TEKA. COMMISSION OF MOTORIZATION AND ENERGETICS IN AGRICULTURE – 2014, Vol. 14, No.

4, 161–168

Calculation of Deflection of One-Layer and Two-Layer Slabs

Supported on Four Sides
Dmytro Smorkalov

Kyiv National University of Construction and Architecture

Povitroflotskyy Prosp., 31, Kyiv, Ukraine, 03680, e-mail: smorkalov@i.ua

Received December 04.2014; accepted December 17.2014

Summary. The article presents the method and results PU5326(2)WORK

of experimental research of deflection of one-layer
and two-layer slabs influenced by short-term lateral The main purpose of proposed work is the
load. The proposed method of calculation is based on comparison of proposed method of deflection
WKH OLPLW HTXLOLEULXP PHWKRG WKH FDOFXODWLRQ RI VODE
calculation for layered slabs under lateral load
defleFWLRQV YDOXHV E\ /,5$-&$' EXQGOHG VRIWZDUH
The comparison is carried out of experimental and with the results of experimental research.
theoretical results of slabs deflection calculation.
Key words: slab, deflection, steel fiber concrete.
(;3(5,0(17$/ 5(6($5&+

INTRODUCTION $FFRUGLQJ to the set purpose of research,

slabs of  series were produced, two slabs for
Layered constructions have been increasingly each series. The scope and outline of experi-
applied in building recently. When appropriate mental studies is set in Table >].
composition of separate layers is selected, multi- The total size of single-layer slabs amounts to
layer constructions with perfect construction 800×800× mm; the thickness of each layer of
properties may be created [9]. Layers are mostly reinforced concrete (concrete and steel fiber
composed of heavy concrete and effective steel FRQFUHWHLQWZR-OD\HUVODEVLV mm.
fiber concrete. Such constructive decisions have Series I (ɉɎUHSUHVHQWVDVODEPDGHRIVWHHO
a widespread use in road and airfield pavements, fiber concrete.
logistics areas, heavy-weight industrial floors, Series II (ɉɁ UHSUHVHQWV D VLQJOH-layer rein-
etc. >  @ $PRQJ WKH PRVW SURPLVLQJ forced concrete slab with single-layer rein-
trends of reasonable usage of steel fiber concrete forcement Ø4 %S-I laid at the bottom of the slab
is its application in composite structures, as a ZLWKSURWHFWLYHFRQFUHWHOD\HU mm thick with
rule, in combination with concrete or ferrocon-  mm pitch
crete, with distinct partition of functions of eve- Series III (ɉȻɎ UHSUHVHQWV D WZR-layer con-
ry material. In particular, the steel fiber concrete crete slab, the top layer of which is made of un-
that is rather thriftily applied along the construc- reinforced heavy concrete and the below one –
tion outline, in a thin layer, provides high crack of the steel fiber concrete.
resistance of constructions, as well as its high Series IV (ɉɁɎRIthe studied samples rep-
durability due to high indices of tensile strength, resents slabs consisting of an upper layer of steel
frost resistance, corrosion resistance and high fiber concrete and a heavy concrete layer rein-
rates of other types of resistance of the steel fi- forced with metal reinforcement mesh Ø %S-I
ber concrete >@ $WWKHVDPHWLPHWKLV solution VHWZLWK mm pitch.
provides necessary preconditions for significant Slab concreting was carried out in two stages.
reduction of strength and value of the primary 6ODEV RI ɉɁ JUDGH DQG FRQFUHWH OD\HUV RI ɉɁɎ
concrete and reduction of the number of rein- DQGɉȻɎVODEVZHUHFRQFUHWHGDWWKHILUst stage.
forcement rod. Thus, there are preconditions for Siliceous sand and giaQW JUDYHO RI … mm
obtaining high indices created while their cost IUDFWLRQ 3RUWODQG FHPHQW RI 0 JUDGH ZDV
drops. used as binding material; water-to-cement ratio
 '0<7526025.$/29

amounted to W/C 0, 3ODWHV RI ɉɁ DQG ɉɁɎ

grade were reinforced with binding wire mesh of
%S-, JUDGH  PP LQ GLDPHWHU DQG mm pitch
in two directions. Concrete protective layer was
 mm. The surface is bush-hammered while the
concrete is immature.
$WWKHVHFRQGVWDJHDIWHUGD\VVODEVRIɉɎ
grade were concreted and steel fiber concrete
layer slabs of ɉɁɎ grade were additionally con-
creted.
Steel fiber concrete contained steel fiber with
diameter df =  mm and length lf  mm;
volume percent of reinforcement was —f = 
Fine concrete without coarse aggregate was used
as a concrete matrix; water-to-cement ratio
HTXDOHGWR:& 0,

Table 1. Scope of experimental studies

Fig. 1. *HQHUDODSSHDUDQFHRIWHVWLQJEHQFK
Slab Composi-
Series Section
grade tion
ɉɎ- –steel fi-
 ber con-
I crete
ɉɎ-
2
–reinforced
ɉɁ- concrete
II
ɉɁ-2
–concrete; a
ɉȻɎ- 2–steel fi-
III ber con-
ɉȻɎ-2 crete

–steel fi-
ɉɁɎ- ber con-
IV crete;
ɉɁɎ-2 2–reinforced
concrete

*HQHUDO appearance of testing bench is

showed in Fig. .
b
The same calculation model was applied for
slab: 1  hinged support, 2  cylinder support
Fig. 2. Distribution of load (aand support (b upon
both the one- and two-layer slabs exposed by
lateral load: a slab is hinge-supported on four
sides and influenced by evenly distributed load ,W LV VXJJHVWHG WKDW  FRQFHQWUDWHG IRUFHV
(Fig.  evenly allocated on the slabs surface show no
significant difference during the operation, as
compared to evenly distributed load, that is why
in further theoretical study of strength, crack re-
sistance, and deformation of studied slabs a de-
sign model was adopted for slabs supported by
 &$/&8/$7,212)'()/(&7,212)21(/$<(5$1'7:2/$<(56/$%6 

hinged bearings on four sides and influenced by $VSURSRVHGE\$ $ *YR]GHY>], it is pos-

evenly distributed load. The load was applied by VLEOHWRPDNHXVHRIFKDUDFWHULVWLFSRLQWVDQG
degrees Pi=2,0 N1ZLWK-minute timing at each on the diagram (Fig. 
stage to take readings from devices. The value of
load was fixed by indices of model forcemeter of
K\GUDXOLFSXPSLQJVWDWLRQ%HIRUHWHVWLQJWKHKy-
draulic system consisting of a pumping station,
jacks, and model forcemeter was calibrated using
model forcemeter of Tokarev system.
During the application of load in the center of
the slab deflections and deformations over sup-
ports were recorded using time indicators with
 mm scale graduation value.
The load was applied by two hydraulic jacks
RI  kN united by common oil circuit con-
nected to a common pump station.
%HIRUH WHVWLQJ WKH VODEV SK\VLFDO DQG Pe-
chanical properties of used materials were de-
termined: those of heavy concrete, steel fiber
concrete and reinforcement 7DEOH.

Table 2. Physical and mechanical properties of con-

1  cracking, 2  appearance of the plastic hinge
Fig. 3. Design diagram of slab deflection:
cretes and reinforcement

Strength, Deflection values in the areas between fcr and

Type of fu are determined by interpolation.
MPa Tensile Starting elas-
concrete or
strength, ticity factor, The general view of the formula f of slab de-
fitting
MPa MPa flection of slab supported on four sides and bear-
cube prism
ing a cracks, may be obtained from the follow-

qu  qcr f u  f cr
ing ratio:

q  qcr f  f cr
concrete     , 

Steel fiber
where:
concrete     qu and qɫr – load at destruction and crack for-
mation;
fu and fcr – slab deflection at the moment of de-
Rein- struction and crack formation correspondingly.
forcement – –   It should look as follows:
ȼɪ-
q  qcr
f cr  ( f u  f cr  .
qu  qcr
f 
$FFRUGLQJ to results of experimental studies
with regard to nature of destruction [], all
slabs collapsed according to normal sections. The value fcr is determined based on elastic
system calculation according to load qcr, which
in turn may be obtained by bending factor, when
&$/&8/$7,212)'()/(CTION first crack appeared in a slab area with the high-
est tension.
7KHSURSRVHGPHWKRGEDVHGRQWKHOLPLWHTXi- The deflections f at the moment of formation
librium method, which may be represented – of plastic hinges may be determined as follows.
GXULQJWKHVWDWHRIOLPLWHTXLOLEULXP– by the sys- Until the conditional yield point of reinforce-
tem of disks united along the lines of fracture ment achieved along all the lines of fracture,
with plastic hinges. cracks are formed and significantly increased on a
VODE$WWKHVDPHWLPHDUHDVZLWKFUDFNVZLOOEH
 '0<7526025.$/29

especially distorted that will mostly determine the 

maximum value of slab deflection. – factor derived from geometrical consid-
2 2
If insignificant curvature of slab areas ne-
erations, which is transition to the angle of rota-
glected bearing no cracks, but rigidity is high,
tion of the disk relative to the support,
the slab calculation model may be represented as
For fiber reinforced structures (such as fiber
stiff disks connected by yielding bracing with
FRQFUHWHWKHYDOXHfy and Es at the point of criti-
width ǻ %HQGLQJ ULJLGLW\ RI DOO MRLQWV LV FDOFu-
cal steel stress is replaced by the value fctf and
lated according to V.M. Murashov theory [7],
elastic modulus Ef suitable to the material. For
though the coefficient ȥ is taken as a one, as the
two-layer slabs the given geometric, strength,
reinforcements reaches instability the influence
deformation properties of two materials are
of stretched concrete between cracks disappears
used.
or becomes insignificant. This assumption sub-
%\WKHFRPELQDWLRQRI(T -(ZHFDQGe-
stantially simplifies the calculation.
fine a final ratio for the calculation of current de-
'
For further simplification the angle fracture
flection of the slab supported by four sides in-
EHWZHHQGLVFVHTXDOWR is assumed as if con- fluenced by evenly distributed load:

q  qcr 
r
f cr  ( '  f cr  ,
centrated along the lines of fracture. The calcu-
qu  qcr r
lated deflection surface prior to exhaustion of f 
the bearing capacity turns out to be similar to the
surface used in calculatLQJ E\ OLPLW HTXLOLEULXP where:
method for the calculation of works on possible fcr – slab deflection at the moment of of for-
movements. Though such likening is not accu- mation of the first cracks in stretched area of the
rate, it allows determining the maximum deflec- element;
tion at a rather decent level. qcr – load, when first cracks were created;
There is a calculation model of VTXDUH slab qu – load corresponding to the limit of the slab
presented on Fig.  and a diagram of angle rota- bearing capacity;
WLRQ DW VODE FHQWHU GHIOHFWLRQ WKDW LV HTXDO WR fu. Thus, the calculated width of deformed area
Owing to symmetry, the fracture diagram of ǻ remains unknown in the (T (7DVDUHVXOWLW
value ǻ for all plastic hinges is the same. is not possible to calculate the value f.
Rigid discs of the slab will turn in relation to In order to solve such problem the following

M
supports by angle: method of finding the desired value of deflection
2 fu f may be proposed. First of all, a boundary value
. 
2 l of the slab deflection is set according to the
standards for that class of structures fu=[ f ] ap-
Mutual angle adjacent discs ply it in the (T (7f=fu=[ f ]. The (T(7LVVHt-
WOHG ZLWK UHVSHFW WR YDOXH ǻ DQG WKHQ REWDLQHG
result is applied to the (T (WKHUHIRUHREWDLn-
M '
2 2 fu 
,  ing the deflection at the time of formation of
l r plastic hinges with regard to real-specified pa-

l'f ym
whereof: rameWHUV$WWKHVDPHWLPHWKHUHVXOWRI(T (7
shows the value of current deflection by intro-
2 2 Es (d  x
fu .  ducing the calculated value of width of de-
IRUPHGDUHDǻ
where:

– curvature-multiplication of which by ǻ
r
HTXDOVWRUHFLSURFDODQJOHRIURWDWLRQRIDdjacent
discs
 f ym
Es (d  x
, (
r
 &$/&8/$7,212)'()/(&7,212)21(/$<(5$1'7:2/$<(56/$%6 

7KHEDVLVRI/,5$-&$'is represented by the

calculation of components and structures by fi-
nite element method [].
The calculation model of a slab (Fig.  is
built out of tridimensional finite elements (type
ɋȿ-The slab is separated into finite el-
ements according to plan. The sectional area of
the slab is composed of  OD\HUV  PP HDFK.
The load is applied in the form of concentrat-
ed forces. The distribution of load and supports
is accepted as in Fig. 2.
a

b a
Fig. 4. Calculation diagram of VTXDUH slab influenced
by evenly distributed load:
a – fracture diagram; b – diagram of disc rotation an-
gle rate

7KHYDOXHRIIFUDQGTFUFDQEHFDOFXODWHGLI
the combine (T (7 ZLWK WKH IRUPXOD
%+*alerkina [2] IRU VTXDUH SODWHV supported b
on four sides, the deflection at the center of
slabs: Fig. 5. Calculation model RIVODELQ/,5$-&$':
ɚ± general view; b – side view

ql  The ɋȿ-finite element is a universal tridi-

f 0, . 
Ei h  mensional eight-node isoparametric finite ele-
ment, designed for the calculation of tridimen-
To determine the width of the deformed zone sional constructions. There is a diagram repre-
ǻVODEVDVHULHVRILVRODWHGSRLQWVRQVHYHUDOUe- sented in Fig  (DFK of finite element nodes
search graphs deflections. Points are usually pre- has three degrees of variance U, V, W defined
VFULEHGLQWKHRSHUDWLRQDOZRUNLQJORDGUDQJH with regard to global coordinates X, Y, Z and are
often – is 0,7...0,8 from destructive load. To find linear displacements according to axis, whose
WKHYDOXHǻXVLQJ(TDQGWDNLQJLQWRDc- positive direction coincides with the direction of
count the specific slabs construction. coordinate axis. $V a result of modeling, there
Results of calculation of one- and two-layer are ,nodes and ,elements
VODEVE\WKHOLPLWHTXLOLEUium method are shown
in Fig. 7, 8.
Considering complexity of mathematical cal-
culations for slab deflection due to analytical
methods [] and unsatisfactory precision of re-
sults, a decision was made to do calculations on
FRPSXWLQJ PDFKLQH ZLWK D KHOS RI /,5$-&$'
bundled software [8].
 '0<7526025.$/29

Fig. 6. Diagram of ɋȿ-finite element

The assumed calculation model makes it pos- ɚ

sible to change the rigidity of materials for both
one-layer and two-layer slabs.
The slab calculation was performed in linear
position for loads corresponding to loading pitch-
es at testing. Non-linear physical-mechanical
properties were taken into consideration by
means of changing the elasticity modulus.
Initial values of elasticity modulus for con-
crete and steel fiber concrete were assumed ac-
cording to [], that is Ec = 22,5·10 MPa,
Ecf = ,8·10 MPa.
In accordance with recommendations [] the
nonlinear behavior of construction is recom-
mended to consider by means of introducing de-
creasing coefficients: 0,2 – in case of any cracks,
or 0,– in case no cracks revealed.
That is why during the calculations the de-
creasing coefficient was 0,before appearance of
any cracks and 0,2 – after the first cracks ap- b
peared. Reduction of slab rigidity as a result of
crack was also considered by means of introduc- Fig. 7. Calculation results for deflection of one-layer
slabs of series ȱ grade ɉɎ ɚ and slabs of series ȱȱ
grade ɉɁb 1  experimental; 2 – calculation by the
ing zero rigidity elements in the tensile area with-
in height of the crack. The rigidity of slab sup-
ports was not reduced to avoid forcing through OLPLW HTXLOLEULXP PHthod; 3 – FDOFXODWHG LQ /,5$
bundled software
the slab thickness
$V a result of calculation of one-layer slabs in
When calculating the two-layer slabs, relevant
/,5$ bundled software, diagrams of deflection
elasticity modulus of material was applied for
of the slab center f of the total pressure Ptot=Pi
each slab layer. The value of decreasing coeffi-
were obtained, which are presented in Fig. 7.
cients and zero elasticity elements were applied
similar to the calculation of one-layer slabs.
$V a result of calculation of two-layer slabs by
means of /,5$ bundled software, diagrams of
deflection of the slab center f of the total pressure
Ptot=Pi were obtained, which are presented in
Fig. 8.
 &$/&8/$7,212)'()/(&7,212)21(/$<(5$1'7:2/$<(56/$%6 

ational overcuts at the expense of increased

strength of materials. The new technologies
make it possible to perform the most complex
design elements made of any materials. Consid-
ering the current trends, it should be noted that
the issue of performance reliability for buildings
DQG VWUXFWXUHV LV QRW DERXW WKH VWUHQJWK UHTXLUe-
ments, but the rigidity of elements and buildings
in general. Therefore, the study of rigidity (most
commonly the deflections RI H[DPLQHG VODEV
seems to be a topical issue.
 $QRWKHU TXHVWLRQ WKLV ZRUN EHDUV DQ Dn-
swer to is the feasibility of using multi-layer
slabs. From the point of view of rigidity (bend-
LQJ RI VODEV WKH WZR-layer slabs have an ad-
YDQWDJH WKH\ VKRZ  OHss deflections, than
ɚ the one-layer ones. Considering the higher level
of crack resistance of tow-layer slabs, the use of
two-layer slabs is absolutely justified.
 &DOFXODWLRQ E\ D FRPSXWHU XVLQJ /,5$
bundled software provides wide opportunities to
deterPLQH ULJLGLW\ GHIOHFWLRQ IRU ERWK VLQJOH-
layer and multi-layer slabs owing to computing
WHFKQRORJLHV+RZHYHUWKH obtained calculation
results indicate a deficiency in accuracy, com-
pared to the experimental data. The fact is that
compliance with regulatory guidelines, imple-
mented to consider nonlinear structure operation
by introduction of decreasing coefficients, some-
times does not match the actual conditions of
slab deformation. Furthermore, there are some
doubts as to the correctness of defining the depth
b crack distribution along the slab height, which
determines its actual rigidity.
Fig. 8. Calculation results for deflection of two-layer . The comparative analysis of experimental
slabs of series ȱȱȱgrade ɉȻɎɚand slabs of series ȱV
grade ɉɁɎ b 1  experimental; 2 ± calculation by
and theoretical diagrams of slab deflection evi-
dences good matching of results. Deviations at
WKH OLPLWHTXLOLEULXP PHWKRG 3 – FDOFXODWHGLQ/,5$
maximum loads were as follows: for one-layer
bundled software
slabs of VHULHVȱ (ɉɎ – for slabs of VHULHVȱ,
(ɉɁ –  for two-OD\HU VODEV RI VHULHV ȱ,,
CONCLUSIONS (ɉȻɎ – for slabs RIVHULHVȱ9ɉɁɎ – 
Such deviation in calculations may be explained
*LYHQWKDWWRGD\¶VFRQVWUXFWLRQLQGXstry is by complexity of defining the real rigidity of
slabs, i.e. the definition of deformation modulus
represented by the vigorous process of increas-
ing the strength of construction materials, par- and the height of crack creation to set the zero ri-
ticularly concrete and reinforcement, due to gidity elements.
achievements in chemistry, there is a strengthen-
LQJ RI TXDOLW\ LQGLFDWors of buildings and con-
structions, in particular bearing construction ar- 5()(5(1&(6
rangements. Thus, systems making it possible to
work in a multi-axial load state (shell structures,  Barashykov $<D, Smorkalov D.V., 2014.
Calculation of solidity of one-layer and two-
slabs, wall-EHDPV HWF DUH JHWWLQJ PRUH SRSu- layer reinforced concrete slabs supported on four
lar. There is an opportunity to significantly re- sides. Resource efficient materials, structures,
duce the cost by reduction of cross-section oper-
 WLRQ RI QHZ JHQHUDWLRQ FRQFUHWHV 7(.$
0RW(QHUJ5ROQ ;%  


 '0<7526025.$/29
, Issue 9, - LQ8NUDLQH
buildings, and constructions: Collected studies.  Snyder M.
ɊȺɋɑȿɌɉɊɈȽɂȻɈȼ
1972. Factors affecting the flexural
Rivne, , Issue 28, - LQ8NUDLQH. ɈȾɇɈofɂȾȼɍɏɋɅɈɃɇɕɏɉɅɂɌ
strength steel fibrous concrete. $&, -RXUQDO,
2. Galerkin B.G., 1933. The resilient thin plate. M, ɈɉȿɊɌɕɏɉɈɄɈɇɌɍɊɍ
, 9RO, No. 2, -
*RVVWURLL]GDW (in RusVLDQ  SP 52-103-2007, 2007. Code of practice on de-
 Gorodetsky A.S., Schmukler V.S., Bondarev Ⱥɧɧɨɬɚɰɢɹ
sign and ɉɪɢɜɟɞɟɧɵ
construction. ɦɟɬɨɞɢɤɚ
Reinforcedɢconcrete
ɪɟɡɭɥɶɬɚɬɵ
cast-
A.V., 2003. Information Technology of calcula- ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ
in-place constructions ɢɫɫɥɟɞɨɜɚɧɢɣ
of buildings. ɩɪɨɝɢɛɨɜ
Moscow,
tion and design of builGLQJ NRQVWUXNWVɿ\ 3URF ɨɞɧɨɫɥɨɣɧɵɯ ɢ ɞɜɭɯɫɥɨɣɧɵɯ ɩɥɢɬ ɩɨɞ ɞɟɣɫɬɜɢɟɦ
2007 (in RusVLDQ
Manual. Kharkiv 178 .3,   (in ɩɨɩɟɪɟɱɧɨɝɨ
 Stages A., 1981.ɤɪɚɬɤɨɜɪɟɦɟɧɧɨɝɨ
Ring fiber reinforcedɧɚɝɪɭɡɤɢ
concrete.
5XVVLDQ Ɋɚɫɫɱɢɬɚɧɵ ɩɪɨɝɢɛɵ ɩɥɢɬ ɫɩɨɫɨɛɨɦ
$&,-RXUQDO9RO ɨɫɧɨɜɚɧɧɨɦ
1R -
 Gvozdev A.A., 1949. Calculation of bearing ca- ɧɚ
ɦɟɬɨɞɟ ɩɪɟɞɟɥɶɧɨɝɨ
ACI Journal. ɪɚɜɧɨɜɟɫɢɹ ɢ
1973. State-of-the-art ɦɟɬɨɞɨɦ
report on fi-
pacity E\ OLPLW HTXLOLEULXP PHWKRG Moscow, ɤɨɧɟɱɧɵɯ ɷɥɟɦɟɧɬɨɜ ɫ ɩɨɦɨɳɶɸ ɩɪɨɝɪɚɦɧɨɝɨ
ber reinforced concrete, , Vol. 70, 729-
6WUR\L]GDW9, 280 (in RusVLDQ ɤɨɦɩɥɟɤɫɚ ɅɂɊȺ
 Tarasov V.P., 1974. ȼɵɩɨɥɧɟɧɵ ɫɪɚɜɧɟɧɢɹ
The use of fiber-reinforced
 =KXUDYVN\L 2.D., Smorkalov D.V., 2002. ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɢ ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɪɟɡɭɥɶɬɚɬɨɜ
concrete in construction. Industrial construction.
Methodology and results of experimental studies ɪɚɫɱɟɬɚɩɪɨɝɢɛɨɜɩɥɢɬ
, Vol. 7, - LQ5XVVLDQ
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ ɩɥɢɬɚ ɩɪɨɝɢɛ ɫɬɚɥɟɮɢɛɪɨ
of two-layer slabs. Steel reinforcement concrete  Tattersall G.H., 1974. %RQG VWUHQJWK LQ VWHHO-
constructions. Research, design, construction, ɛɟɬɨɧ
fibre-reinforced concrete. Journal of Concrete
operation: Collected research papers, Issue , Research.  Vol No. 87, -
Kryvyi Rig, KTU, 2002 LQ8NUDLQH
 Korolyov A.N. Calculation methods for slab de-
flections supported on four sides at short-term
load. Concrete and reinforcedCollected
concrete studies.
Vol. , ɊȺɋɑȿɌɉɊɈȽɂȻɈȼ

, 9- (in RusVLDQ   LQ8NUDLQH. ɈȾɇɈ- ɂȾȼɍɏɋɅɈɃɇɕɏɉɅɂɌ
ate. M, ɈɉȿɊɌɕɏɉɈɄɈɇɌɍɊɍ
7. Murashev V.I., 1950. Crack resistance, stiffness
*RVVWURLL]GDW VLDQ
and strength of concrete. M, Mashstroyizdat,
 Ⱥɧɧɨɬɚɰɢɹ. ɉɪɢɜɟɞɟɧɵ ɦɟɬɨɞɢɤɚ ɢ ɪɟɡɭɥɶɬɚɬɵ
 (in RusVLDQ ndarev
Information Technology of calcula- ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɣ ɩɪɨɝɢɛɨɜ
8. LIRA-CAD 2011. 7XWRULDO(OHFWURQLF(Gition ,
ɨɞɧɨɫɥɨɣɧɵɯ ɢ ɞɜɭɯɫɥɨɣɧɵɯ ɩɥɢɬ ɩɨɞ ɞɟɣɫɬɜɢɟɦ
LQ5XVVLDQ GLQJ NRQVWUXNWVɿ\ 3URF
 178 .3,   (in ɩɨɩɟɪɟɱɧɨɝɨ ɤɪɚɬɤɨɜɪɟɦɟɧɧɨɝɨ ɧɚɝɪɭɡɤɢ
9. Rudenko N., 2010. The development of concep- Ɋɚɫɫɱɢɬɚɧɵ ɩɪɨɝɢɛɵ ɩɥɢɬ ɫɩɨɫɨɛɨɦ ɨɫɧɨɜɚɧɧɨɦ
5XVVLDQ
WLRQ RI QHZ JHQHUDWLRQ FRQFUHWHV 7(.$ Kom. ɧɚ ɦɟɬɨɞɟ ɩɪɟɞɟɥɶɧɨɝɨ ɪɚɜɧɨɜɟɫɢɹ, ɢ ɦɟɬɨɞɨɦ
 0RW(QHUJ5ROQ, Calculation of bearing ca-
Vol. ;%, -.
E\ OLPLW HTXLOLEULXP PHWKRG Moscow, ɤɨɧɟɱɧɵɯ ɷɥɟɦɟɧɬɨɜ ɫ ɩɨɦɨɳɶɸ ɩɪɨɝɪɚɦɧɨɝɨ
 Smorkalov D.V., 2003. Crack resistance and ɤɨɦɩɥɟɤɫɚ ɅɂɊȺ ȼɵɩɨɥɧɟɧɵ ɫɪɚɜɧɟɧɢɹ
6WUR\L]GDW VLDQ
fracture modes for two-layer slabs. Resource ef- ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɢ ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɪɟɡɭɥɶɬɚɬɨɜ
 =KXUDYVN\L 2
ficient materials, structures, buildings, , 2002.
and ɪɚɫɱɟɬɚɩɪɨɝɢɛɨɜɩɥɢɬ.
constructions: Collected research papers. studies
Rivne, Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ ɩɥɢɬɚ ɩɪɨɝɢɛ ɫɬɚɥɟɮɢɛɪɨ-
, Issue 9, - LQ8NUDLQH concrete
construction, ɛɟɬɨɧ
 Snyder M. 1972. Factors affecting the flexural
Issue ,
strength of steel fibrous concrete. $&, -RXUQDO,
LQ8NUDLQH
 9RO  



  VLDQ
VLDQ

$&,-RXUQDO9RO 1R  
 VLDQ

7XWRULDO(OHFWURQLF(G
 
LQ5XVVLDQ

WLRQ RI QHZ JHQHUDWLRQ FRQFUHWHV 7(.$
   LQ5XVVLDQ
0RW(QHUJ5ROQ ;%  
 %RQG VWUHQJWK LQ VWHHO

   

   LQ8NUDLQH


$&, -RXUQDO
 9RO  


VLDQ

$&,-RXUQDO9RO 1R