the
per.imeter of the outside walls and across the structure above the columns
in tlte prefabricated ceilings and floors; insurance of monolithic nature
- of the prefabricated ceilings and floors.
In ordez~ to avoid the transmission of additional forces to the structural
elements of the built-in shelters when the above-ground part of the build-
ings i.s deatroyed, !.t is necessary to connect the columns of the building
rigidly to the shelter ceiling.
The structural elements of the shelters are designed for high-intensity
Zoading, and improvement of the technical-economic indexes of the struc-
tural deaigns is a complex problem. One of the ways to improve the
structural designs of the shelters can be the application of prefabri-
- cated monolithic pre-stressed reinforced concrete structural elemen~s
which are highly efficient with respect to material consumption.
Significant cost benefit can be obtained by considering the increase in
strength of the concrete with time. For exa~ple, in 2 years the relative
ultima.te compresaive strength of concrete will increase by 1.75 to 2 times
by comparison with the trademark strength after 25 days used in the
- des~.gn calculr~tions.
Technical-Economic Indexes of Shelters
Ix~ oxder to discover the optimal designs f~r m~~i*Pra is necessary to
Astadl~.sh a method of determining the technical-economic indexes.
When designing buried structures adaptable for shelters, it is necessary
:~einforce the structural elements, and provide special internal equip-
rnfa~:t, Thi~ gives rise to additional expenditures. At the same time the
strucrures and their internal equipment also are operated and maintained
in pear_etime,
Ii: wo~a3.d be possible to establish the additional ex~enditures on shelters
by comparing two desi.gn solutions; the design of an ordinary buried
structure (for example, a garage~ and the design of a buried garage-
shelter. However, for this purpose in each case it would be necessary to =
make two designs, which is unrealistic. A more accessible method is
determination of ~he additional expenditures and the difference between
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the estimated cost of the facilities adaptable to shelters and the average
cost of the ordinary buried facilities (basements).
Here it is necessary primarily to decide whether the ob~ects are comparable.
The f irst condition of comparableness of the ob~ects is unity of purpose.
It ie possible to compare underground garage-shelters, but it is impossi- -
ble to compare a garage-shelter and an underground storage-shelter. The
aecond condition is unity of the design norms connected with the given
climatic, hydrogeological and other conditions of the construction site.
In addition, it is necessary to consider not only the difference of the
indexes with respect to the buildings themselves, but also the difference .
of the indexea with respect to the master plan of the enterprise (including -
by the waterline, sewerage, power supply, transport and other networka),
inasmuch as on making the transition to the uae of the underground space
the denaity of building increases, and the extent of the networka is
reduced .
As the calculated units of ineasure it is expedient to select one location
( for those sheltered in the shelter) and one m2 of useful (common) area,
for these units insure comparability of~the analyzed indexes: cost
having the greatest comparability, natural (concrete and metal consumption)
and relative (the system of coeff icients K: K1 ratio of the area of
the basic facilities to the total area of the shelter; K2 ratio of the
- construction volume of the shelter to the total volume of the shelter;
K3 ratio of the area of the basic facility to the shelter area;: K4 _
ratio of the useful area of the shelter to its capacity).
- The gradual accumulation by the design organizations of technical-economic
indexes for shelters will permit establishment of standard indexes which
can be used to eatimate the quality of shelter designs and the preliminary -
calculations.
The basic space-planning parameters of the shelter are its length, width,
height and the ratio of its areas. The effect of these parameters on the
construction expenditures can.be'characterized by the above-indicated
relative indexes.
The ratio of the area of the basic facility to the total (useful) area of
the shelter (coefficient K1) is an important criterion of the economical-
ness of the planning solution. The greater the area isolated in the
shelter to protect people, the more economical it is. The coefficient
K1 varies from 0.4 to 0.6 (in shelters with utility corridors along the
outside walls) to 0.75--0.83. The value of K1 increases with an increase _
in the capacity of the shelters~ which indicates their better economic
indexes by comparison with the small-capacity shelters. The coefficient
K2 characterizes the efficiency of the composition and use of the shelter~
space. When comparing the veraions of the space-planning solutions, the
smaller value of K2 indicates that per m2 of useful area, a smaller
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construction volume of the shelter ts required, and, consequently, the
given vergion is more economical under other equal conditions. The
volun~etr~~.i coeff icient can serve as the criterion of economicalness of the
designa oiily in the case where the eatimated coet of 1 m3 0� the compara-
ble sheltera is identical or close. However, the estimated coat of 1 m3
of shelter with changes in the planning and design solutions can fluctuaCe
within significant limits. Therefore even for equal volumetric coefficients,
tk:ta co:t ~f i m2 of useful area in the compared designs can be different.
Thus, the coefficient K2 is not a suff iciently accurate index characteriz-
ing the econornialness of the design, in e}~ite.of the fact that it
reflects the effect of the height of the facilities, the wa11 thickness
and it considers the efficiency of planning of the shelter.
The coefficient Kz varies from 4 to 8; the lar~er values correspond to
shelters with so11d foundation s1ab.
The ratio of the area of the basic facilities to the basement area (the
coefficient K3) makes it possi.ble to determine the possible capacity of
the shelter beginning directly with the building (basement) dimensions.
The coefficient K3 takes into account the area of the various unprotected
facilities which are provided for in basements for technological needs.
- In the shelters with utillty corridors along the walls K3 is equal to
~J.25~G.~F, increasing to 0.5-0.65 with an increase in their capacity.
The total actual consumption of usable area per man (the coefficient K[F)
is 0.65 to 1.2 m2, where the smaller values of K!} are characteristic of
the high-capacity shelters.
For the developed standard designs of shelters, the best indexea are the
following: K1=0.82, K2=3.8, Kg=0.6 and K4=0.64.
The natural indexe3 are also required to estimate the economicalaess of
th~ plarin~ng solutions, especially the strucrural designs of the shelters.
The concrete consumption in the atypical designs will be the following:
pez man 1.3-3 m3; per m2 of usable ares, 1.2-3.2 m~; and in standard -
des:.!.gns the best indexes are 0.94 and 1.05 m3, respectively.
The metal consumption in the individual designs of built-in shelters will
be .:i: ~~llows: f or 1 man 80-380 kg, for 1 m~, 70-370 kg (the indexes
incr2ase witli an increase in the degree of protection of the shelter).
Theae ::x;dexes also in:clude the concrete aiid metal consumption for the
st�ruct~+r~:l. c:l _,irf~nts rayu3~ted by the condi~ions of opexation and maintenance
s.;:E �h~~ 'o~~;~r;ci~t in peacetime and not designed f.or the effect of the shock
wa ve ioads.
The cost indexes of the individual designs of built-in shelters vary with-
in broad limits.
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The cost of the basement shelter is approximately 1.7 to 2 times greater
than the cost of the ordinary basement. The increase in cost with respect
to type of operations will be in -
For general construction work 55-60
For water lines and sewerage 5-10
For tnsulation and ventilation 15-25
- For electric lighting 3-6
Strong and weak current electrical equipment 2-8
For diesel electric power plants 7-15
The increase in cost of industrial buildings with a basement shelter by
comparison with buildings with ordinary basement (for ~.dentical purpose
of the basements in peacetime) will be 5-20%.
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CHAPTEK II. DXPdAMIC LOADS ON THE STRUCTURAL EL~'IENTS OF CIVIL DEFENSE
SHELTER5
The dynamic loads and the structural elements of the shelters are created
basically under the.effect of the shock wave frou? a nuclear blast or the
explosion of chemical explosives. The maximum magnitude of this load and
the law of ita variation in time depend on the placement of the atructure
with respect to the ground surface (above ground, semiburied, completely
buried), the location of it in the builtup area (built in, separately stand-
ing) and also the dimensions, shape, orientation of the investigated atruc-
tural elem~nt with respect to the blast center and the parameters of the
3szcident shock wave (the magnitude of the maximum excess pressure and the
da~ration of its effect).
The term "incident" wave is applied to the waves moving from the blast cen-
ter to tne investigated surface at dip angles of 0-90�. In this case the
dip angle is:measured between the normal to the inveatigated surface and the
dir~~tian of motion of the wave. With respect to magnitude it is equal to
the slope angle of the shock wave front.
In addition to_the.loads:from the shock wave, some structural elements of
the sheltzrs (for example, the ceilings in the basements of buildings) can
al.ao be sub~ected to the effect of short-term dynamic loads from the impact
of fal:lLng fragnients of the buildings on destruction of them by the ahock
wave. Usually the.loada from the air shock wave effect are def ining for the
she?*_ers.
1. Parameter.s of the Air Shock Wave
'SMe shor_~: wave has a sharply expressed front, on which the temperature, den-
sity, presr~ure and velocity of the medium increase discontinuously. The
wave is made up of a compression phase and the rarefaction phase following
cii.rF�c~=;~ S~: `P.~' it (see Figure 13) .
The basic characteristics of the compression phase are the excess pressure
4p~ of the-shock wave front and the compressure phase duration ~r+ called the
time. of effect of the shock wave.
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rva vrrt~ieu, u~r. Vi~LI
The pressures on the air shock wave front from the nuclear blast or the
explosion of chemical explosives can be defined by the formulas of V. P.
Korobeynikov [29] which after aimplifications, have the form:
for 0.16 < ~p~ < 100 kg/cm2 or 0< R2 < _
~P~ 3(Vl -F 29,8R~ -I) kg/cm2~ ~1) -
(a)
Key: a. front
for ~p~ < 0.16 kg/cm2 or R2 > 2
0,227 2
nP~ Rz Vlg Re-I-U.158 k8/cm . (2)
In these formulas R2 is the dimensionleas radius of the shock wave
Rs - ~ kEo ~ (3)
where r2 ia the distance from the blast center in meters; pl ~ 104 kg/m2 is
atmospheric pressure; E~ is the blast energy with respect to~:the shock wave ,
in kg-meter; k ia the coefficient equal to 2 for the ground blast and equal ~
to one for the air blast. -
P P -
. q
d t
F --~LP .
~ c+ ~
- Figure 13. Variation of the pressure at a fixed point in the
terrain as a function of the time t(~p is the ma:~imum rare.= ~
faction pressure). '
The blast produced in the air above the ground or the water at an altitude ,
at which the f ireball does not tauch tlae earth's surface the time of its
maximum brightneas is called an air blast; a ground (water) blast is a blast _
where the fire ball touches the surface of the earth (water) or the blast
occurs at the surface of the earth (water) [21]. For high-energy air blast
the dimensions of the fireball at the time of its maximum brightnese are
less the fireball radius Lfireball of the fireball, the magnitude of which
according to [39] is:
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Lo,~, 10 ~ q
, ~a)
Key: a. f ireball
where q is the TNT equivalent of the blast energy in tons.
The energy E~ communicated to the medium (air) is part of the total energy
released during the blast. For a nuclear blast, approximately 50% of a11
of the released energy is expended on the formation of an air shock wave;
for a TNT blast, 65 to 70% [7].
The power of the nuclear b2ast is compared with the equivalent amount of TNT
with respect to energy. The heat of explosive conversion of the latter (the -
' explosion energy) is approximately equal to 1000 kcal/kg.
For a pressure on the shock wave front with a ground bl~ast of the nuclear
material with total TNT equivalent q in tons the formulas (1), (2) are
written as follows:
for
R G 78 m/ tonl/ 3~
2,333
~A,6--~ ,-e kg/cm2~ ~4)
~~l 0,49~� !0- R~ - 1
for R > 78 in/ton1J3
8, 9
np~ RVigR-1,44 k$~Cm2� ~5)
Here R i4 the reduced distance
R- s~rz m/tonl/3, ~5a~
V Q
For~ulas (4) and (5) are represented graphically in Figure 14.
~or an air nuclear blast the pressure at the front of the incident shock wave
is defined by the formulas:
2,333
~1+10-~.R3 -1 kg/cm2; ~ ~Pfront ~ 1 kg/em2
' ~ = d -I- ~ ~ ~
(1 + 8 j+ J ,
where ~
d=-1,33~p,y f or ~p~ < 3;
d=-5,6 + 0,63c1p,~ f or 3< ap~, < 10;
f= s, ~ ep,~; g= o, 72~on,~. ~
The solution to the equation of motion of the structure (etructural element) ;
under the ef f ect of the shock wave load is signif icantly simplif ied if the ,
effective load variea in time according to linear laws. Accordingly, in the
calculations frequentl.y instead of the effective diagram ~p(t) a line dia- '
gram (Figure 16) is used with replacement of the time of effect of the com-
preasion (rarefaction) phase by the effective time 6[40] .
;
If the maximum deformation of the structural element comes at the end of the
compression phase or after completion of the load effect, the effective time ~
is determined from the condition of equality of the pulses: ~
(a) ~
spMex~
e- i=. Op di~ ~13~ I
. 2 ~ ;
Key: a. max
wh~re Opm~ is the maximum pressure. ;
Depending on ~pfront' the effective time of the compression phase with re-
apect to the condition (13) is expressed by one of the approximate formulas:
f or ap,p < 1 kg/ cm2 !
8 (0,85 - 0,20p,~)r.+ sec; (14) ~
for 1 C ~p~, C 3. kg/cm2 ~ .
0 (0,72 - 0,08~p~,)r~ $ec. (15) ;
t
For loads from the shock wave of the nuclear blast the maximum deformation
= of the structural element takes place during the initial period of loading
in a time which in the majority of cases is two orders less than 'r+. There-
fore in the calculations it is possible to assume that the pressure varies
with reapect to the transient to the effective curve ~p (t) at the point
t ~ 0 (Figure T6). ;
34 -
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The effective time here is determined hy the formulas;
i+
Qo=- (1,5-~-ep~,) sec for QP~ 5 it is possible to set 51~17 = 1. Here the error will be less
than 1%.
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The maximum value of ct~ for which the structural element still operates in
the elastic stage will be found from the expression _
( arc tg ~A
6~` 2 ` ~ T (Us ) Qo.
~:ence, considering (5~) and (51) we ~liall have
Q 0,5:56((qlo)~/I; b~/li ao �
o=
l arctg w9 ' (52~
c~d
for a= 17, t~ = 895 sec
o,ss2~",7 a~r~~ Qo
6�Y arctgw6 ' ~53~
w8
Let us consider the eff ect of the dynamic load of the type of (23).
The minimum dynamic ultimate strength can be reached.only for t> 6 when a
decrease in the load occurs. Therefore the time of its occurrence lwill be
equai to the time t* at which the function T2(t) reaches the first maximum.
From expression (38) we have the following equation:
~o ( ~0 )4 = S~ (T~ ~~)1 �`dt ~ (Ts (t)]a dt.
Qo o e~ (54)
_ In order that it be possible to obtain the analytical function from (54),
1et us replace the functions (24), (25) by linear functions. Let us set
T~ = kl e~ , (55)
where k~ is found from the equality T1(01) = T1(61), that is, -
kl -1 Sw0'~l and T~ s(n c~~~ 1~ (56)
~ c~0~ 1 ~
,
The f.unction T2(t) is replaced by
7~z ~t) = 7~~ -F- k~ (t -O1), ~57 ~
~ where ic2 ~aill Ue found from the equality T2(t*) = T2(t*), that is
1
h~ (l'-el) ~Ta ~l')-Ti ~~i)~.
Let us substitute the linear approximations (56) and (57) in (54) in p'lace
of T~ and T2, and after integration we obtain
67
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~ r~utt ur~r~lc:i~t., ua~; UNLi
to ( Uo )a = ~a + je~ [T~ (e~)]a -f- (t*-91) [Ts (t`Tz (t')-Tl ~d~j)12+ ~ 1, ~58~ -
~ ~
Hence, for ot = 17 and t~ = 0.895 we find
1,1776w~/~~ o0
aa -
: R ~w~~~ ~ (59)
wher e
. R - {w~i [Ti (Ai)1" -~-~wt*-u~01) ~T' o the reinforcing operates in the elasti~ stage, but
c - c
here the stresses can exceed the static yield point.
Inasmuch ae the value of 6 corresponds to stress in the reinforcing equal
to the minimum dynamic yie~d point Q*, this stress Q* must be taken as the -
~imit when calculating the structural designs by the limiting state lb.
Let us briefly touch on the problem of the effect of the deformation rate on
the stresa-strain state of the concrete [6, 30]. The increase in loading
rate leads to an increase in the ultimate strength of the concrete and the
change in its ct-e deformation diagram. A characteristic feature of the dy-
_ namlc o-e diagrams is their approximation to t he linear diagram (Q = Ee) as
the deforznation rate increases. This fact is explained by a decrease in the
role of the plastic deformations during f ast loading.
'rherefore when calculating the reinforced concrete structural elements for
the effect of short term dynamic loads, the determination of the moments of
the internal forces will be correctly made beginning with the triangular
stress diagram in the compressed zone of the concrete.This especially per-
tains to the calculation of the structural elements in the elastic stage
(by the limiting sate of lb).
The effect of the loading rate on the concrete strength can be considered
with sufficient accuracy by increasing the calculated values of the stresses
in the concrete by 20-30% [6].
4. Characteristic Limited States of Bent Structural Elements
The limiting state with respect to the total bearing capacity (la) of bent
and extracentrally compressed (with large eccentricity) re3nforced concrete
structures occurs as a result of their operation in the plastic stage, that
is, when the atresaed reinforcing in the ~ross sections most stressed
68
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normal tc~ the longitudinal axis is in the condition of plastic flow. In
these cross sections a significant opening of the cracks takes place almost -
to the enti.r.e height of the beam. This leade to the splitting of the entire
structu:e :r~:o individual, slightly deformed sections.
The achievement of the limiting state la is characterized by the beginning
of destruction of the concrete in the compreseed zone in the crosa aections -
operating in the plastic stage. Here it is proposed that the reinf orcing
;~as ~ufftc.Lent plaszic deformation reserve and doea not break before complete
rupture of ~he compressed concrete and that the crosa section is not extra-
_ reinforced, that is, the compressed concrete is not des=royed before the be-
_ ginning of yield of the reinforcing.
It is known that for reinforced concrete structures the destruction with re-
spect to inclined cross sections from a transverse force is especially dan-
gerous. Therefore in order to prevent a large opening of the inclined crack
it is expedient that the transverse reinforcing which takes the transverse
f.orc~ ;,perate only in the elastic stage.
The destruction o� thP concrete in the compressed zone occurs at the time
when the atresses in the concrete reach the ultimate compressive strength with
bendtng. At this time the dieplacements of the structure must be the largest,
that is, the speed of the structure fs equal to zera.
Th~ lim~tir.~ state la is normalized by the values of the deformations which
ar2 eelected in such a way that they can be found by dynamic calculation of
the atructure and at the same time they will be convenient for experimental
deCermination. For bent reinforced concrete elements the most conveni~'nt
normalizir.g value is the angle of opening of the crack at the plasticity
hinge [15]. In this case the strength condition of the structure in which
n plasticj.ty hinges are formed has the form
i < ~{'ur ~1 = 1, 2, n),
(61)
where t~ is c.he angle of opening in the ith plasticity hinge obtained from
i
the dynam3.c calculation; ~,~i is the limiting angle of opening in the ith
plasticity hinge.
The m~~.gnitude of the limiting angle of opening ~ essentially depends on the
rela~~.v~~ heighr a in the compressed zone of the~conerete taken at the time
of ruptur.e in thep cross section with the crack and equal to the following
for the 3-ectangu~ar cross section
F
ap~ RY F~ 610 ~ (62)
~a~ ney: a. i
where Ra and Ri are the calculated resistances of the reinforcing and the
concrete,
69
~nn n~FTrreT TTCF nrJr.v
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r~n urri~lru. uoc v,.L~
~1 dp
0,J 0,46
0,6P6 '
0,664
0,660 0,4 ~ -
0,616 - -
OSSYJ '
~ O,S64 S , ~ R
' ~ D,S,lB Yi�~~ `~p'~ =R
a'
O,S>0 ,3
0,48?
O,fS?
D, 4??
0,.~5~1
0, Q?
0,3?8
0,294
0,~150
' 0,7?6
0,190 0,1 -
O,1S4
O, lf6
0,07B
0,040
' Q02 0,0f D,OB 0,12 0,16 0,2
Figure 28. Graph for dete~ining the limiting angle of opening
in the plasticity hinge.
The values of must be determined from the experiments by dynamic bending ;
of the beam elements. ~
Figure 28 gives the graph of as a function of ap and the characteristic of -
the cross sectiun Sb/SO conatructed by the results of auch numerous tests .
with diff erent beam atructures. When using them the values of aP and Sb
must be determined without considering the compressed reinforcing (F'), for
its effect on the value of t~ is not invesrigated. The graphs in Figure 28
is well approximated by rhe ~unction
'~II - 0~035 a~ ,
p (63)
The calculation of the bent r.einforced concrete structural elements by the
- limiting state lb~d~t~rmines the operation of the structure without residual
elongations of the stressed rein�orcing. The achievement of this limiting
atate is characterized by the beginning of the occurrence of plastic defor-
mation in the stressed reinforcing in the most stressed cross aections.
- .
The normalization of the limiting state lb is accomplished by the stresses;
at the time the maximum displacements are reached, the stresses in the -
atressed reinforcing of the most stressed sections will reach the yield point
(dynamic).
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S. Calculation of a Hinge~Supported Beam in the Plastic Stage
The plastic stage occurs after th~ stresses in the stressed reinforcing of
the most ~,Lrt~ssed Cross section reached the dynamic yield point, and plastic
flcw begit:s in the reinforcing. This state of tY!e cross section is called,
as is known, the plasticity hinge. It is depicted in the form of an ordinary
hinge at which a concentrated bending moment of constant magnitude is
- applied. When caLculating the reinforced concrete structuress the position
of the plasticit}� hinges is assumed to be constant (the stationary plasti-
city hinges).
The plastic stage of the entire structural element is considered to occur
after the plasticity hinges form, and it is converted to geometrically
variable, that is, to the mechanism. In thie case all of the elementa into
which the etructural element is broken down are assumed to be absolutely
rigid.
In the case where after formation of the plastlcity hingest the structural
element still remains geometrically unaltered, the elastic operation of its
individual elements is taken into account, and the structural element is
considered ~o be in the elastic-plastic stage.
We shall consider that the beam is loaded uniformly under a distr.ibuted load
of intensity
P tl) = t1P (t)b,
where b is the calculated width of the beam. ~~~en calculating the beam slabs, -
it is taken e~lual to one meter or 1 cm.
1. Ca.lculation f or an instantaneously increasing dynamic load of the type of
~17~ -
P~t) =P(1 - ~ l ~ P== Llpb. (64
~ ~ )
The bending moment in the middle of the beam span in th.e elastic stage in
accordance with (18) is:
- ~y1(!) = M1, ( 1_. cos wl Si wQ 1= M~,7~
~ ~
f~jp _ P~' , u~ - l ~ (65) ~
8 1 ~ rn
Z'he time at the end of the elaatic stage will be found from the equations ~
(3$) or (4Q). In these expressions we sha11 replace the stress ratio in tha
reinforcing Q~/c3~ by the ratio of the bending moments M~/MP, where Mp is the
moment of the internal forces in the middle cross section of the beam at
71
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rux urrt~l~u. ua~ u~vL~
the time ihe atresses in the stressed reinforcing reached the static yield
point Q0.
The ratio
k,~ _ ~!0
- .ti1r~ ' ( 66)
is called the dynamicity coefficient with respect to the bending moment. For
_ known 1cM, the magnitude of MD is f ound by the formula
Mo = k,,, Mp. (67)
Equation (38) is written in dimensionless form
sy ~~a
k+~ ~w~o)~ l a = S ~ ~S) ds ` . (68}
where s= t~t is the dimensionleas time; sy =~T;
?J(S) T C ~u (69)
for a load of the type of (17) we have
y~s)= 1- ~8 -coss-~ S~8 (70) -
For the linear approximation of tnis expression from (39( we obtain
kM~~o(a-f- 1)J~/Q~sy/c~J(~,) ~ ~~1~
- and for the special case of steels classes A-I, A-II and A-III, from (40)
. 1~1;~gY_SYi~~~~sy), (72)
where �
Y_kh~~~/t~_ Mow'~'~~ M - pl~ .
Mv n R (73) -
The bending moment in the plasticity hinge will be equal to;
lIn the presence of a static load the value of M~ is understood as the dif-
ference between the moment of the internal forces in the most stressed cross -
section and the bending moment from the atatic load in the same crosa section.
72
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M~ � MPy (Sr) � ~?'1~~Yi: Yi = ~(Sy)� (74)
~a~
� Key: a. hinge
If the beam is reinforced with steel, the strength of which can be considered
not to depend on the def ormation rate, the end of the elastic stage ty is
determined by the condition M(ty) = M~, that is,
k,u = y(sy) and M,~, = Mo. (75)
For 6=~, from (75) and (70) we obtain
sy = arc cos (1 - kM). (76)
The bending of the beam at the end of the elastic stage is:
� ~ x\
~o ~x) e pl? ( 2 lx - I y (Sy) _
~ 2 ~
~ ~2M,t, z� Ixy !''x
3B19 ( 2 � 2 ) ' ' (77)
Hence, we obtain the maximum elastic bending (f or x= k/2)
h1~ l~ 5p13 . (78)
~ s,s a ~sa~ Y''
In the plastic stage the total bending of the beam is:
~~(z, 1)=~(~)X+ 381a ( 2-lx3-f- l~zl ~O -
I~ence it is obvious that the coefficient k is equa.l to the ratio of the
total deflection of the beam to the elastic deflection caused by the fast
- loading of intensity p. Therefore k can be called the dynamicity coeffi-
cient with respect to the displacements. From the formulas (92), (72), (74),
(87) it is obv~ous that with a specific function y(s) the coefficient k
depends only on two parameters Y and w6. This f act off ers the possibil~ty
of easily constructing the graphs which greatly facilitate the calculation.
Figure 29 also shows the graphs of the dynamicity coeff icient with respect to
the dieplacement k as a function of Y for different values of w6. The func-
tion ~
y(s) was taken in the form (70), and for determination of the time a
of the end of the elastic stage, the approximate equation (72) was used. y
The plastic deformations in the beam occur if
< Yo~ . (94)
wherr~ Y~ is the maximum value of Y when plastic deformations are still pos-
sible in the reinforcing. The value of y~ can be f ound from (73) and (53):
yo~hMOw~ii~= Q~ w~lir ~1,915S-i~i~(1 - arctg~~0l
�0 u~e / ' ~95)
fiere the limiting value of the dynamicity coefficient with respect to the
bend:tng moment .
knro = yow-~~i~ (96)
75
FnR (1FFTf'TAT. TTSF ~1NT,Y
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1:V1~ VLL'tViaW VJL v~~ua
K ~
?0 ~ '
~
18 c~Ba30101
16
/4 ~
SO
~ ~Z - orrl
2
JS p -
!O
10
8 e
B
6
4
5
?
0,4 O,S 0,6 0,7 0,8 0,9 f,0 l,f 1,? f,d f,4 l,5 r _
Figure 29. Dynamicity coeff icient in the plaetic stage for a ~
hinge-supported beam cons{dering the effect of the deformation
rate (instantaneous increase in load).
x,,~ -
I.li
l~6
1,4 wg, ~ ~
s
f~ 2S
I
1
/1 ~
l,f
~ S
~Pad/ax ra~
0,925 f00 200 3Gb . 400 ~
Figure 30. 1cM~ as a function of wa
~ Key; rad. / sec ~
determines the limit of elastic working of the beam, Chat is, for 1~ > 1~,~~
the beam works only in the elastic stage. In Figure 30 we have the
76
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graph ot l~~ as a function of w for different values of w6. As is obvious,
the value ot 1cM~ can be appreciably less than the value of
k =2 ( _ arc tgw0' _
MO `1 ~e (97)
This value determines the limit of elastic working of the structure, the me-
chanical properties of the material of which are not influenced by the de-
for~*..ation .xate.
The graphs in Figure 29 show that for c~8 > 200 the values of the coefficients
k ia practice do not diff er from their values of the case where 8=~. For
t~iis case we obtain the calculation formulas using (85) and (86). The dimen-
sionless working time of the beam in the plastic stage will be:
' ),08sinsy
~ SMeKO =
Y,-~ ' (98)
and the dynamicity coefficient with respect to displacement
ko = yl (1 -}-0,694 2"~''
~ yl-1))' (99)
Here yl = 1- cos s where we should have yl > l.
Y
Let us define the transverse strength in the beam. Since in the plastic
stage the bea_~ elements are asswned to be absolutely rigid, the transverse
force can be found only from the condition of equilibrium of the beam under
the effect of the load and the inertial forces. For the beam cross section
witi? the coordinate x we have:
r~2
Q~z, ~-p(1 e 1( 2-xl- f m~xdx=
/ J
s
=P~j- e /l 2 -xl- 2~ ( 41-x~1.
~ ~ 1
Substituting ttie value of ~ from(84) in this expression, we obtain
Q~x~ = 2! L\ 1 e 1\~ 21 / 3\ 1 1" 1'i / 11 _ 4x� / J~
4 e !a (100)
Hence, it is obvious that the transverse force has maximum value at the
beginr.tng of the plastic stage. Thexefore the calculated value of the trans-
verse force must be determined at the end of the elastic stage. In the tn-
~~esr_i~~~~~d case tha transverse force on the support for t='t is:
Q = 2` yi? (101)
- that is, the dynamicity coeff icient with respect to the transverse force
will be:
77
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rUtc urrtLirw u~n v?vt,i
k~' - ~ (102)
In Figure 31 we have the graphs of kQ as a function of ~ for several values
of w. As is obviou~, the coefficient kQ can ~ignificantly exceed kM. The
value of k,. for fixed w increases to certain conatant values (horizontal
~
li~ea on the gra~h in Figure 31) which depend on w6. -
~Q ~a~ -
~ npuWBi7A0- -
t~9-100--
CJ = SO - -
~9 wB =2S -
_ l,8
f,7
tJ � 4
~OC! '
_ ~~6 200
i
f,S ~
~ 2S 1
~ J,4 ~
I
I
l,3 ~ ~ _
i
_ 1,2 ~
_ ~
- J,f
- 1
;
/,0 ~
i
, ,
Kp i-
09 0,7 0,8 0,9 . l,0 1,2 f,3 f,4 ~,S f~6 I
- I
Figi:re 31. kQ as a funct'on of 1cM. i
;
' Key: a. for
I-
!
When calculating the strength of the incZined cross sections with respect to ;
the transverse force it is necessary that the stresses in the transverse i
reinforcing not exceed the values of the minimum dynamic yield point (46).
_ . i
- The formulas obtained in this way can be used also when calculating beams j
without considering the effect of the deforniation rate on the strength I
characteriatics of the steel if we replace equation (72) by (75) and set ~
~71 _MlYaYl~ Y~a~~~ i
~ I
~a
~
~
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In Eigure 32 the graphs aze constructed for the dynamicity coefficient with
respect to displacement of k,~ as a function of the dynamicity coefficient
with respect to bending moment 1~ for different values of w8.
K .
20 "
IB - ~B~J00
- ~
Jb ~ _ ?
p(r)
~
u _ roo ~ ~
i1 f
~SO e
i
/0 I
B ; `--I- ,
75' . '
6
10 . .
4
S
? - ~
O,S 0,6 0,7. 0,8 Q9 I /,f ;3 14 >,S~~
- Figure 32. Dynamicity coeff icient in the plastic stage for
the hinge-support beam without considering the effect of the
deformation rate (the instantaneously increasing load).
2. Calculation for a load with build-up of the type (23)
P d~ ' 0 < ! < Al;
P _
r-e,
, P(1 - e~ ~ 8, < t< Al 8z.
In thys r:ase the function (69) of t'~e dimenstoless time s= c~t has the form
s-sin s
~s~ y~(S)= ~,a (o 6 (s > W6 The
- i y- i
eqnation for the time of the end of the elastic stage will be obtained from
(59) and (60), replacing Wt* by sy = wT: -
- 1,1776y = R (sy), (108)
where ' R (sY) _ _
� {~i I~~ ~c~9i)jl~-~-(sy -W91)~?Js ~Sy~lls-~?Ji ~w0i)11�tt/t~
b~ csy~-y~ t~,~e~ 1
_ (109)
Since the variation of the load with time during operation of the bea.ID in the
plaetic atage is analogous to (64), the general view of the relations ior
the dynamicity coeff icients coincides with (92), (99), (102).
- 80
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It is eas;~ to f ind that
kn =1'! ~,591 ( 0- S~e= ) SMaK~ -f- 1,28~sn~aKO~ -
(110)
Here Y1 - y2~sy~
~ ~ cos s 1
r = -~--l COS (Sy - W81~ - ' - i
w6, m02/ ~~~A, c~A~
2.17r l
sMeKO = ~~e, e + + ~Z 1 '
where
~ =1 - Sy-w9, _Yi. .
we,
For A2 a m we wi11 have
~n ~ Yt + 0,694ra
Y~-1 ('Yi> 1), (111)
where
Y~ =1-I- ~e [sin (sy-we~)-sin syl; _
~
~ _ ~e, y '
! (cos (sy -wel)-cos s j
(ii2)
2. If Y< y, the plastic aatage occurs at the time < A1 (s < w61). The
Y
dimensionless time sy of the end of the elastic stage is determinef f.rom the
equation obtained from (40) for ..T(T) ~ T1(T), that is,
i~i~ ~
1,1776y= ~e (sy-sinsy). (113) -
,
In the plastic stage tiie equation of motion of the beam in accordance with
(80) and (23) has the f orm:
forT T1, the elastic-plastic stage of working of the beam begins. The
expression for the deflectiona wi11 be r epresented in the form
~x~ ~ ~ PG ~x)T~ -F pF (x)Y~? (145)
where G(x) is the static form of the def lection of the hinged beam, and the
function T2(t) ia def ined from the equation which wae obtained from �(130):
~',-}"w T,-C1- e -Y~)~', (146)
where
_ 90
.
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. ~
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. ~
f G (x) dx
_ ~ .
rn~'rGz (x) dz (147)
0
The initi~l v-alue (for t~ T1) of the function T2 o~viously is zero. The
value of 'I'2~�rl) will be determined from the cond3tion of equality of the .
momen~um at the end of the elastic and the beginning of the elastic-plastic
stage
1
' f F (x) dx
Tatzi) = 7'? (zi) .
f G (x) dx (148)
u
Then in order to simplify the calculations let us consider that the rigidity _
of the segments of the beam near the supports influences the deflection of
the hinged girder insignificantly. Therefnre as G(x) let us take the form
of the deflections of the hinged girder with constant rigidity with respect
to the entire span equal to Bspan~ that is,
~p
G~X~ 128 ' 2 2z)- ~149)
Then
. ~e ~np
W = ~ m (150)
and Ta ~T~ = wlr,v9. (150a)
where t- cos ~u1 zt
r1=sinc~,i1- ,
~ W' 8 (151)
v- 0 184 -0,13k 0 184k -0 071 ~
9 ~ i -f- ~ , i ' ) ~i ~
(152)
The cc+efrfcient v3 has the following values: for Sl = 1, kl = l, v3 = 0.167;
for 3i = 2, kl = 1.125, v3 = 0.105.
Solvi.nq equation (146), we obtain
7'2(~=c,sinc~(1-iil-c,cosc~(t-r1)+1-Y,-e, (153) -
~ri~ w~ 1
~l ~e ' =1- y~ - T' .
e (154)
The bending moment in the middle of the span will be:
91
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_ ~ ~ _ ~ _ ~ _ ~

~ ~~t'~~ ~~I~~I~~~lw I'l~r 1~. ~"'3C~1~~H~~'ir' ET R~ ~ ~F ~
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L' VL\ VL'L' lUl[1L UaJli VLIL~a
M~p >,2 " _
~ir =1,38 , ;
8 ~ kQ k� k4 '
B
s ~ ~32=~,52 ~ ~3z=,,, ~,SZ ~
6
:
4 r ~ i 2 y Z
2 .T 1 2 ~
0 ~'S2 ~ kn
p ~'S1 kN ~
0,6,f Q8 ! 1,2 Q6 0,8 1 1,2 , j
{
J�r S ~~s i
q~ ~ ks Bn k4 :
i~
I=1,76 2~34 I
6 f32=1,76 2 34 6 t
4 24 Z ,
Z 1 2 i ~
p on ort
0,6 0,8 } ~~Z kn D~~ D'B r ~Z kn
Figure 38. Dynamicity coerficienta in the plastic stage for a -
clamped beam without c nsidering the effect of the deformation
rate. Solid line k~ear~ dotted line k~Pan, dash-dotted ~
line kQ.
, '
Key: a. bear
where ~ +(1-a2)e (1 +3�,) I 1- ~Q ) �
_ ~1 \ Y~ ~
ka _ , ~
~t + -aa)9 i 1+ ~i / I
i
a Q Bp4 (
a2 = ~ + 1'1 - B~p ~ ,
(191) ,
The bearing reaction on the hinged support will be: ~
( s1 ~
A 2 l i w 4 J (192)
!
98
i
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The ben~ing moment in the span will assume the maximum value in the cross
section with the coordinate
xo= 2 r~'_ 4! l.
~ ~ (193)
Tte value will be:
Mnv` BC~~_k~ ~y
' (19k)
If the following expression is satisfied
( Mon I - ks M�n
_ ~p ~ m ~
M 4' M~ ~ (195)
the first plasticity hinge wi11 occur on the support. Then we shall c~nsi-
der the case itself. The calculations show that it is possible to set
a2 ~ 0.3. Then
0, 26 0, 74~,
. k9~' o,57s+o,4?2~,' (196)
The dynamic load is assumed in the f orm of (17).
The expression f or the elastic deflection of the beam will be represented,
as before, in the form of (131), where F(x) ie the corresponding static for
of the deflections:
Tl ( t) -1- e- cos cuZ t+ S n g r.
~ , ' ~ (197)
~ ~ 15,45 B"p
" 4 l' m ~ (198)
In the elastic stage the bending moments on the suppor and in the span (for
x=x0):
- ll'1 j" (t) _ - kQ T, (t};
M;�(t)= 81~(1- 4 }2T~(~�
(199)
After the occurrence of the plasticity hinge on the support, the elastic-
plastic stage of working of the beam comes. The time Tl of the end of the =
elasti~ sta~e is found from the equation (139), in whic~i:
on
Y_kMWZii~_ ~tii~ wiii~~ S _c~ T .
MA2 2 r s i~ (200)
~J~~S)= 1- ~;e -~oss-}- ~~e ; R~la2 = 8~ kz. (201) -
99
� FnR n.FFT('.TAT. TTSF. (INT.V -
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rux urrl~ltu. u~G uivLx
_ In the elastic-plastic stage the beam is considered as a hinge-supported
beam with concentrated moment on the right end equal to:
Y
M~ = 8 k9 T~ (i,) = MDS Y~~ Y~ = Ty(t~)� (202)
The angle of opening in tihe bearing plasticity hinge will be:
' ~on ~t~ _ ~on ,.F ~on _ p~a v2 t 4,45T (203)
~ y Y� 1~,88"P ~ Y 9~~~.
where
_ Ta = cl si n cu (t sl) - Cq COS G) `'Cl~ -I-' Y 1-
'u~, - - 1 - tl
c~ _ ~ r,y8 + ~e ; cZ =1- Y~ - e ;
v8 = 0,168 -O,F (1- 4 ) -f- 1,225v1-}- (0,46ka-0,018) ; -
~
v1=0,637-0,245k~-(0,06?--0,175k21 ;
v$ (14-4Em)~E~n-2(1-k4 ~~m-f-~2--1.5kz)~m-I-ks-l~.
~ /
Here ~m = m/!~ is the relative coordinate of the cross section with maximum
deflection determined from the equation
4~8~6~1~ 4 /~~+v'-~' -
0 >.2~ ~a6 ~,8 1 t,2~
Figure 40. Dynamicity coefficients in the plastic stage for a
beam with one clamp support and the other hinged support con-
sidering the effect of the deformation rate. In Figures 40-
43, 46 and 47 the following are denoted: the solid line de-
notea kRe8r; the dotted line denotes 1cRPan; the dash-dotted
l.ine is kQpan on the clamped support.
The following relation exists between the dynamicity coefficients with re-
spect to the bending momen2 ~
k~ _ kon k ~ M
s p
M ~s k! lp ' ~s~ ~Op ' (217)
Then l 4 ~
~ ~
_ Y~-Y k' ,
. ~~(1- q! J
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t~a9=50 � lJyB=SSO
p,=o,sas
k~ h�" k h�~
a a p
8 8
~2'~,82 ~,12 JJ2=i,lZ
g Q,53 6 0,82 -
4 Z 4 OSJ �:2
. p ~ `
Z 1,12 ~ Z ' i I,12 ~
' on
h k
~0,6 0,8 1 1,2 n~Q7 0,9 1,1 1,~ ~
I~r' ~
~ � k~ kQn k~ kQrt
8 8 -r-
6 ~2-~- - 6 f~i2=1
4 q~ Z 4 07 . 2
2 1 ! Z 1
,
U fr�n~ 0 hti
0,6 0,8 ; 1,2 0,7 0,9 1,1 1,3
� J3~=1,45
kn � kQa Krt : ~ kon
8 8 a
s ~ ~/~z = t,s _ ~ 6 - I~z = ~,s
' ~.a, C'
4 ,,16 , _ 2 4 ~ 16~ " 2 -
Z .i61 f 2 I j,6 1
0 - kti ~ kn
0,8 0,8 1 ;Z 0,7 ' 0,9 !,1 .1,3
. ~ ~ _
Figure 41. DynIImicity coefficients in a plastic stage for a
beam with one clamped support and the other hinged support
without considering the effect of the deformation rate.
and rherefore the values of the dynamicity coefficients k~e8r~ k~pan are ~
completely determined by the values of the parameters S1, S2, w26, y. 2n
Figur~ 40 the graphs are constructed for k~ear~ k~Pa~ and k4 on a clamped
support for certain values of the parameters sl, S2, w26. When calculating
beams wj.thout considering the effect of the deformation rate in the expres-
a;.ons o~~~~_iz...~l :it is necessary to set ~y = kMear, and the equations for de-
_ termining sy anci sy,~ are taken as foilows:
. kM = y~ ~SU); ke~ = yz fsya)�
(218)
105 ~
~(1R l1FFTrTAT rTC~ n*nv
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rux ~rrt~l~w uan v1vLx
In Figure 41 graphs are presented for kbear~ kspan and kQ on a clamp support
obtained without considering the effect of the deformation rate. The check- -
ing of the beam strength when calculating it by the limiting state la is
done ~ust as for a beam clamped on both supports.
8. Calculation of Continuous Beams
_ The dynamic calculation of continu~us beams is appreciably more complicated
that the calculation investigated in the preceding items, especially if it
- is necessary to consider the nonsimultaneous loading of the spans under a
dynamic load. This sj.tuation occurs in the case of movement of the shock
wave front along the longitudinal axis of the beam. Later a more detailed
investigation will be made only of the case where simultaneous loading of
all of the spans take~ place.
In the elastic stage it is possible to perform the calculation in accordance -
with item 2. The displacements and forces are f ound by multiplication of
their static values from the load of intensity p{determined, for example, ~
by the tables for calculating continuous beams) times the dynamicity func-
tion T(t) which satlsfies equation (7). The value of W entering into this
function is taken to coincide with the angular frequency of the natural vi-
brations of the continuous beam which corresponds to the shape of the vibra-
tiona closest to the elastic line from the static load p. For continuous,
_ equal-apan beams with edge hinge supports it is possible to take:
for two spans -
w~ - 1 ~45 ~~Rmp ~ (219)
f or three spans
~ we = 3 ~Bm� ~ (220)
_ f or f our or more spans -
~e = 2~ 4 Bmp . (221)
V
Here Bspan is the bending rigidity of the beam cross sections in the spans.
Af ter formation of the plasticity hinges on all of the supports the contin-
uous beam is converted to a sst of single-span, hinge-supported beams with
concentrated bending moments in the bearing cross sections. Therefore each
span can be approximately calculated by the relations obtained above for the
single-span beams with different supporting conditions on the ends. Here
the middle spans of the continuous beam must be calculated by the formulas
for the single-apan beam clamped on both supports, and fihe edge span with
� the hinge support, by the formulas for the beam with one clamped and the
other hinged aupports. In these �ormulas, the values of the coefficients
(128) and (196) can be changed, taking into account the redistribution be-
tween the bending moments on the supports and in the span and ~~e':inagn~.tude
of the angular frequency of the oscillations wi (i = 1, 2).
- 106
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The value,of the coefficient ki must be changed in accordance with the values
of the ratios between the supporting and spanning bending moments in the con-
tinuous ba�:zn. For unequal values of the momenta on the supports it is pos-
- sible to ~a?~e their mean values. Let us obtain the values of these coeffi-
cients whi~h will be d~noted by k~i, for the equal-span continuous beam with
edge hinge supports. From the tables for calculating the continuous beams
we find ttiat on the second supports from the enda for any number of spans
;reatez than two, the bearing moment is: Miear ~-O.lpk~. Therefore when
~~a1c.ulaCii~g the edge spans it is necessary in place of k2 from (196) to take
k~ _ k, = 0:8k,.
8.
For the two-span beam Miear =_0.125 pSL2, and therefore k2 = k2. When calcu-
lating the middle spans the value of coefficient ki will depend on the number
of spans :
for three apans Miear =_O.1pR2, and therefore instead of kl from (128) we
take kl = 1.2k1;
for four spans Miear 3_0.107 p�~'~; MZeSr =-0.071 pk2;
0,~07+0,071 _ 0;089 and k; - ~~~s k- l,p7k �
2 ~ 1 ` i,
12
�or five spans Mlear __0.105 p~,2; M~ear _ M3ear __0.079 pQ2.
Then:
for the second ~pan
k~ ~ o, las 'o,o~s ki =1,105k;;
� 2.12 .
fo.r the third span ki = 0.948k1.
The oscillation frequencies wi (i = l, 2) are replaced by the corresponding
osci~..'~ation frequencies ~H of the continuous beam defined for the equal-
span bear~ with edge hinged supports by the formulas (219)-(221). In the
dimensionl.ess expressions (180), (182), (211), (214) here it is necessary
~n rnultiFl.y t~:e third term times the ratio (t~i/w )Z and the fourth term by
U~i/t~H. T.e anglF~s o~ opening in the bearing pTas~icity hinges ~~ear af the
cont~.nuoua beam are taken equal to the sum of the angles of opening in the
aupparting hinges of the ad~acent spana.
10~
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. . . ~ . . . _
1~H9=SO ~ fJHB=SOO
~ ~f~ �0,585
~ kn kqR kn kQ ~a~
8 1,12 B 0.82~~ 1,12
6 ~ 53
6
~~82 f3g =0,53
~ 053 2 4 2
Z j~iZ 1 2 ' ~ f
~0,6 , 0,8 1 1,~ ~0,7 0,9' 1,1 1,3~
Ai= t
kn . f k o gkn . kQ~
8 '
6 z�~,~ 6 J~
_ ~ p~ q
4 Z 4 ~ 2
2 1 Z 1
_ ~ 0,6 0,8 1 1,2~ ~Q7 0,9 1,1 1,3 ~
~ ~ . p,=ts~ .
8k~ k~n 8 n ka
,
6 0.9 /3t=/,38 6 p~g- ~2=1,38
4 0 9 Z 4 0,9 ?
~
2 ~~~8 ~ 1 Z ~ T3g ~ I
~0,6 Q8 f 1,2~ ~ Q7 0,9 !,1 1,3~
Figure 42. Dynamicity coefficients in the plastic atage for
the edge span of a continuous three-span beam with edge hinge
aupporta considering the effect of the deformation rate.
Key: a. bear
When performing the calculation with respect to the~.limiting state la the
strength condition of the continuous beam is: for the s ans SPan ~
< gpan� for all of the su orts exce t the edge supports, ~bear~~ ~bear~
- ~.~i ' PP ~ � i max '~i
for edge clamped supports ~bear ~ ~.5 ~bear.
max - ~r
The dynamicity coefficients k~e8x, k,~an, kQ for continuous three-span beam -
are presented in graphs in Figures 42-45; for the edge spans of the contin-
uous beam with more than three spans, on the graphs of Figures 46, 47. For
the middle spana of continuous beams with more than three spans it is _
possible to use the graphs of the dynamicity coefficients for the c3~amped
beam in Figures 37, 38.
108
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~�e=so c~�e=soo
J3i =0,585
g n k�Q" K~ ko~n .
8
b ~~82 - t 1,12 6 fJz �Q 53 0,82 1, i2
. ~
f,iZ=R53. Z 4 p,53 Z
Z 1
1,12 2 1,12 ~
~~~s 0~8 r ~rZ kn 0 ~~7 0,9 1,1 ~ ~13kn
f~,=1 � ,
~ ~ N Q ~ ~ kQn
_ 0,7 Az=1 ~,7 ~2�1
6 6
4 p 7 2 4 p,7 Z
''O ''O .
Z 1 2 ~
0 hM kan
0,6 0,8 l 1,2 ~0,7 0.9 l,f 1,3 ~
f3! =1,37
8kn k�" 8k~ t kq~
a
0,9 fd2~I,38 0,9 f3t=1,38
6 g
~ . 2 k 0,9 2
0,9
2 1 2 }
1,38 1,38
0 k� p hoe
0,6 O,b _ 1 1,Z 0,7 ~0,9 1,1 1,3 n
Figure 43. Dynamicity coefficients in the plastic stage for
the edge span of a continuous three-span beam with edge hinge
supports without considering the effect of the deformation
rate.
9. Calculation of Rectangular Slabs
The rectangular slabs for which the ratio of the side lengths b and a satis-
fy r_he conditions b/a > 2 usu~lly pertain to beam slabs and they are calcu-
latec3 by the above-discussed methods. If
1 ~ a C 2� (222)
the calculation of the slabs must be made considering the bearing conditions
with respect to a11 four sides.
The dynamic calculation of the slabs is much more complicated than the cal-
culation of the beams. Even under the effect of static loads the solution
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~ vu vrr t~.icv.. u.~u Vt'ILL
G~NB=S~ ' GJNB=Jrb~
. ~ie~'
k~ kQ k~ kQ
8 8
6 ~Z=1 ~2=1
Q~ 6 0,7
4 Z t~ Z
2 l Z /
~
~Q6 0,8 f 1,2~ ~0,6 0,8 1, 1,~
f~~' 1,54 ~
k~ kQ krt kQ .
8 8
~i=~,~ ~z=1,7
6 1,23 6 1,23
2 4 Z
2 ~ Z ~
~0,6 0,8 1 ~~Z~ D p~s p,8 ' l~d"
Ar=2,s .
8 n k� 8~ kQ `
s f,1Z=3 ~f3Z=3
_ l~6 . 6 ~ 6
4 2 .4 2
Z ~ Z r
~ 0,6 ' 0,8 i 1,2~ n~
,6 0,8 1 1,2~
Figure 44. Dynamicity cosfficiencs in the plastic stage for
the middle spans of a contin.uous threa-span beam with edge
hinged supports considering the effect ~f the deformation -
rate (the dash-dotted line is kQ).
of the slab equilibrium equation can be obtained only in infinite series and
not for all of the support conditions along the edgea [44~. Therefore the
calculation of the slabs in the elastic stage for the effect of the inves-
tigated dynamic loads will be made oy simplif ied methods discussed in item
2, as systeme with one degree of. freedom; the times (bending and torsional
moments) and transverae forces will be found by multiplying their static
_ valuea from the load with intensity p= ~p defined by the slab calculation
tables times the dynamicif:;~ coefficient T(t) satisfying the equation (7). ~
The angular solution frequency of the slab w entering into T(t) is taken
equal to the lower angular frequency Qf the natural vibrations of the
slab and is also determined by the reference data.
110 '
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. A=50 G~� 9 = 500
f3r=1
kn kQ h~ kQ
8 ~2_~ a . AZ_~ _
6 6
4 Q,1 - 2 4 2
2 1,0 1 Z p !
0 hri p kon '
0,6 0,8 1 ;Z 0,6 0,8 1 ~2 M
f3r =1,54 .
8 n k0 8~ kQ
~2- ~3Z =1,7 .
b G
4>>Z3 ~ 2 4- 1'23
2
Z _ I.. ~ 2 _;T �I . ~
koa on
~0,6 q8 1 t~Z �~p~6 ~~8 ~ 1~2kn
� Ai = 2~6 - -
8kn kQ ~ a kQ'
ffz "3 f,it = 3 -
s ~'6 6 ~ I 6
4 16 Z 4 I fS ' 2
~
Z 1 Z ' I
~3,0
kn ~ ~ i 3'~ kon
Q6 0,8 1 ~Z 0,6 0,8 1 1,2 y
Figure 45. Dynamicity coefficients in the plastic stage for
the middle soans of a continuous three-span beam with edge
hinged supports without considering the effect of the defor-
mation rate (dash-dotted line kQ).
Here~fter we shall consider the calculation of a slab which is hinge suppor-
ted on all f our sides (see Figure 48) . In this case the vibration frequency
is:
~-~2(a ~ bp )V m ~
(223)
where m is the mass per unit area of the slab; D= Eh3(12(1 - v2) is the
cylindrical rigidity of the slab.
Let us denote the bending moments per unit length of cross section parallel
to the sides of length b and a, respectively, by M1 and M2. The largest
111
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I
rux ~rrt~ir~, u~~ u1vL~
mNe=so ~vNA=30o .
fd,=0,585 � ~
ka ka Kn kq
8 8 0
ma ~
6 ~ s ~ ~
4 + 4sJ 2 4~,53 o.,r a
Z~~33 ~~~12 1 Z j~12 ~ 1
, ~0,7 0,8 l,l t,3~ ~0.7 0,9 iyt ~3~
. . ~f~=1
~
n kan 8
~ ke
s \~r.- 6 ~ o ~~i
o~ - ~ ~ p,7
q Z 4 2
2 f,0 1 2 ~ 1,0 ~ l
~ 0,7 0~9 1,1 ;3~ ~0.7 0,9 1,1 1~3~
A~=~,4s ~ .
~
8~~ keR g ~ k~
~~l 6 ~
6
~Q` i
4 kon kep >>~s Z 4 kQ~'h" ~'16 Z
n+ n
2= ~,ae r a ' i
a ~
~ 0,7 0,8 ~,1 1,~3 ~ co,~ o,s ~i ~,a ~ -
Figure 46. Dynamicity coefficients in the plastic stage for
~ the edge span of a continuous beam with edge hinged supports
~ with a number of spans greater than three considering the
eff ect ~.of the def ormation rate.
values (at the center of the slab) are:
Ml s~,m~ (Figure 61,c,d) the unloading process caused by tlie wave from
the structural element ceases before encounter of this wave with the unload-
in wave running from the surface of the ground. Beginning with some time
sH~~ lesa than the time the deflection reaches a maximum, the pressure on ~
the surf ace of tne structure exceeds its value of s= s,~ + s~, that is, the
148
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FOR UFFICIAI. USE ONLY
s s s 6) -
- s s s
fj~f ~ ~ ~ (r - ry~ s~: = _ - -
' ~ _ ~yj i f (s)~{f~s1 ~ -
{ c gx sn
~~~0 Q2 0,4 QG 0,8 ~ 0 1 Z p( 2 J PnuKC p 0,2 Q4 0,6 0,8 1{ 0 1 2 0 1 Z
. f~aJ !"~aQ~~ , sn~~~4 ~~~q3
pR) . , ~ d) ~S S
S S
_ 6 , 6 ~~s(s) _ S� -
4---�:~._�s" 5~_ ~~^S 9
^s - 4 f~(S ' Qx
~ f:(s) B,Is - ..s^ - ~ S~
$ ~o,~ t ~
(a~ P~`�"~0 q2'q40,6q9 1 0 1 2 0 1 Z 3 0 O,Z0,4Q60,8 1 0� f 2 0 t Z 3
Srt=Z ~ce=0,3 g~~J f,'p,3 6i
- Figure 61. Processes of the interaction of a compression wave
with a flexible structure from various va.:.ues of the parameters
- s,~ (s = wt) .
Key: a. max
weight associated with the structural element is again converted to a defor-
rn3ble medium. The derlect;on of the beam y*(s) calculated approximately, as
~s obviovs, in practice coincides with y(s) at the times when there is an
associated mass on the beam with a thicknesa less thari the total thickness
of the soil laye Tne noted circumstances explain the cause o� the fact
that when s> s~m~ the layer of soil when calculating the structural element
can be cons~dere~ as an infinite linearly elastic medium.
Tn reference [40J a similar conclusion regar~ling the possibility of consider-
ing the soil as a linearly elastic medium in the problems of the interaction
of waves with structural elementa was drawn on the basis of experimental data.
Using the graphs of lcgen (Figure 60), we obtain the condition of correctnesa
of each of the two noted cases of calculating the burried structural elements.
For this purpose we assume that these cases are delimited by the values of
s,~, uZ corresponding to ~he points of intersection of the solid and dotted
lines. Applying the coordinates s,~, u~ of these points in the coordinate
- plane ul, we find ttiat the "boundary~' values of s,~, }tl 1.ie on a straight
?ine
. Sn -i- 3�1 = 2,5, (76)
Therefore when calculating the buried structures in the elastic stage for the
effect of a shock wave of sufficiently great duration applied to the surface
of the ground, it is possible to cons~der the soil as an infinite linearly
149
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1'vL~ vi'1�t\,LCU, u.)a. v?Ya,�
i
_ elastic medium if
(77)
so 3�, > 2,5.
In the case where ~
so 3�, < 2,5,
. . ~78~
- it is necessary to consider the effect of the free surface of the ground and
the propagation of the unloading processes in the soil layer. ;
From Figure 60 it is also possible to find that the buried structure can be ~
calculated without considering its interaction with the ground which is taken ~
into account only as the associated mass, if
� so < 0,25. . ~79) ~
Here the error in determining the dynamicity coeff icient does not exceed
2. 5~6. ;
i
. ;
150
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CHAPTER V. CALCULATIQN OF ROCK WALLS AND COLiJNINS FOR THE; EFFECT OF SHOCK
WAVE LOADS
Depending on the location in Che structure the walls of the shelters are
divided into outside and inside walls. The operating conditions of these
walls under the effect of external loads on the structure from a shock wave
diff er significantly, for the outside walls talce the vertical and horizontal
external loads, and the inside walls, only the vertical (mass forces occur-
ring during horizontal displacement of the structure are not considered here).
In addition, the working conditions of the oucside walls under the eff ect of
a horizontal extternal load depend in turn on their location with respect to
the other bearing structures of the shelter the floors and ceilings,
columns, inside walls, and so on. Accordingly, the schematics of the opera-
~ion of the outside wal~.s in the horizontal direction can be different the
schematics of beams structures or slabs with different supporting conditions
around the outline.
From this chapter a discussion is presented of the method of calculating the
ou~side unreinforced rock walls working by the beam structural schematic.
For calculatton of the outside stone wall with longitudinal reinforcing, the
_ methods of calculating reinforced concrete structures investigated in Chapter
III can be used. The calculation of the inside walls and columns is pre-
sented in the ~ase of the effect of only vertical forces causing central
compression or extracentral compression with small eccentricities.
The calculated resistances of the column and wall masterials made from con-
crete and rock masonry are taker. with a hardening coefficient equal to 1.2
[6, 30], and the calculated pressures for nonrocky soil in the foundation,
- equa.~_ to the normative pressures of the soils of the foundations in accord-
ance �~r~`tth the main SNiP II-B. 1-62*, augmented by several times if by the
opera~ing conditions of the structure its ~ettling under the effect of the
- dynam~.c lo~d is permitted.
r'or. st:ati~w Zaying of the foundai:ions having relatively small dimensions, it ~
- is necessary to consider the poss~ibility of squeezing of the soil from under
their footings under the effect of the load. In order to prevent this phe-
nomenon, the calculated pressures for soft ground must not be designated as
more than 15-kg/cm2.
151
Fl1R IIFTiTf'TdT TTCF r1A1TV
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r~ux urr~l~i~., ua~ uriLz
The static loads are taken into accaunt with a coefficient of 1.2.
The strength calculations of the inside walls and columns are performed by
the static methods with respect to the corresponding chapters of the SNiP
by the general formula
NII < NB,
where N~ is the maximum value of the large general force, N,~ is the reduced
longitudinal force from the external force effects determined from the ex-
preseion -
' Nn = N,~eKO -f- 1,2N~T~
~a)
Key: a. static
in which N~X ia the maximum value of the longitudinal force from the effect
from the dynamic load; Nst is an arbitrary �orce from the static loads.
1. Loads -
All the loads are divided into horizontal and vertical. The vertical (or
longitudinal) forces are transferred to the walls from the floors and ceil-
ings. These forces are taken to be equal to the support reactions of the
ceiling from the dynamic load acting on it which is variable with respect to
the law (17) of Chapter II and applied only within the limits of the ceiling
span purely. Here the effect of the deformation of the ceiling is not taken
into account, which can be permitted when calculating the walls of structures
on soft ground.
When determining the cost of vertical load on the walls of t:~e builtin shel-
tera the mass of the walls of the building supported on the walls of the
structure is not taken into account for ~p> 1 kg/cm2, where ~p is the preasure
on the shock wave front, for under such loads the building is entirely de-
stroyed [14]. For ~p < 1 kg/cm2, the caeight of the part of the building
- walls resting on the calculated wall can be taken inCo account.
~he basis load causing the bending of the outaide wall is horizontal. The
magnitude of the hvrizontal dynamic load p2(t) is determined without consid-
ering the nonsteady state transient processes occurring during the buildup
time of the load to the maximum value and the time of loading of the wa11
by the compression wave with respect to the entire height (the arrival time). ~
This load is therefora taken to be unif ormly distributed with reapect to the
height of the wall and equal to the following; , '
~ Ps (t) - $P C 1- 0 l ~ . ~1) ,
~
_ ,
152
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~ ~
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where p ls the maximum pressure detertu:tned hy the formulas in Chapter II, and
~ is the coefficient, the magnitude of which is determined by the conditions
of the interaction of the compres,sion wave in the ground or the transmitted
shock wave'aith the structural elements of the wall.
For the walls which take the load through the ground, the value of ~ depends
on the fill diagram and the type of soil. For complete burial of the wall ~
ie equal to the coefficient of the lateral pressure kb, the value of which
vari?s frcm 1 to U.3 [35, 40]. In the calculations it is possible to set
icb = 1 for water saturated soil and lcb = 0.5 for soil with natural humidity.
Far the walls enclosing the shelter from the f acilities not protected from ~
the shock wave it is approximately possible to set ~ m 1.
For the elements of the walls erected above the level of the ground and tak-
ing the load from the air shock wave directly, the value of the coefficient
t, is taken according to the data of Chapter II in accordance with the con-
ditions of reflection of the air shock wave from the surface of the walls
and f.low over the structure.
2. Characteristic of the Limiting States
Unc]er the effect of the calculated forces, deformation of the walls and the
soil in the foundation takes place as a result of which the points of the
~�~-"_1 r.ake the vertical and horizontal displacements.
Depending on the relation between the bending moment and the longitudinal
force the wall can work with respect to two different diagrams: either under
the conditions of compression of a11 of the cross sections or on the occur-
rence Ln some of the cross sections of the tensile stresses leading to the
occurrence of horizontal cracks.
'The first case is extended to walls centrally and extracentrally compressed
with small eccentricities of application of the longitudinal force, and the
second case, is extended to the extracentrally compressed walls with large
eccentricities of the force. In the general case for any cross sections the
bour.ciary between the regions of small and large eccentricities under static
loading is the ratio sk/s0 = 0.8, where sk is the static moment of the com-
pressed zone of the cross section with respect to its stressed face and s~
is th{_> static moment of the entire cross section with. respect to the same
face.
Let us ccu;~i:.lF~r r_he conclusions obtained duriag static loading for the case
~~~.~u?.:~ ;.r:,,c:~.~,~. Then for sk/s > Q.8 we sha7.1. have a region of small
ecce:i~rlci~ies. For a rectangularQcross section of height d, this inequality
is equiva3.ent to the fnllowing:
eo ~ 0,225d, ` (2) -
153
T./1D l~T.~TTl~T A7 �Tnn /~*rr
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run urrtt,ieu, uor~ vivLi .
where e~ is the eccentricity of the longitudinaZ force with respect to the
center of gravity of the cross section.
On satisfaction of this inequality the calculation of the wall for strength
can be made by the methods discussed in item 4 of thia chapter.
Great eccentricity will occur under the condition sk/s~ < 0.8 or rectangular
- cross section
. eo > 0,225d. (3)
When calculating the walls extracentrally compressed with la�rge eccentricities ~
for dynamic load, the magnitude of the eccentricity is not limited. ~
The numerical value of the eccentricity of the longitudinal force is deter-
mined by dividing the moment of all of the forces with respect to the wall
axis by their resultant. During the process of displacement of the points of
the wall under the effect of external dynamic loads the magnitude of the
eccentricity of the longitudinal force changea. For the beginning of loading,
_ the relative eccentricity of the longitudinal forces can be approximately de-
termined (without considering the static loads) by the formula
2eo _ (2-a)~ . ~4)
- d 4d 1 t ~.5 d~ ,
. where H is the wall height; R, is the span of the ceiling; a is the dimension-
iess level defined from the expression
;
' a - I + ~ (o,5d-e)
~ya ? (5)
in which e is the distance from the axis of application of the bearing reac- ~
tion of the ceiling to the inside face of the wall. ,
The cross aection for which the eccentricity is defined by formula (4) will
be found at a distance Ha/2 from the lower plane of the ceiling. For the
values of a< 1.172 the maximum bending moment will in practice be in the
~cross section of the middle of the height of the wall, for a differs insig-
nificantly from one. For a> 1.172 the maximum moment with respect to the
absolute value will occur in the cross section on the level of the ceiling, ~
and the eccentricity of the longitudinal force in this cross section will
be defined by the formula
~0 0,5 l ( l -2e/d?
d d(1 -~-0,51/d) ~ ~6~
This case requires special investigation. In the first approximation for
a> 1.172 the wall can be calculated by static methods. Then we shall con-
sider the bearing outside walls with a magnitude of the dimensionless parame- _
ter a < 1.272.
15k
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~ For self ~bearing outside walls the maximum bendi.ng moment will be in the
middle of tt?e height of the wall, and the magnitude of the relative eccentri-
city of the longitudinal force in th~,s cross section will be defined by the
formula
( kT d
2eo ~ya . 1 1 2h1 )
~ ~ 4d~ . r j + ~krHl ' (7)
~ 2d ~
where ic,r is Che friction coefficient of the wall m3terial with respect to the
ceiling material. Formula (7) wae obtained considering the longitudinal
forcea occurring in the aelf-bearing outeide wa11s as a reeult of friction
' of the upper end of the wall against the bearing structures (the ceiling).
T~e f irst limiting state of stone walls with large eccentricies ia reached as
a resu.lt of the appearance in the most stressed cross sections of horizontal
cracks opening with an increase in the displacement of the wall and decreas-
ing the working part of the compressed cross section. These cracks divide
up the wall into individua~ blocks which rotate relative to each other which
are deformed at the points of contacts with each other, with the foundation
and the ceilj.ng. The horizontal cracks occur in the masonry after dsstruc-
tion of the wortar from tension. As a result of low tenaile atrength of the
mortar, the elastic deformations of the wall turn out to be negligibly small
by comparison with the residual, and when calculating walls by the first
iimiting state the elastic stage can be neglected.
The achievement of the first limiting state is characterized by the beginning
of destruetion of the masonry ma.terial of the compressed zone in the cross
section with open horizontai cracks at the time the wall receives the great-
est dieplacements.
The first limiting state (state la) is normalized by the magnitude of the ~
_ total angle of opening of the horizontal cracks. The strength condition of -
the wa11 is =
~Maxc < ~n~ ~8~ -
where is the angle of rotation of the blocks obtained from the dynamic -
calcu~.ation; -
i~ half the li.miting angle of opening of the crack (the seam), the magni- -
tude of which is determined fr.om the experimental data or by the formula
~ , . ~ ~,2R j'd%yp,y~ . . . (9)
i~:.. ~ , . . ~ D . E~, yo . ~ ,
. ~ . . .
in which R is the calculated resistance to compresaion of the masonry; H' is
the tieight of the row of masonry; y~ ~s the height of the compressed zone of
155
T/~Tf AT.TTnT~T ttnr. n+.7~�
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masonry in the cross section with tI~ crack; Ek is the modulus of deforma-
tion of the masonry determined by the formula -Ek = 0.5 E~, where E~ is the
modulus of Alasticity (the initial modulus of deformation) of the masonry
defined by the chapter SNiP II-B. 2-62*~ The value of y~ is determined by
the formulas (37), (38).
The calculation of the stone walls with respect to absence of large residual
deformationa (state lb) insures preservation of the initial seal of the walls.
The achievement of this~limiting state is characterised by the beginning of
reduction in seal of the walls as a reault of the appearance of cracke in the
atressed zone of the moat stresaed crose section of the masonry at the time
the wall receives the greatest displacements.
The condition of calculation of the walls with respect to absence of large
residual deformations is the following: at the time the wall reaches the
maximum displacement, the height of the opening of the seam must not exceed -
the admissible values. This condition is reduced to the f ollowing:
. _ : ; " ~
.
~F~+~HO ~ ~Pn ~ ~ ~ 10
d-yo; . . ~ )
,r,~;v. ~ � : i v . ~ ,
where cp~X is the angle of rotation of the block obtained from the dynamic
*
calculation; is half the angle of opening of the crack f rom the condition
of preserving the seal of the wall; n is the maximum width of opening of the
crack.
According to the experimental data, for a value of n= 0.4 mm the seal of the
walls witl~ atucco is maintained. When it is necessary to maintain the ini-
tial aeal of ~the wall the calculation with~reepect to absence of large resi-
dual deformations is mada in the case where where is determ~ned
by the formula. (9). If c~~ then this indicates that the width of open-
_ ing af the cracks from the strength condition is less than (or equal to) that
_ germittred with respect to absence of large residual deformations, that is,
the initial seal. of the wall is maintained for the calculation by the first
limiting state.
3. Calculation of Outside Stone Walls
, The wall is assumed to be broken up by the horizontal crack into two identi-
cal blocks 1 and 2 which rotate relative to each other (Figure 62). The
angle of rotation of the longitudinal axis of the block with respect to the
vertical is denoted by The height of the compressed zone, the crushing
zones at the point of supporting the wall on the foundation and the point -
of supporting the ceiling elements on the wall is denoted by yp, yl and y2 '
reapectively. The value of yl can be taken equal Co y0; the values af
y2 is defined by the formula
/ y9 - R"7 , ~11~
156
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k
where R is the normative resistance to compression of the masonry; R is the
ceiling span.
It is po~~it~le to take y2 approximately equal to the length of the finfahing _
of the ceiling elementa which uaually does not exceed 12 cm. For the walls
filling the~frames, y2 = y~,
The lumped masses mb and m* are equal to the mass of part of the ceiling with
which the.-load is transferred to the wall and the mass of part of the wall
located above considered in the calculation, respectively. By Q(t) the load
from the ceiling is denoted:
. Q = Pi (r) 2 ~ (12)
where pl(t) is the dynamic load acting on the ceiling;
R (t) = R, + p (t)d, (13) _
where R is the weight of part of the walls located above considered in the
calcula~ion.
The horizontal load p2(t) is taken by the formula (1). The pressure diagrams
obtained in the experiments at various points in time with respect to height
of the wall indicates the significant effect of the interaction of the com-
pression wave with the wall blocks rotating with respect to the direction of
effect of the load on the magnitude of the horizontal load. In the central
part of the wall the pressure can be half the pressure at the upper and lower
points of the wall (after loading of the wall by the compression wave with
respect to the entire height). This interacCion on derivation of the calcu-
lation formulas was taken into account by the generally accepted procedure:
by subtraction of the term pa~ (au/8t) from the horizontal load where u is
the horizontal displacements of the points of the wall, p is the soil density, -
a~ is the propagation rate of the elastic waves in the soil (see Table 6).
The displacement of each point of the wall (for exatnple, A aitd B) is expanded
into the horizontal u and vertical w displacements. The displacements of the
- poinc:: of the lower disc of the wall caused only by rotation of the discs by
a sm~.:!..i. angle c~ (Figure 63) are defined by the formulas
u = -zcp; ~ = x~p. (14)
Analogously, for the points of the upper disc (Figure 63)
u = -~H - z)~; ~ _ ,x~ (15)
157
FQR . (1F.FT('.TAT. . TTGR f1NT.Y
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R~t~p(~ z
nr* m6
I ~ yi ~
- ~ e~
~
~ 2
~
t Pi~tr~
ye
h
~ Ar
~ ~ -
Yi ~ _
x -
m~
D
TZ -
Figure 62. Calculated diagram of a stone unreinforced wall. _
z z ,
�-~0/2
. y
A~ u~ B~ w Z =
~ N
I ~
x X ~ x 0
Figure 63. ~isplacements of the points of the wall in the lower
and upper disc.
The total displacements of the points of the discs will be:
a) for the lower disc ~ .
ul ~ -z~p' . (16)
~ � w~ ' ~CII X~ a ~ lx - yl~~f ~17~
where W~ are the vertical displacement of the foundation (without rotation);
rt~~" = yl sin cp cos cp ~ yl ~ is the vertical displacement of the disc caused by
crushing at the point of supporting the wall on the foundation;
b) for the upper disc
u, _ -(H - z)~~ ~1~~
c~, = In6 to~`~' - z~ ~ We -I- (y~ - x)S~,
(19)
158
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where Wb is the vertical displacement of the ceil~,ng element; -
~ o ~ y~~�
The value ::f Wb can be expressed in terms of W~ and ~ from the condition that
wl = w2 For x s d- yo, z~ H/2. Hence;
, [~6 = W~ (2d - 2yo - Ji - ~Js)~~
` ~2 - ~ + ~2d - 2 (20)
~ ~Jo-y~-X)~. ~21~
The equations of motion of the wall will be obtained beginning with the ,
principle of the possible displacements. As the possible displacements let
us tak+e the displacements caused by small variations of ~ and W~, that is,
the values ~nf and L1W~.
- Then in accordance with the principle of possible displacements we write
AA� -I- aAp DAB = 0,
(22)
- Key: ~a~i
where the indexes for operation of the forces (L1A) on the adopted displace-
ments denote the following: i-- force of inertia; p-- external and B--
internal forces.
The work of the internal forces on the possible displacements will be d~eter-
mined under the assumption of the rectangular compres�ive ~tress diagram.
We shall assume that at the point where the wall is supported on the founda-
tion this work is half the work of the stresses at the point of ^ontact of
two blocks. L~t us a~so assume that the vertical displacement of the wa11.
foundation tJ~ has 13tt1e ~nfluence on its horizont~l dispZacements.
Then consider~.ng th~ assumptions made by formula (22) we obtain the desired
- equations oi ~not~on:
a) for. bearl,ng walls
~P -F ~a ~ -F Ae ~ - PAb ( I fl ~ -R, (1,5d-3yo) (23)
- where
A,~ma~~~
~9,+ ~ d ~1- d l
~
- Q nQ+,_' 1'~ d� ~~~'-}-2,25ma\I-2 l~l�
~ ~
(24)
mc ~~dH is the small mass; m* = R~/~, ~where g is th~e gravitational accele-
ration;
1S9
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rvn ~rrtVieit. ua~ U1VL1
Aa ~ ~ : (25? -
p, a~ are the soil characteristics of the wall;
1,SE~c 90 . �
A~= H, , ~26~
A6 = d~ f 4N~ -~(1- 1,5 d- 0,5 d' 1-
l ~ ~
-1,5 (1-2 d (27)
~
b) for self-bearing walls =
A' A A 1- Rt (1,5d-3yo)~ ~28~
A~W+ e~D+ ~~==P 6( .b
where ~ . ~ ~ ~
ql=Al-k,,m~'~4 (1- 1 5 d)(1 H(1-2 d)~; (29)
~ ~
. s ~ ~
,Qa- P�04i d( 3 � d -kT (1- 1~5'~-~� ~ (30)
~
. ~ Ab=d~IE~--k (1-1,5 d.~1~,~'+.~/
l 4d~ , ~ H ~ . �
. , . _ t'~ . yo'1 ..i~,:, ~ -
('1-2 (31)
_ ~ ~ d ~
A4 = A4, where A4 is defined by formula (26) .
The equations of motion of the walls (23) and (28) are also valid for walls
that are not banked with soil. It is only neGessary to assume that pa~ a 0
in them and the value of thQ coefficient ~ from formula (1) in accordance
with the indications of item 1 is taken. The prc:sented equations are appli-
cable for 0.1 rad. For large angles of rotation it is necessary to in-
troduce the term ER(t) - Q(t)] H$ into the left-hand side of the equations ;
(23) and (28) . ;
In the equations (23) and (28), A1, A3 and A4 are the inertia, resistance and
rigidity coefficients respectively.
The equations (23) and (28) are noneuiiform second-order diff erential equa-
tions with constant coeff icients.
The solution of equation (23) as a function of the type of roots of the ~
characteristic equation
, ~ .,...,~A~~, +~A3S A4~_.~,, , � ,.,~;1 (32)
160
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will be:
a) for the effective roots sl and s2
` . ' � , r .
A~� (1- e' 1- Dl es` r-E- Dz es,�r -F- A
0).r
~ ~
-A~ (1,5d-3yo),. (33)
where , . s.. . , , . ~ � ~ , ~ ,
. . ~ ~ ~ D1:= ~-i � ~ s . rs2� (1 AA
~ ~ -F- ~ ~ ; . ,
~~'~l \ ~ ~8
. . ~ ; , ~ ~
. D= _ .l � ~(S~./ ~ + A3 1,+ t 1 ~ . .
. . s=- sl. ~ t A~6 ~ 0 ~ . . . .
b) for the complex roots
pA~ r r A3 ~ ,q
A~ i I- e-}- A,
e- es' t L\ 1~- A g) Cos 52 l-
a
' ' ~ (34)
-DZsinszflJ-Aa (1,Sd-3yo),. .
where � ~
_ . , jya:_ ~(slll+'AOI+g . '
` < < 4 / '
For the walls not banked with soil A3 = 0; therefore the solution of equation
(23) hae the form
q~ = A~' (1- e-cos ~t si~e},r 1_ A~ ~ 1,5d-3ya)~ (35)
1 ,
where � '
- ~ A1 �
The solution of equation (28) is determined from the expressions (33)-(35) in
_ which the coefficients Ai are replaced by Ai.
The strength condition of the wall when calculating by the first limiting
state r.,ill be the expr.ession (8); for the calculation with respect to absence
of large residual deformations, expression (10).
_ '~'he li~~tt.izg ina~nitude of the load on the walls is determined in accordance
~ ,.:.ti, ~he �"spression (8) from the equality �~~X = ~here is determined
by the formula (9) and ~~a~ is determ~Lned ~y the formulas (33)-(35) for t=
= t~X, the magnitude of which is determined from the equation c~(t) = 0.
Hence the lim~iting value of the load for Rl = 0
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r~n urrtt,iat, uo~ ULVL1
~
~n ~901
p� 8 (yo) f (t~axc) ~ � (36)
where ~,~(y~) is defined by formula (9j; g(y0) = A5/A4; f(tmaX) is the maxi- ~
mum value of the time function from the right-hand side of the expresaiona
(33)- (35) . -
From for~~::la (36) it is obvious that p,~ depende on the value of y~. Let us _
take the value of y0 to be that for which p,~ has the least value. Then from
the condition of the minimum ~imiting loada
(dpnl _ ~0
\ ~yo ~Maxo ~
the height of the compresaed zone is; -
a) for self-bearing walls
yo=1,25d rl-? C2e~IJ
~ 3 d f or Q� ~ 1,35;
yo=0.125d for d�> 1,35; ~37~
b) for the bearing walls
2 ! ~H~
yo= t,2sa s~ d a~ for < t,35-}-
1
~ ~ ~ +o,s a
: 0~925 d ;
(38)
1
yo = 0,125 d f or 4Q~ > 1,35 0,925 a.
For th~ walls filling the frames including quite rigid bars, between which
the ma~onry is located, the height of the compressed zone is determined from
the equation (20) for Wb = W~ = 0 and yl = y2 = y0'
yo = 0,5 d- 8;~ ~ o,s d. ~
~ In the solutions of (33)-(35) f or these walls R1 = 0, ,
. A' = m~ d ' + y
12' ( d~ 6= 9 �
the remaining coefficients, ~ust as in formula (23). -
162
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4. Calculation of Columr?s and Inside [~ails _
The formulas presented below for determining the longitudinal forces in the
column an3 4Ta11 cross sections and under the footing of their foundations
are obtairied considering the verticai displacmeents of the structure inves-
tigated as a rigid body. -
_ Let ua consider the frame of the structure made up of columns (walls) with
foundations and part of the ceiling from which the load is collect:ed which
is transmitted to the column (the wall) tlYrough the span supportecl on it.
Under the ef~fect on the structure of the dynamic loading, verti.:al shifting
of the columns (walls) and the foundations takes place. The e~.~uatian of
motion of the entire structure as a rigid body will be -
P -I- N~ (t) - Mii = 0, (39)
where P(t) is the total dynamic load acting on the covering of the structure;
N(t) is the total longitudinal force under the footing of the foundation
c~used by the resistance of the soil to the movement of the structure; M is
the mass of the entire structure; u is the vertical displacement c?: ~~he
structure.
The expression N~(t) can be represented in the form
. z
P� i F~h
N~p pa, F~ u- 2D u, (40)
where p is the density of the soil in the base; al is the propagation rate
of the elastic-plastic wave in the foundation soil taken by Table 6 from
Chapter IV; F~ is the area of the foundation footing under the column (wall);
D is the large side of the foundation footing of the column or the width of
the footing of a strip wall foundation.
When calculating the wall the area of the foundation footing is defined by
the f ormula
F~ = bD,, , . (41)
wher~. b is the distance between the ax:es of the beams (slabs) resting an the
wa~z
tls a r~~s=~J.t of solving equation (39), the dis,placement, the velocity, the
a,.~:~ ~ e;: =~,e force under the foundation footing are found, and i~rom the
co~d-~.zior: o� d}maml_c equilibrium~ the force in any cross section of the
column (wall).
The longitudinal force in any cross section of the column (wall) at the time
t is:
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rux UrPll.ltil. u~~ ~ivLx (
~
N (f) = N~,~ (t), . . . i
~ . C42)
where i1p is the longitudinal force cauaed by the load p applie~t atatically,
and ~(t) ia the function which depends on the time and is defined by the ~
- f ormula i
~
. ;
~
~~t)=S(1)- "-`'rS~n,l~+ .l . .
~ , M ~ . e,.~ (43)
~
, ~
in which ml = m~ + ml~, M a m~ + mk + m,, mk, m~, m,~ are the mass of the ~
column (wall), the foundation under the colimmn (wall) and part of the ceilin& ~
_ respectively, from which the load is collected on the column (wall); mlk j
is the mase of part of the colum (wall) from the foundation to the investiga- ;
ted cross section; i
~
r. , ~ i
S(t 1 S~ E-?+ f (COS 9~ 1-.~ 1~ ,SI i~ 9s r' !
~ ~ . ~ . ` . . . ~ Aq~ ~ . ~ ` (44)
where ql, q2 and r are def ined by the formulas respectively: ?
- ~ ~i , = . ' � kM '
q1 F'Q O~ 49 -~Qir N'~. ~1'a a~, D~
P~
where k is the coeff icient equal to 2 for columns and 1 for walls.
~
As is obvious from formula (43), with an increase in ml, that is, with an in- i
crease in the distance from the foundation to the investigated cross section, ;
the dynamicity coefficient decreases. 4
i
~
The maximum value of the longitudinal force in the column (wall) is: ~
,
NMaxo = Np ~Masa? (45) ~
I
i
where ~ is the dynamicity coefficient which respect to the force in the ~
max ~
column (the wall) determined by the formula (43) on substitution in it of j
t= tm~ found from the solution of the equation ~
i
dm(~~ ~ +(1 M le-v~~~~2+e~ ~ cosq,t+ !
dl - q~A ~ 1 9~
. ..F(r-! - ~ )singat~=0.. (46) ;
~ � ~q, j
For 6=~, the time t~X is defined from the expressi,on ;
i
tg 9s c � - ? ~ ~
- i ~47~ ~
,
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The longitudinal force under the �oundation footing is;
N~ = N`p'S (r)~
(48)
where NP~~tis the longitudinal force caused by the load P applied statically,
- and S(t) is defined by the formula (44).
The maximum value of the longitudinal force under the foundation footing is:
N~ ~~AK~ = N~o'S~~eK~. (49)
where S~~ is the clynamicity coefficient with respect to the force defined by
~ formula ~4k) on substitution in it of t= t~, which we find from the solu-
tion of the equation
ddtt) aa,+~Q ~~~2+ eqt~cosqst+
~--(r- ~ -~Di ) slnqat~=0. _
Q~ (50)
For 6=~ the time tm~ is determined from the expression (47).
ln Figure 64 the graph is presented for Sm~ as a function of r for various -
values of 8q1 constructed by the formula {44).
The longitudinal f orces can be defined more exactly by comparison with the
investigated method if the column (wall) with the foundation is considered a
rigid body loaded by the longitudinal f orce from the ceiling elements, which
is found considering the deformation of these elements under the effect of
dynamic loads. When calculating the deformation af the ceiling elements the
equation of motion of the column (wall) with the foundation of the rigid
body has the form
T (t) N~ (t) - mic = 0,
(S1)
where T(t) is the longitudinal force from the ceiling elements:
m=m�~-m~.
Thr ~..~~~1~,~~i:~n3 ~ force for the column on which two spans are supported with
- ~ fMo~rc
(6
By~tSO _ _ - ~
!0
- 4 -
- -y
f~
~ I
~l!6
~ ~ , 0,5
- ,4.
0,4 '
_ BQ~=O,d
~
0 t
~ ? � ~ 4 S
Figure 64. Graphs of the dynami~ity coefficient with respect
to force in the centrally compressed and extracentral3y com-
pressed columns and walls with small eccentricities considering
the vertical displacement of the entire structure as a rigid body.
Key: a. max
Tne forces in the column (wall) cross sections and under the foundation foot-
ing are defi,ned by the formulas (42), (43) and (45) in which the form of the
function S(t) depends on the investigated stage of operation of the ceiling
elements.
The expression for S(t) is complex and awkward and is not presented here.
The consideration of the deformation of the ceiling is reco~nended for use
in etructures loca~.ed on foundations made of sufficiently dense soil, with
ceilings that have low frequency of natural oscillations, with a magnitude
of the dimensionless parameter n2 > 1.7 defined by the expression:
R - ~'n3) . ~i . n ~ ma
I ~ , �o ~ wD' 9 mx-I-?~ ?
where W is the angular frequency of the natural vibrations of the beam on
stationary support.
In Table 8 for three values of nil~ the values of the dynamicity coefficients
are presented for the force under the foundation footing calculated as the
maximum values of the function S(t) for working of the ceiling elements in
the elastic stage for a suddenly applied load with 6=~ for the following
valuea of the dimensionless parameters:
ne=2~ n~n ~ sa~nsi~~ n i~ _~t sn it~
aa, ~ ~ca, ~ ;
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where the value of n21~ i$ assumed to he equal to 0.57.
Table 8
N~r~~er Valuea of S~~ for values of
t Q~ parameter nll)
a,
0.093 I ~,078 I 0,(183
I 1 1 1,237 I.208 I, I 82
2 !,5 1.248 1.222 !,2
_ 3 2,25 1,296 1,259 1,23
4 3 1,.~36 1.278 1,241
5 4,5 1,586 1,528 1,486
6 6 I,736 1.686 1.646
7 l2 1,938 1.9I 6 l,gg5
For the values of n2 < 0.57 the dynamicity coefficient S~~ varies little,
remaining close to 1.1-1.2.
From Table 8 it is also obvious that on variation of the value of n2(SZ) the
_ dynamicity coefficient varies within the limits from 1.2 to 1.94.
5. Example Calculations
~xample 17. Let us determine the magnitude of the limiting load which the
aelf-supporting wall withstands (d = 0.5 m, H= 3 m) laid from concrete blocks
uaing type 25 mortar (R = 430 tons/m2, Ek = 705,000 tons/m2, p~ = 0.18k ton-
sec2/m4). The wall is completely buried in the ground = 0.124 ton/sec2/m4,
a0 = 100 m/aec, 0.35). The friction coeff icient of the wall material
against the ceiling m= 0.6. The height of the masonry row H' = 0.6 m.
The relative eccentricity accordtng to formula (7) 2e0/d = 1.83 > 1.35; there-
fore by formula (37) y0/d = 0.125 or y0 = 0.0625 m. When calculat~.ng by the
f irst limiting state according to formula (9) is:
1,2�430�2~0,6 ' _
~n=705�109(62,5�10-a)-19�10-~e rad;
for c-~lc;~lation with respect to absence of large resfdual deformations from
formula (10)
, 0,4
500--62,5 -0,915�10-a rad.
Calculation by the f irst limiting state. Let ua determine the mass of the
wall m~ = 0.184�0.5�3 = 0.276 ton-sec2/m. Let us find the coeff icient Ai of
the equation (28);
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L VL~ VL ~ iV~.{~~.A V~IY VL\L/?
A1=0,276�0.5=I ~ +1,33-0,546,-0,6�0,276~~4~3x
~
x(0,8125)(1+~0,751=0,242-0~05~Q,237 ton-m-sec2; -
\ ~
~ A' 0,124~100�3~�0.5 ( g ~0,6�0,81251=21,! ~ _
O 1 / ton-m-sec�
`
A, 1~5�705�l0a(6,25�10-3)s _4~. ton m;
~ 0~6
AF=0.5~f ~~~~~-~6 (Q,8125)(0,33�36+1)-1,1251=0,206 m3.
~ .l
The characteristic equation (31) 0.237s2 + 21.1s + 430 - 0 gives the real
roots sl x-32 and s2 ~-57.
The maximum value of ~ X from (33) for 6= 1 sec will occur for t= 0.135
sec (the term with es2~wi11 be neglected in view of its smallness)
~~axa � 0,48� ]0-a P (0,882).
From the condition (8)
' 14�!0-~
p~ 0,48� 10-'�0,882-~ ton/m3 or ~p = 3.3 kg/cm2.
For the calculation with respect to absence of large residual deformations
the limiting load will be:
0,915�10-~ 2 2
p~ 0,48.10-~�0~882~2'6 tons/m or ~p < 0.26 kg/cm .
In thi8 case it is gossible to consider the mass of the masonry located above.
Here the value of A1 is more precisely def ined, the roots s1 and s2 are de-
fined, and the further calculation is performed by the f ormulas (33), (34).
Exaruple 18. The structure is given with the basic characteristics; column
good 6 X 6 meters; coating made of reinforced concrete slabs 40 cm thick over
' bars with a cross section of 40 X 85 cm (w = 50 radians/sec); the constant
_ load on the covering without considering the natural weight q= 0.128 kg/cm2;
reinforced concrete columne 240 cm high, 40 x 40 cm in cross section made of
type 300 concrete (Rl~b = 130 kg/cm2), with longitudinal reinforcing made of
- class A-II steel (Ra~~ = 3000 kg/ctn2); U= 0.02; the foundation 40 cm high,
240 x 240 = 57,600 em2 in area; subcolumn 40 x 120 x 120 cm; foundation soil
sandy (p = 1.6�10'6 kg-sec2/cm4; al = 5~103 cm/aec).
Let us determine the li~iting magnitude of the dynamic load on the column of
the structure. '
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In accordance with the above~presented data we find the values of the masses
(the dimensionality y~] = kg~sec2/cm); mk = 0.92; m~ = 7.6; m~, = 85.53;
M= 0.92 + 7.6 + 85.53 = 94.U5.
Let us det~rmine the dimensionless parameter
~ ~n 2�0,416 85,53
n 1 }
s Na +mx-+-~! 8,5 (1+8~52)=1,~,
where . ,
A= a1 =0,416; 2M _ 2�94,05
wD �c =pF,p ~ -1 10-"�240II�240 -8,5. �
- Since n2 = 1.08 < 1.7, the calculation is performed by the formulas (43)-(49). `
Let us find the parametera
- ~ r� �c-1=2,74;
a
q~ =�c
p= 2,45 1/sec; 4a=~�4~=6.7 t/ sec.
By formula (43) let u3 determine the dynamicity coefficient with respect to
the force for the lo~er cross section o� the column. For this cross section
m~ = m~ = 7.~i kg-sec /cm.
For simplification of the calculations let us take 6=~. Then by formula
(47)
2
tH 9 tn~aKC !/i `-0,892~ Q7fM9KC ~ 2~44
or
2,44
tuaNC=-=0,3G4 sec.
6,7
By formula (44)
Smaxo=l-e- o.8s rcos 2,44-2 79 sin 2 A4~ ~
` ~ ~
=1-0~91 (-0,999)~1,41.
T11e sr~me r~sc:li. will be ohtained by the graph in Figure 64 for Oql > 50 and
- 7, %u. -
The dynamicity coeff icient ~m~ is;
mnaaxc�1,41-~4,06i1~41-1),^; 1,38
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and the maximum value of the longitudinal force
- N~+axc =(~p � 600=) 1.38 - 496. l0a Ap kg .
- The limiting magnitude of the longitudinal force for the coefficient of lor,-
gitudinal bending ~ 9 1 in accordance with the chapter SNiP II-V. 1-62* is:
N,=1,2RayFH-~-Ra,Q�Fx=1,2�130~90�40+3000�0,02�40�40=
~ 346� !0' kg. ,
The longitudinal force from the static loads
' Ncz=(~n-I-~~x)8=(85.53-FU.92)~81a84.8�10~ kg:
then from the condition N~ + 1.2 N8t < NB the limiting magnitude of the
dynamic load on the ceiling which can be taken by the column is:
Na-1,2N~i (346-1,2 . 84,8) t0~ 2
~P ~ 496 � 10~ a 49G � 10~ 0, 5 kg / cm .
The maximum value of the longitudinal force under the �oundation footing from
the dynamic load by formula (49)
N,p. am~c m(0. 5� 600~) I. 9 I~ 254 � 10' k$ .
The maximwa stress in the ground under the foundation footing considering the
static loads will be:
N~,.r,nr,c-{-Mg 254~10~+94,05�981 2
ac.raKC= F � 240" ~5 kg/cm < Rg.
- a
where Rg is the calculated pressure on the foundation soil under the effect
of the ~iynamic loads.
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CHAPTER VI. CALCULATION OF THE ~NCLOSING S~'RUCTURES FOR THE THERMAL EFFECTS
OF MASS FIRES
l. Calculated Thermal Effects
The effect of fires on the enclosing structures of shelters is connected
, with the variation of the temperature of the environment with time depend-
ing on the type of fire. Inasmuch as the temperature conditions of mass
fires have been discussed Iittle [3, 11, 45], it is expedient to take the
standard temperature as the initial curve for determining the calculated
thermal 2ffects of mass fires. In contrast to the temperature regime
represented by this curve, for any single fire the temperature distribution
is more cortiplicateti. If m2asures are not taken with respect to extinguish-
ing or localizing it, four periods are distinguished: the initial period of
_ ignition, fhe period of complete combustion, the period of afterburning and
the coolirig off period. In the case of the occurrence of fires in buildings
sub~ected to the effect of a nuclear blast shock wave or damaged after
ordinary air bombing, the duration of the initial fire period does not
exceed 10 to 15 minutes [11, 45].
At the end of the initial period, as a result of the combustion of the
volatile products tne temperature at the center o� the fire reaches 500 to
600�C.
During the second period combustion of the basic mass of the combustible
materials takes place. Under the effect of high temperatures, the structural
elements of the building the ceilings and floors, the supports, the
~iartitiohs are heateil up, deformed and collapse. A charred heap is
formed at the place where the building stood in which the charred remairis
burn completely up. -
The third period is characterized by the afterburning o~ the solid carbon
reaidue in the pile of incandescent fxagm2nts of building materials.
- The fourth period the coolin$ o~f o� Clie charred pile is charact~rized
by the fact that the combustion is in pxactice absent in the body o~ tYie
pile.
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The curves for the temperature regimes o~ mass fires of the first three '
types (see Table 2) without consider~ng the duration of the cooling off
period of the charred piles are presented in Fig 65. Curves (1, 2, 3)
were calculated under the following prerequisites: 1) the duration of the
initial period of the fire was taken equal to 15 minutes; 2) the burned '
' load for 1 m2 of pro~ection of a floor of the building will be 50 kg;
3) the nature of the temperature variation during t3~e first and second
periods of the fire corresponda to the standard temperature curve; 4) the
temperature at the end of the third period of the fire arops to the initial
temperature.
The temperature drop in the burned piles to 100-70�C can continue for several
days [69, 71, 73].
t;
C '
f00
� ~0 � ?
6~0 ~ 3
400 y
c1~D �
0 1 2 3 4 S 6 ~ B 9 a0 2, v(1~
Figure 65. Variation of t~emperatures at the center of the .
fire. ;
1-- type I(see Table 2); 2-- type II; 3-- type III;
4 type IV (KV-III) ~
Key:
1. hours ~
The specially performed studies made it possible to eatablish that the temp-
erature regime of the fires considering the cooling off period of the
charred piles can be represented in the form of a curve made up of three
sections (Fig 66).
The first of them is limited with respect to�time ta 1 hour from the time
of beginning of Che fire to collapse of the structures (basically~ the
ceilings and floors). The calculations:show that at the time of collapse the
surface temperature of the unburned structural elements can reach 700 to 800�C
at the same t~me as the average temperature of the fragments does not exceed .
250�C.
During the second period as a result o� redistribution of the temperatures
with respect to the thickneas o~ the fxagments, the total temperature drop
in the pile is from 700 to 250�C. The duration of this period defined
by calculation does not exceed 8 hours and depends on the thickness of the
fragments and their thermal physical characteristics. ~
~
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�
The third section of the curve is characr_eristically the cooling off period
of the pile. Its duration is a��ected by the significant ef~ect of the
hol.lowness of the pile, its height and the temperature of the fragments at
the beginn~.ng of this period. However, the convective air temperatures
formed in~the piles as a result of the t aaperature difference of the pile '
(to 250�C and the environment (less than 50�C) have a decisive effect on
the shortening of the time of the third period. The intensity of the con-
vective currents of the pile increases significantly if unobstructed
through channels passages are created under them in advance.
- t'C
,
~ ~
, 6A7 , -
' 4L~
?00 ~
D / 2 J� 4 S 6 1 B 9 10 1f 2; v(1)
Figure 66. Gl:aphs of the temperature variation
- 1-- in the center of the fire; 2-- �ragments of the
structural elements after collapse; 3-- surfaces of the
structural elements before reaching the limit of their
f ireproofness
Key:
1. hours
- The temperature variation of the lower surface of the pile in time consider-
ing the effect of the ccnvective currents can be determined using the
expression
- r(o,zt-~-eo:i:a~ i _o.2sl ~I~
i= t,~
q~-}- 230 e L n J
Key: 1. init
where t is the desired temperature of the lower surface of the pile in �C;
tinit is the temperature of the collapsed structures before the fire in �C;
e is the hollowneas of the pile in fractions of a unit; T is the time from
the beginning of the ~ire in days; n is the number o~ floora.
In contrast to the charred piles, the temperature regime o~ the obstructiona
formed as a result of the destruction of buildings and other above-ground
structures by a nuclear blast shock wa~re ~s signif~,cantly more ~avorable
for the shelter enclosures (Fig 65, curve 4~. At the same time the dura-
tion of the f'ires in the piles (type IV) can exceed by 2 or 3 titnes the
duration of other types o#~u~ass ~ires. The combustion in such piles will
be ceater type combustion. On the average, no more than a third of the
combustion materials which also:support the high temperatures for 10 to
12 hours, burn actively in the pile.
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During the process of development of mass f ires, the air temperature also
varies, which reaches 800, 500 and 200�C for type I, II and III fires,
respectively [69, 73]. The maximum increase in the air temperature in
the territory where the type IV fires occurred does not exceed 10�C.
E,�C
800 - - ~
- - � ' - -
601~ ~ .
40D �
.
. ZQO . ' -
O f 2 3 4 S 6� 7 8 9 f0 , v 1~
Figure 67. Variatio~ of the air temperature during the fires.
1-- type I(KV-I); 2-- type II (KV-T~);3 type III (KV-IV) ~
Key:
1. hours
The actual temperature conditions in each individual case can differ some-
what from those presented in Figures 65 and 67. However, hefore taking the
field measurements, the proposed temperature curves with small margin can i
be used when calculating the enclosing structures of shelters and designing ;
the internal equipment. '
From what has been stated it follows that the enclo~ing structures of the ;
shelters during m~ss f ires occurring as a result of nuclear blast can ~
experience the following thermal eff ects: '
i-
Short-term effect lasting up to 12 hours directly from the center of the ;
fire and the heated air;
Prolonged effect last~ng mor2 than 12 houxs from the charred pile farmed i
in the building as a result of collapse of the ceilings and floors, and
the partitions when the limit of their f ireproofness comes.
The floors and ceilings not in contact with the ground or unburied outside ~
walls and the entra~~ces of tite built~in (under an above-ground structure) -
and separately standing shelters can be sub~ected to the short term thermal ~
ef f ect s . ~
i
Only the ~loors and ceilin~s o~ the bu~].t-in shelters can be sub~ ected to
long-term thermal effects.
The short-term thermal e~~ecta, depending on the type of mass �ire and the ~
temperatures occurring in ~t can be o� ~ive types: KV--I, KV-II, KV-III, ;
KV-IV and KV,V.1 '
1
[Translator's no te: KV stands for short term.]
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The caleulated thermal effect on the enclosing structures of shelters is
defined according to Table 9 depending on the expected type of f ire~ the
type of construction (built-in or separately standing) and the calculated -
structural design. For mass fires of the f irst three types the calculated
thermal,effects presented in the aecond column of the table are realistic
only for calculating the floors and ceilings of the ahelters located under
, two-story or higher buildings of I, II and III degree of fireproofness.
If there are passages in the f loors and ceilings of such shelters, the
prolonged theriaal effect (DV) must be replaced by the short-term type RV-V
- during the calculation (see Fig 68), and in all remaining cases (other ~
degrees of fireproofness of the above-ground buildings~ the one-story resi-
dential or j.ndustrial structures) the DV is replaced by the KV-I type
effect. When the separately standing shelters are located in the zone of
the formed piles their enclosures are designed for the short-term thermal ~
effect type KV-III.
t,'C
700
S00
.~A'J 2
_ , 100 ~ f
~ O! 2 d 4 S 6 7 B..9 Z;4' -
Figure 68. Variation of the temperature of the outer surface
of the ceiling of a shelter.
_ 1-- with passages (KV-V); 2-- without passages
~ Key : _
1. hours
Table 9
Calculax~d Thermal Effects
_ Thermal eff ects on the structures
Type of fire Built in Separatel~standint~
Floors and LJalls and Floors and ceilings, -
ceilings entrances walls and entirances
I DV KV-I KV-I
II DV K~II KV-II
III DV KV-IV -
IV KV-III - _
2. Protection of the Enclosing Structures of Shelters from Heating
The admissible heat and humidity parameters of the air in the shelters in
the case of limited air supply are basically ma.intained as a result of the
accumulation of heat released by people and equipment and the enclosing
- structures [14].
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Under the conditions of the thermal e�fect of a mass fire the removal of _
heat from the structures becomes worse, and with insufficient thickness of
the enclosures, an inflow of heat from the outside will be observed which
undoubtedly will worsen the protective propertiea o~ the shelter. Accord-
- ingly, the encloaures and the entrances o~ the eheltera muet be thermally
ineulated in the correaponding fashion, which ie achieved by increasing the
- thicknesa of the bearing structure or the structure of the thermal insula-
tion layer, the shields and ducts.
The thermal insulating layer can be b~~ilt both on the outside and on the -
inside surface of the enclosure. Tr~e outside thermal insulation is made
from incombustible materials with low coefficient of thermal diffusivity
(slag, sand, crushed claydite, pumice, slag concrete and so on).
In order to construct the internal thermal insulation, special heat insulating
wadding, slabs, panels, and sheet thermal insulating materials are used [55].
The shields dPSigned to protect people against radiant heat emitting from
the heated inside surface of the enclosure are made of sheet building
materials (metal, asbestos, plywood, and so on), and they are installed on
the inside surface of the enclosing structures at the points where people
are constantcly present. The greatest eff ect is achieved with double shields -
inatalled at a distance o� 10-15 mm from the inside surface of the enclosure
and from each other.
J
The ducts can be construcred to lower the thermal load acCing on the floors
and ceilings of the shelter and decrease the required shielding. They are _
= channels that are open at the top (or closed by any burned structural ele-
ment) arranged perpendicularly to the length of the building. The width
and spacing of the ducts (the distance between their axes) must be no more
than 0.4 meters, and the depth, no less than 0.3 meters. In the bulldings,
the floors and ceilings of which are made of reinforced concrete panels,
the width of the ducts can be increased to 1 meter. It is expedient to
construct the ducts in the floors and ceilings of built--in shelters located
in buildings, the floors and ceilings between floors of which wi11 collapse _
when the ljmit of fireproofness ar_rives. .
3. Designing Enclosing Structures of Shelters for Heating During Fires
- The problem of designing the enclosu~es for heating consists in determining
the temperature of their inside sur~aces.
The desired temperature in general ~arm can l~e repzesented as a function of
_ many parameters:
, ~3~- ~4~ ( )
1= f (t~ B~ h, lI; B: H; `H84r ~H84i al[t aee~ (2~ 2
~2) ~2) (5) (6)
Key: 1. external effect; 2. init; 3. enclosure; 4. soil; 5. outside;
6. inside
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where texternal effect is Lhe external thermal effect; h is the thickness
= of the cal`culated structural element; a is the coefficient of thermal
diffusivity of the ericlosing materia~; B is the width of the shelter;
H is the~height of the shelter; tinit is the initial temperature of the
calculated enclosure; tinit is the initial soil temperature surrounding the
- ehelter; ag ia the heat exchange coefficient for the outside aurface of
the enclosure; aint ig the coefficient of heat exchange for the inside
surface of the enclosure.
It is extr~mely difficult to obtain this function u~ing the Fourier equation
of thermal,conductivity. The calculation formulas that are presented below
were found after mathematical processing of the results of the numerical
solution of a series of problems on a hydrointegrator.
The experimental checking of the proposed calcula~tion formulas demonstrated
that they insure a precision of calculating the enclosures for heating with~
in the lituits of +10%.
Design for Snort-Term Thermal Effect
The design of the enclosing structures for short-term thermal effect is
carried out from the condition
tn~CKC~< 1rtp~Q~ ~3~
xry: 1. ma.x; 2. lim _
where tlim is the limiting temperature (�C) on the inside surface of the
_ structure, the magnitude of which for the shelters is taken at 30�C [lk,
55] (when constructing double shields on the inside surfaces of the
enclosures the value of tlim increases by 10�C); t~X is the maximum
temperature ( C) on the inaide surface of the structure defined by the -
formula
Qi,s ~
rrtanc-=~' ~ ~P~~iHUH~2 ~4~
(~1) h' ' ~ )
Key; 1. max; 2, init
A is l~he coefficient characterizing the total amount of heat acting on ttie
enclosure taken according to Table 10 as a~unction of the type of calcu-
~ lated thermal effects;
Table 10 ~
Table .o~ .Coeff~.cients..A-and K...
Thertnal ef~ect KV~I KV-II KV~ZII . KV-I:V KV,V
A� 10'"3 16 . 7 11~.~8 11 7. 2 13 . 9
K�10'3 13 17.2 17.8 21 15.4
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h is the total thickness of the calculated structure in meters; !
~n
h _ hi, (5) ,
i=i
where hi is the thickness of the individual layers of the multilayered
structure in meters; m is the number of layers;
a ia the coefficient of thermal diffusivity of the structure in m2/hr
m i
~ y~ h ~ 1 I
a _ r= i 1 ~ (6)
� ; h~ m
~ ~ ~tYih~
~a' '~j 1
where ~i, ~yi, ci are the coefficients of thermal conductivity, the specific
nass and the heat capacity of the i-th layer taken in accordance with
SNiP II-A.7-62 [56];
~ is the coefficient taking into account the effect of the size of the
structure on the temperature of the inside surface af the structural ;
element; when calculating the wa11s of a shelter the coefficient ~ in all ,
r_asea is taken equal to one, and when calculating the floors and ceilings,
it is determined by the formula
~ ~=1 + a,as-e~H ~7~ .
1,45 ~
T~a~co/, ~ _
\1
Key; 1. max
where B is the width of the facility (the spacing between the outside wall ~
adjacent to the ground or the inside concrete walls no less than 0.4 meters
thick) in meters;
A is the height of the facility (the distance from the surface of the floor
to the upger point of the inside surface ~f the ceiling) in meters;
T~X is the time of occurrence (days) of the max~num temperature on the
inside surface of the structure during the short-term thermal effect taken ~
equal to:
T~~ = 0.15 days ~ox T~0.15; (8) ;
_ T~X = T days ~or T>0.15.
Here
j = K aQ~s da~s ~$a)
17$
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K is th~ coeff icient characterizing the total amount of heat acting on the
enclosure; it is taken according to Table 10 as a function of the type of
calculated thermal effect;
~y is the coeff icient taking into account the effect of the amount o~ heat
exchange on the temperature of the inside surface of the enclosure defined -
by the formula
~1)TO. iaz
~=1- 5�K~ (l-O,la), ' ~9)
3,31
Key : 1. ~tax
a is the heat exchange coeff icient on the inside surface of the enclosure
in kcal/m2-fir-d'e~,�
tinit is the initial temperature of the calculated structure takex~ equal to
the maximum calculated temperature of the soil for the given terrain, but
no less tlian 15�C.
When calculating the enclosure with internal thermal insulation, the maximum
temperature of application of the therma.lly insulating material [55] must
be above th~e temperature of the inside surface of the enclosure in contact
with the hea.t insulating layer defined by the formula (4) under the condi- ~
tion ~_hat a=0 and ~=1.
fihe ca].culation of the heat insulation of the outside door of the shelter
(for example, in the lock chamber) is made in accordance with the dtscussed
method of thermal calculation of the floors and ceilings; the difference
, only consists in the fact when determining the coefficient ~ it is not the
height of the faCility that is taken as H, as when calculating the floors
and ceiTinga, but the len&th of the lock chamber; correspondingly B is
equal to the height of the lock chamber.
In the structures where it is possible ta neglect the heat influx through
the entrances, the calculation of the thermal insuiation of the outside
door is not made.
_ In order to facilitate the calcula!-ions by formu].as (4)-(9), a nomogram
(Fig 69) is constructed which makes it possible to determine the maximum
temp~ra~ure'on the inside surface of the enclosure t~X and the time of its
onset T~ depending on the type of short~term thermal e~fect KV. In
additiori~ the nomogram permits calculation o~ the enclosures under the
_ thermal e~~ects differing from the KV type ef~ects. ~or this purpose the
- nomogram has the scales S, the div3.sions of whi.ch correspond to the areas
(deg-hr) included B~etween the x~axis and the cuxve ~or the temperature
variation on the outside surface of the enclosure. When calculating the
enclosures for the e�fect of the thermal load simila~r witli respect to aut-
line But d~ifferent with respect to absolute ~nagnitude from the calculated
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loads, from the corresponding points of the scales S straight lines are
drawn at an angle of 45� to the horizontal axis. These straight lines
will also be calculated for the given thermal load.
It is possible also to use the nomogram when determining the valuea of
~ and ~ used in the formulas (4) and (11).
The determination by the nomogram of the maximum increase in temperatures
on the inside surface of the enclosure is made in the following procedure.
On the a-axis, from the point corresponding to the thermal diffusivity of
- the calculated structural element (point 1) a perpendicular is drawn to the ~
intersection with the curve h corresponding to the thickness of the ~
enclosure (point 2). From point 2 a horizontal line is drawn until it
meets the given calculated thermal load of the KV type (point 3), from which
a vertical is dropped to the iatersection with the curves a(point 4) and
b/H (point 7). In addition, on the T~X axis at the point of intersection
of it by the vertical, we determine the time of occurrence of the tempera-
ture maximum on the inside surface of the enclosure. Drawing horizontal
straight lines from the points 4 and 7, oi; the lefthand scales when
necessary [for calculation by formulas (4; and (11)] we determine the
values of � and On the right of point 4 we find the point 5 from which
in the given sector we draw the straight line to the intersection with
contir:uation of the T~X axis (point 6), and then the inclined straight
line to the intersection of point 8 with the horizontal straight line drawn
from the point 7. We find the point 9 at the point of intersec�tion of the
perpendicular dropped from the point 8 in the horizontal axis of the nomo-
gram. Drawing the ray of the process (X) from the point 9, we return to ~
the point 1 trom which we now drop the perpendicular to the intersection at
the point 10 with the curve h, ~ust as when finding point corresponding to
the thickness of the calculated enclosure. From point 10 we draw the hori-
zontal line to the intersection with the inclined straight line X drawn
from point 9. Reproducing or dropping (depending on the location of point
11) the perpendicular to the given thermal load of the KV tSTpe and drawing
the horizontal straight line from the point of their intersection (point 12),
on the lefthand scale we determine the desired increase in temperature
(~t~X). Summing the value of Ot~X obtained with tinit~ We determine the
maximum temperature on the inside surface of the encl.osure during the time _
of the f ire.
When using the ~iomogram it is necessary to consider two peculiarities.
First, the straight line drawn ~rom po~nt 2 does not always intersect the _
- straight lines of calculated thexmal loads. This case is possible when
calculating the enclosing structure, the thickness of which is equal to
- or lesF than 0.5 meters. In this case the horizontal straight line clrawn
from point 2 intersects with the vert~,cal straight line bounding the
straight lines KV on the right; the points 4 and 7 are shifted to the same
vertical straight line, and the point 5 merges with point 4. The time of -
180
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J
Re-i 3)
Ke-9
J rajnl KG�C
~1 S Ka-~
xe�ID
A,~l
t
a,n~~yS~{c�yo _ _ _ _
� ( ) ~ a~ _
~ ~
a~ . 'ie _
~
. 4e -
r + ; I ,
4! ` .
T e� at ~ RS�!0~ I�10"~ !5�0'~ 2.p�~ p� " 4~'
"M` , I
\Z~ t ~ 9( ~ � ~
a
p f ~Ie I
~ ~I ~ ~ ~u
~
'91 _ i_
00 F,n
pR.,., I 4f
(5) ~ I ,
, ~ .a ~
~ ~ ~
~ i 1 - - ~
~ ~ ~s
p r~~- ' ~ ' ~--1~-~~~-~ ~
~ ' _ , ' -i-._~_~ 09
- -
~ � ~ ; ~ !s
ron Rs�m~ ' '
i
Ke�m-~ -
�e�Q. i '
xe-v~
- Ke-i i
�S,r~pod ~J7J p001000
Figure 69. Nomogram for determining the value of
~t~X tmax~'tinit �f the inside surface of the enclosure under
the thermal effects of the KV type
Rey: -
1. S, hr~deg 5~ Qti~
2. T~, days 6. S, hr~deg
3 . ICV-T . . .
4. a, kcal/m2-hr~deg
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occurrence of the maximum temperatures with an enclosure thickness to 0..~ m
and a coeff icient of thermal diffusivity of the enclosure material is no
lesa than 1.0�10'3 m2/hr will always be equal to or leas than 0.15 days.
Secondly, when calculating the heating of the shelter walls in connection
with the fact that ~ is taken equal to one, the point 7 is shifted to the
T~X axis, and the point 8 merges with point 6. For the rest, with the
exception of the indicated peculiarities, the procedure for performing the
calculations by the nomogram in both cases does not change.
Simplif ied Method of Calculating the Floors and Ceilings of Shelters for
the KV-III type Thermal Eff ect
For designers the f ires in the piles (IV type mass fires) are uncondition-
ally of the greatest interest. Therefore, along with the investigated
method of calculating the enclosures for thermal effect of the KV type, 1et
us present a simplified version of it.
The minimum thickness of the single-layer reinforced concrete structure
providing protection from heating during type IV fires must not be lese
than 0.6m*. The thickness of the two-layer structures can be taken accord-
ing to Table 11.
Table 11
Thickness of the Heat Insulating Layer Providing Protection
from Heating of ~ao-Layer Structures
~ Material of the Thicknees of the heat insulating layer in m
heat insulating for a thickness of the bearing layer of
layer reinforced concrete in meters
0,4 0.35 0.3 0.25 0.2 0.15
Sheet asbestos 0.05 0.06 0.08 0.09 0.1 0.1
Asbestos cement slabs 0.09 0.11 0.13 0.16 0.18 0.2
Concrete with gravel 0.21 0.26 0.31 0.35 0.4 0.45
Concrete with brick rubble 0.17 0.21 0.25 0.29 0.33 0.37
Planted ground 0.23 0.28 0.34 0.39 0.45 0.5
Sandy clay (wet) 0.3 0.38 0.45 0.52 0.59 0.67
Brick masonry and crushed
claydite 0.12 0.14 0,17 0.2 0.23 0.25
Claydite concrete, dry sand
and slag concrete 0~13 0.16 0.19 0.23 0.26 0.29
Boiler slag 0.11 0.13 0.16 0.19 0.21 0,24
Blast furnace slag 0.1 0.12 0.14 0.17 0.19 0.21
*In accordance with the norms adopted in the Federal Republic of Germany,
the thickness of the reinforced concrete ceiling preventing its heating ,
during f ires in piles also must be zqual to or greater than 0.6 meters [7~a].
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When making the heat insulation from material not presented in Table 11,
`the~thickneas of the layer can be determined by the formula
Key: 1. ~ins
wherp hing is the deaired thickness of the heat insulating layer in meters;
A1 ia the coefficient determined by Table 12 as a function of the actual
thickness~.af the bearing reinforced concrete atructure; ains ia the co-
efficient of thermal diffusivity of the heat inaulating material in m2/hr.
Table 12
Values of the Coefficient A1
Values of A1 for the thickness of the
Structure reinforced conGrete structures in meters
0.4 0.35 0.3 0.25 U.2 0.15
Without shield 82.5 102.5 122 141 160 180
With shield 27.5 47 66.7 86.3 106 125
_ Calculation for the Long-Term Heating Effect
- The calculation of the ceilings for long-term heating is also made from the
c~ndition of ~satisfaction of the inequality (3) in which t~X ie the higheat
temperature ( C) of. the lower surface of the ceiling reached while people
are in the structure ~qual to:
~1) (2)
r (3)
1M8H0 -/~~BNC) f~~ taKO C Tnpen~ 1 10
t - f~Ta e~ f or~ - t ~ TQ ~ ~
'
- Key: 1. max; 2. ext; 3. lim
where rl~ is the limiting time people will be in the structure, days;
T~~ is the time (in days) for the lower surface of the ceiling to reach
the maximum (extremal) ~alue of the Cewperature determined by the graph
o� functian f(T); f(T) :.s the function o� the tempergture variation on the
lower surface of the ceiling as a~unction of the time T;
t - / ~i~ - ~OW~~ Tb iH(99 ~~~5 < z < 10 r dag~; (11,
T
\1~
iCey: 1. init
T is the time fro~ the beginning of the f ire in days; d are the
empirical coefficients equal to:
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aa 0,112h,
oo.~i + (12)
1,1 n-}- ah 0,05 Q~'~~ -0,25~
, (13)
where n is the number of floors in the building above the shelter; h, a are
calculated by the formulas (5) and (6); ~ are the coefficients
determined by formulas (7) and (9) for any time within the limits from 0.5
to 10 days; tinit is the initial tanperature of the calculated structure
defined ~ust se when calculating for a short term thermal effect. -
When constructing the inside thermal insulation, the surface temperature of
the enclosure in contact with the heat insulating layer, 3ust as during the
short term thermal effect, must not exceed the limiting temperature of the
application of the selected heat insulating materia]. in accordance with the
SNiP [55].
The temperature of the inside surface of the enclosure in contact with the
heat insulating layer is defined by the formula (11) under the condition
that aa0, and ~=1.
4. Effect of Heating on the Bearing Capacity of the Ceilings and Floors and
Seal of the Structure
The thermal e�fect of mass fires is not only the cause of heating the ,
enclosing atructures, but in a number of cases leads to a significant
reduction of their calculated bearing capacity and to destruction of the
seal of the structure.
The heating is of special danger for the cailing of a shelter as the most ~
outer structural element. Therefore the bearing capacity of the ceiling ~
must be determined not only by the f orce effect but also by the thermal load.
- When calculating ceilings it ie asaumed that:
The bearing capacity of the ceilins before heating (ql) corresponds to the
calculated equivalent static load ~or the given shelter;
The bearing capacity of the ceiling after the calculated thermal effect of
f ires (qz) must not be less than 0.15 kg/cm2 in buildings up to 3 stories,
and 0.3 kg/cm2 in buildings over 3 stories high inasmuch as the baeic load
on the ceiling o~ the built~in sheltera in the case of a mass Pire is
the load from the collapsed st~ucture;
The ef~ect of the shock wave when determining the value of q2 is not taken
into account, for the occurrence of the masa fires is poasible only with
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_ a pressure on the shock wave front of no more than 0.5 kg/em2 for which the
enclosing structures of the shelter do not reach the calculated limiting
state.
The value of q2 within the limits of accuracy of the engineering calculationa
can be determined from the condition that the compressed concrete layer hl,
the temperature of which reached 500�C and higher does not participate in
the working of the cross section [60].
Therefore the calculation of the bearing capacity of the ceilings consider-
ing the thermal load must be made for thermal effects type KV-I, KV-II,
KV-V, and DV for which the outside temperature can exceed 500�C (see Fig 65
and 68). In other cases (thermal eff ects of the KV-III and KV-IV type) the
effect of heating on the bearing capacity of the ceilings can be neglected.
- The procedure for calculating the ceiling considering its heating during a
fire can be considered in the example of a single-span beam. -
For a single-span reinforced concrete structure q2 is defined by the formula
8Fa Re ho ~ 1-}i Ra)
, R�
9~ - bt~ ~ ' ~14}
where Fa is the area of the transvers2 crosa section of the reinforcing
in em2; Ra is the calculated resistance of the reinforcing in kg/em2;
Ri is the calculated resistance of the concrete compressed with bending in
kg/cm2; k is the calculated span of the structure in cm; b is the width ~f
the cross section in cm; U is the reinforcing coefficient
~ _ 6ho ' h� a h0 _'hl'
F� ' (1S)
where h'0 is the working height of the cross section after heating of the -
structure in cm; h~ is the working height of the cross section before heat-
ing of the structure in cm; hl is the height of the cancrete layer in cm
with a temperature of 500�C or more. It is determined by the formula
obtained from expression (4):
h, = 652 a�.6S cM. (16)
In the presence of heat insulation over the ceiling hins the iaagnitude of
tne layer hins heated to 500�C will be ~ound by the formula (16) after !
substitution of the coefficient o~ the~qal diffusivity of the heat insula-
tion in it in place of the coefficient of thermal diffusiv~ity of the rein-
forced concrete.
If hins,hins~ the value of hlns is calculated again with the coefficient of
thermal diffusivity found by formula (6)~ after which q2 is determined by
formula (14), the magnitude of which must not be lesa than that indicated
in the present item.
In~order to prevent losa of aeal of the strt~.cture, it is necessary to pr~-
vide for protection of the sealiag materials by the heat insulating layer
ineuring maintenance of t,heir properties during heating of the enclosures
during f ires. The thicknesa of such a layer can be determined by the nomo-
gram in Fig 69 under the condition that a~0, and ~~1. The surface te~npera-
ture of the heat insulating layer bor