Estimation of the cracking probability in road structures by modeling of external influences/Plysiu atsiradimo keliu dangu konstrukcijose tikimybes vertinimas modeliuojant isorinius poveikius/Plaisu veidosanas varbutibas novertejums cela konstrukcijas modelejot arejas ietekmes/Teekonstruktsioonide pragunemistoenaosuse hindamine valiste mojude modelleerimisega.

Leonovich, Ivan ; Melnikova, Irina ; Puodziukas, Virgaudas 等

1. Introduction

The network of public roads in Belarus exceeds 418 km by 10 000 [km.sup.2] of the territory. Roads with asphalt concrete pavements are the predominant. Reconstruction and works to improve transport characteristics of the main roads linking the major cities of the country with its capital--Minsk--are carried out nowadays. Furthermore, as the Republic of Belarus is a transit country, road service is working on increasing the road capacity up to 13 t/axle.

Durability of the road structure is one of the most important parameters of technical and operating conditions of a road. It depends on an accepted base layer construction, the used materials and their aging, degradation of the road pavement, the external traffic load, capacity of traffic, hydrogeological factors, solar radiation and other climatic factors, geometrical parameters of a road, etc. Considering the non-rigid pavement, it should be noted that the influence of temperature is very significant too. Indeed, with the considerable temperature fluctuations changes occur within the physical and mechanical properties of asphalt and other road building materials with an organic binder (Teltaev 2007). It may cause the appearance of damages in a pavement.

The most common type of asphalt pavement damage is cracks of different nature, size and location. Researchers are constantly offering new constructive measures to prevent the formation of cracks and repair of already formed temperature, reflection or technological cracks (Blazejowski, Styk 2004). However, theoretical studies using modern calculation methods make it possible to accurately determine the genesis of cracks in the pavement in order to make the right decisions in the design and repaire of road structures that enhance their durability.

2. The choice of design schemes to estimate crack resistance of road structures under the influence of temperature and traffic load

Cracks usually appear under tensile or bending stresses in pavement layers under the action of traffic loads and temperature fluctuations, and especially in their combined action. Thermal cracks are initiated at the top of an asphalt layer and grow from the top to bottom, as the crack in the bend zone of the pavement under the wheel load. Reflected cracks grow up from the bottom: from the crack of and old lower asphalt concrete layer or a joint of cement concrete slab (Vasiliyev 2004). It is assumed that the cracks are formed in asphalt concrete layer when tensile stresses exceed the tensile strength of asphalt concrete.

For theoretical studies of the road structure mode of deformation the finite element method (FEM) should be applied as a calculation method (Elsefi 2003). This paper presents a model of pavement made in the analytical design system SolidWorks.

The geometric model of the considered structure has 5 layers: 2 layers of pavement (a dense asphalt concrete layer 2.5 cm thick and a porous asphalt concrete layer 7-10 cm thick) and 3 base layers (gravel--20-50 cm thick, sand--30-60 cm thick, soil base--not less than 80 cm thick).

The database of road building materials properties allows defining the values of the next physical and mechanical parameters: modulus of elasticity, mass density, Poisson's ratio, thermal conductivity, specific heat capacity, coefficient of thermal expansion, tensile strength, compressive strength. Moreover, values of the modulus of elasticity, density, tensile strength and compressive strength of asphalt concrete, sand and gravel depending on the temperature are also taken into account when setting the properties of the materials to calculate the mode of deformation of the structure (Melnikova 2012).

Geometry of the three-dimensional pavement model is as follows: every layer is a box 900 x 900 mm to avoid the influence of the edge effect. The thickness of the structural layers may vary.

Traffic impact on the road surface has been modeled as a wheel load from a heavy truck KAMAZ-65117 which has the load of 115 kN/axle. This load has been modeled as a pressure of 0.43 MPa to the rectangular area 28x23.8 mm.

The initial conditions for air temperature effect estimation are as follows: geographical location--Minsk (53.89[degrees] latitude), season--winter, January, air temperatures were taken according to data of the Republican Hydro Meteorological Center for Minsk. Surface temperatures were taken in accordance with the obtained mathematical relation between air and surface temperatures. The formula was obtained by statistical analysis of measurement data from the road measurement stations provided by Belarusian Road Engineering and Technology Center (Leonovich, Melnikova 2012).

Several analytical models of pavements were considered to predict the mode of deformation using FEM (Zholobov 2000). Design models reflected the work of a pavement before cracking, after temperature or reflective cracking, as well as before/after the repair activities of different kinds. Furthermore, two base types were taken into consideration: solid (discrete) which does not result in pavement deformation and cracked slab causing additional horizontal deformation of the pavement due to an adhesion with the base (old cracked asphalt concrete layer, concrete slabs) under cyclic deformation.

Design models for estimation of pavement crack resistance before cracking as well as the connection between pavement layers and adjacent sections (hinged movable support) between asphalt layers and lower construction layers (hinged-fixed support) are shown in Figs 1-3.

Fig. 1 shows the asphalt surface layer without cracks on solid base at the beginning of road service period. Fig. 2 presents a design model when the top asphalt layer is laid directly on the cracked asphalt basis or cement slab with joints (crack-interrupting layer is absent). Model in Fig. 3 takes into account crack-interrupting layer arranged in the lower area of the upper asphalt concrete layer over the existing cracks in asphalt base or joint in concrete slab. In Figs 1-3 L--length of the considered pavement fragment and [delta]--joint width in concrete slabs or width of the existing crack in asphalt base layer.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

In all 3 cases (Figs 1-3) it is assumed that the length of the considered pavement fragment remains the same. The base of the pavement for modeling consists of 3 layers:

--fractionated gravel layer;

--medium size sand layer;

--clay loam layer as a pavement basis.

Design models for estimation of pavement crack resistance after cracks appearance are shown in Figs 4 and 5. Fig. 4 presents the process of thermal cracking in the upper zone of asphalt layer due to the appearance of max tensile stress in the zone as a result of temperature and traffic load. Fig. 5 shows the development of reflected cracks at some distance from the existing cracks in the lower asphalt layer or joints in concrete slabs.

Different pavement repair technologies are taken into consideration in design models from Figs 6-11 (Verenko 2008). Fig. 6 presents small cracks (0.5-0.7 cm width) repair method: filling a crack with a sealant (emulsified asphalt, liquid bitumen) followed by crack powdering with friction material without making a protective asphalt concrete layer. Pavement after its milling (width A = 10-20 mm, 10-40 mm depth) and sealing the crack is shown in Fig. 7; width-todepth ratio is taken 1:1 if crack width is up to 25 mm and width-to-depth ratio is taken 1:2 if crack width is more than 25 mm. Figs 8 and 9 correspond to asphalt pavement on a cracked basis crack sealing without/with milling. Crack sealing in asphalt concrete pavement on a cracked base when applying a wearing layer is presented in Fig. 10, the same thing with an additional application of crack-interrupting layer of geosynthetic material 10-50 cm width - in Fig. 11 (Gorszczyk 2004).

Formation of thermal and reflected cracks may also take place after the repair. These design models are presented in Figs 12 and 13. The formation of cracks in the upper zone of the asphalt layer over the crack-interrupting layer is shown in Fig. 12. The crack formation in a sealant material is shown in Fig. 13.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

Further researches were related to learning of the pavement models mode of deformation to reveal shortcomings in the road pavement design (selection of materials, their properties, later thickness, etc.), structure itself and choice of repair activities.

3. Modeling of the temperature and traffic load impact on the road structure

For the modeling of the temperature and traffic load impact some physical and mechanical properties of the materials were set according to the mean values, but properties of the upper layer's asphalt concrete were defined experimentally in order to obtain the most reliable modeling results. Modulus of elasticity and tensile strength of asphalt concrete were defined after the testing of beams (4x4x12 cm) on elastic supports at three different temperatures (-20[degrees]C, 0[degrees]C, +20[degrees]C) using the dynamic load press (central point load). Thermal conductivity was defined by laboratory tests on the HFM 436/3/1E Lambda[TM] device.

Pavement structures for simulation were chosen according to the requirements of the normative documents of the Republic of Belarus. Design models are corresponding to the schemes presented above in Figs 1-13 and detailed information is given in Table 1.

Thermal and traffic loads:

a--wheel load was modeled as a quarter of a tire print and a pressure of 0.43 MPa (Fig. 14);

b--thermal load - (-20)[degrees]C;

c--the simultaneous impact of temperature and transport--(-20)[degrees]C and the wheel load;

d--3 of the coldest days of the year - Minsk (from 6 pm of the 22th of January to 6 pm of the 24th of January 2011);

e--3 of the warmest days of the year - Minsk (from 6 pm of the 23th of August to 6 pm of the 25th of August 2011).

[FIGURE 14 OMITTED]

The modeling results are represented below. Calculations of compressive stress, tensile stress and deflection were carried out using finite element method.

Research results of the mode of deformation for Figs 1-3 are presented in Table 2 (layer 1 is dense asphalt concrete, layer 2--porous asphalt concrete). Stresses in asphalt concrete upper layer under the thermal load for 3 coldest and warmest days are presented in Figs 15 and 16. Fig. 15 presents stresses during the days with max negative temperatures, Fig. 16--with max positive temperatures.

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

The calculation results for the scheme in Fig. 5 are identical to the results of the scheme in Fig. 4 (Dave 2007). Research results of the mode of deformation are presented in Table 3 (layer 1 is dense asphalt concrete, layer 2--porous asphalt concrete). Stresses in asphalt concrete upper layer (dense asphalt concrete) under the thermal load for 3 coldest and warmest days are presented in Fig. Figs 17 and 18. Fig. 17 presents stresses during the days with max negative temperatures, Fig. 18--with max positive temperatures.

The calculation results for the scheme in Fig. 8 are identical to the results of the scheme in Fig. 6. Research results of the mode of deformation for schemes in Figs 6-9 are presented in Table 4 (layer 1 is dense asphalt concrete, layer 2--porous asphalt concrete, layer 3--sealant).

Research results of the mode of deformation for Figs 10 and 11 are presented in Table 5 (layer 1 is dense asphalt concrete, layer 2--porous asphalt concrete, layer 3--sealant, layer 4--wearing layer, layer 5--geotextile layer). Stresses in asphalt concrete upper layer under the thermal load for three coldest and warmest days are presented in Figs 19 and 20. Fig. 19 presents stresses during the days with max negative temperatures, Fig. 20--with max positive temperatures.

Research results of the mode of deformation for Figs 12 and 13 are presented in Table 6 (layer 1 is dense asphalt concrete, layer 2--porous asphalt concrete, layer 3--sealant, layer 4--wearing layer, layer 5--geotextile layer). Stresses in asphalt concrete upper layer under the thermal load for three coldest and warmest days are presented in Figs 21 and 22. Fig. 21 presents stresses during the days with max negative temperatures, Fig. 22--with max positive temperatures.

Key simulation findings are presented below. These are recommendations on how to improve crack resistance of asphalt concrete pavements.

1) Road structures with the thickness of asphalt pavement of at least 10-12 cm are less exposed to cracking in the climatic conditions of the Republic of Belarus. Pavements with less than 10 cm thickness are not enough resistant to thermal cracking.

2) Modulus of elasticity of the upper layer material (asphalt concrete) should be small at low temperatures below zero.

[FIGURE 17 OMITTED]

[FIGURE 18 OMITTED]

3) Material for a membrane type crack-interrupting layer should have the smallest modulus of elasticity as it is inexpedient to apply these layer in case of a close modulus of elasticity values of pavement materials and membrane type layer itself.

4) The use of a geosynthetic material as a crack-interrupting layer and as repair material allows to reduce the resulting tensile stresses in the top layer of a pavement, but only if it was laid in the bottom zone of the pavement. It is allowed to make a reinforcement of the top pavement zone with geosynthetics with an additional apply of surface treatment or wearing layer 2-3 cm thick.

5) The most effective measures of crack repair are: milling and sealing the cracks with a wearing layer construction; sealing the cracks with a crack-interrupting geosynthetic layer construction over the crack with a wearing layer (dense asphalt concrete).

Further research will focus on a more detailed study of road structures using FEM. It will allow to substantiate the use of materials, choice of layers thicknesses, etc., to increase crack resistance of asphalt concrete pavements.

[FIGURE 19 OMITTED]

[FIGURE 20 OMITTED]

[FIGURE 21 OMITTED]

[FIGURE 22 OMITTED]

4. Conclusions

1. If design decisions are made on the basis of the theoretical research base--crack resistance of road structures will be increased. It is possible today because a large amount of calculations of temperature and traffic impact on a pavement is done using FEM.

2. Recommendations to improve crack resistance of asphalt concrete pavements in the load conditions of the Republic of Belarus were made as a result of the modeling process. First of all there are recommendations on the choice of pavement layer thickness, physical and mechanical properties of construction materials, material type to make a crack-interrupting layer, on the choice of a repair activity to repair the cracks in the top pavement layer effectively. These recommendations should also be considered when designing flexible pavements, its maintenance and planning of the current and capital repairs of the Republican roads.

Caption: Fig. 1. Design model for the calculation of stresses in asphalt concrete pavement before cracking on a solid basis.

Caption: Fig. 2. Design models for the calculation of stresses in asphalt concrete pavement before cracking on a cracked basis.

Caption: Fig. 3. Design models for the calculation of stresses in asphalt concrete pavement before cracking on a cracked basis with a crack-interrupting layer.

Caption: Fig. 4. Design model for the calculation of stresses in asphalt concrete pavement after crack appearance with thermal cracks in the upper zone of the asphalt pavement on a solid base.

Caption: Fig. 5. Design model for the calculation of stresses in asphalt concrete pavement after crack appearance with reflected cracks in asphalt layer on a cracked base.

Caption: Fig. 6. Design model for the calculation of stresses in asphalt concrete pavement after repairing activities after sealing cracks of a small width in the top layer of asphalt concrete pavement on a solid base.

Caption: Fig. 7. Design model for the calculation of stresses in asphalt concrete pavement after repairing activities after milling and sealing a crack in asphalt concrete layer on a solid basis.

Caption: Fig. 8. Design model for the calculation of stresses in asphalt concrete pavement after repairing activities after sealing cracks in asphalt concrete layer on a cracked basis.

Caption: Fig. 9. Design model for the calculation of stresses in asphalt concrete pavement after repairing activities after milling and sealing cracks in asphalt concrete layer on a cracked basis.

Caption: Fig. 10. Design model for the calculation of stresses in asphalt concrete pavement after repairing activities after milling, sealing cracks and applying of a wearing layer.

Caption: Fig. 11. Design model for the calculation of stresses in asphalt concrete pavement after repairing activities after milling, sealing cracks, applying of a crack-interrupting layer and a wearing layer.

Caption: Fig. 12. Design model for the calculation of stresses in asphalt concrete pavement after the repair and re-crack formation after applying of a crack-interrupting layer and a wearing layer.

Caption: Fig. 13. Design model for the calculation of stresses in asphalt concrete pavement after the repair and re-crack formation after crack formation in a sealant material.

Caption: Fig. 14. Pavement structure model adopted for the calculations.

Caption: Fig. 15. Mode of deformation for the upper asphalt concrete layer - stresses during the days with max negative temperatures (schemes in Figs 1-4).

Caption: Fig. 16. Mode of deformation for the upper asphalt concrete layer - stresses during the days with max positive temperatures (schemes in Figs 1-4).

Caption: Fig. 17. Mode of deformation for the upper asphalt concrete layer--stresses during the days with max negative temperatures (scheme in Fig. 4).

Caption: Fig. 18. Mode of deformation for the upper asphalt concrete layer--stresses during the days max positive temperatures (scheme in Fig. 4).

Caption: Fig. 19. Mode of deformation for the upper asphalt concrete layer--stresses during the days with max negative temperatures (schemes in Figs 6, 7, 9-11).

Caption: Fig. 20. Mode of deformation for the upper asphalt concrete layer - stresses during the days with max positive temperatures (schemes in Figs 6, 7, 9-11).

Caption: Fig. 21. Mode of deformation for the upper asphalt concrete layer presents stresses during the days with max negative temperatures (schemes in Figs 12 and 13).

Caption: Fig. 22. Mode of deformation for the upper asphalt concrete layer presents stresses during the days with max positive temperatures (schemes in Figs 12 and 13).

doi:10.3846/bjrbe.2013.31

References

Blazejowski, K.; Styk, S. 2004. Technologia Warstw Asfaltowych. Wydawnictwa Komunikacji i Lacznosci. 408 p. ISBN 9788320615401.

Dave, E. 2007. Reflective and Thermal Cracking Modeling of Asphalt Concrete Overlays, in Proc. of the International Conference of Advanced Characterization of Pavement and Soil Engineering Materials: selected papers, vol. 1. Ed. by Loizos, A.; Scarpas, T.; Al-Qadi, I. June 20-22, 2007, Athens, Greece. Athens: Taylor & Francis, 1241-1252.

Elsefi, M. A. 2003. Performance Quantification of Interlayer Systems in Flexible Pavements Using Finite Element Analysis, Instrument Response, and Non Destructive Testing. Virginia: Virginia Polytechnic Institute and State University, 429 p.

Gorszczyk, J. 2004. Modeling of Asphalt Pavement Structure with Geosynthetics Interlayer Using of Finite Element Method, in Proc. of the 10th International Conference "Durable and Safe Road Pavements": selected papers. May 11-12, 2004, Kielce, Poland, p. 271-278.

Leonovich, I.; Melnikova, I. 2012. Innovacii v sisteme ekspluatatsii avtomobil'nykh dorog, napravlennye na preduprezhdenie i likvidatsiiu poverkhnostnykh defektov, Trudy BGTU 2(149): 130-133.

Melnikova, I. 2012. Modelirovanie vozdejstviia temperatury i transportnykh nagruzok na vozniknovenie i razvitie treshchin v asfal'tobetonnykh dorozhnykh pokrytiiakh, Nauka i Tehnika 4: 44-52.

Vasiliyev, A. 2004. Spravochnaia enciklopediia dorozhnika. Moskva: Informavtodor. 507 p.

Verenko, V. 2008. Deformatsii i razrusheniia dorozhnykh pokrytij: prichiny i puti ustraneniia. Minsk: Belaruskaja encyklapedyja imia Brouki. 304 p.

Teltaev, B. 2007. Prognoz temperaturnogo rezhima dorozhnoj konstruktsii metodom konechnykh elementov, Nauka i Tekhnika v Dorozhnoj Otrasli 2:18-21.

Zholobov, A. 2000. Prognozirovaniepovedeniia tekhnologicheskikh sistem na stadii ikh proektirovaniia. Mogilev: MGTU. 150 p.

Ivan Leonovich (1), Irina Melnikova (2) [mail], Virgaudas Puodziukas (3)

(1,2) Dept of Road Construction and Road Management, Belarusian National Technical University, Pr. Niezavisimosti 65, 220013 Minsk, Belarus (3) Dept of Roads, Vilnius Gediminas Technical University, Sauletekio al. 11, 10223 Vilnius, Lithuania

E-mails: (1) [email protected]; (2) [email protected]; [email protected]

Received 21 December 2012; accepted 10 September 2013

Table 1. Road structure of modeled schemes (schemes in Figs 1-13)

Layer                                 Layer thickness, cm

                                             Figs

                                   1    2     3    4    5
Wearing layer
  (dense asphalt concrete)
Wearing layer (dense a/c) with
  a crack of 1 cm width
  and 1.6 cm depth
Dense asphalt concrete             4    4     4
Dense a/c with a crack of          --   --   --    4    --
  1 cm width and 1.5 cm depth
Dense a/c with a crack of          --   --   --    --   4
  1 cm width and 4 cm depth,
  2 cm away from the base crack
Dense a/c with a crack of          --   --   --    --   --
  0.5 cm width and 3 cm depth,
  filled with sealant
  (liquid bitumen)
Dense a/c with a crack
  of 2 cm width and 4 cm
depth after milling, filled        --   --   --    --   --
  with sealant (liquid bitumen)
Geotextile Dornit                  --   --   0.4   --   --
Porous asphalt concrete            8    --   --    8    --
Porous a/c with a                  --   8     8    --   8
  crack of 1 cm width
Gravel                             35   35   35    35   35
Sand                               40   40   40    40   40
Silty clay loam (base layer)       80   80   80    80   80

Layer                               Layer thickness, cm

                                          Figs

                                   6    7    8    9
Wearing layer
  (dense asphalt concrete)
Wearing layer (dense a/c) with
  a crack of 1 cm width
  and 1.6 cm depth
Dense asphalt concrete
Dense a/c with a crack of          --   --   --   --
  1 cm width and 1.5 cm depth
Dense a/c with a crack of          --   --   --   --
  1 cm width and 4 cm depth,
  2 cm away from the base crack
Dense a/c with a crack of          4    --   4    --
  0.5 cm width and 3 cm depth,
  filled with sealant
  (liquid bitumen)
Dense a/c with a crack
  of 2 cm width and 4 cm
depth after milling, filled        --   4    --   4
  with sealant (liquid bitumen)
Geotextile Dornit                  --   --   --   --
Porous asphalt concrete            8    8    --   --
Porous a/c with a                  --   --   8    8
  crack of 1 cm width
Gravel                             35   35   35   35
Sand                               40   40   40   40
Silty clay loam (base layer)       80   80   80   80

Layer                               Layer thickness, cm

                                           Figs

                                   10   11    12    13
Wearing layer                      2     2    --    --
  (dense asphalt concrete)
Wearing layer (dense a/c) with                 2    --
  a crack of 1 cm width
  and 1.6 cm depth
Dense asphalt concrete
Dense a/c with a crack of          --   --    --    --
  1 cm width and 1.5 cm depth
Dense a/c with a crack of          --   --    --    --
  1 cm width and 4 cm depth,
  2 cm away from the base crack
Dense a/c with a crack of          --   --    --    4
  0.5 cm width and 3 cm depth,
  filled with sealant
  (liquid bitumen)
Dense a/c with a crack
  of 2 cm width and 4 cm
depth after milling, filled        4     4     4    --
  with sealant (liquid bitumen)
Geotextile Dornit                  --   0.4   0.4   --
Porous asphalt concrete            --   --    --    --
Porous a/c with a                  8     8     8    8
  crack of 1 cm width
Gravel                             35   35    35    35
Sand                               40   40    40    40
Silty clay loam (base layer)       80   80    80    80

Table 2. Results of the mode of deformation research (Figs 1-3)

Model number:    Layer        Max         Max     Deflection,
load             number   compressive   tensile
                            stress,     stress,

                              MPa                     mm

Fig. 1: load a     1        0.4919      0.0569      0.1985
                   2        0.2296      0.0776
Fig. 1: load b     1        0.8327      2.3499        --
                   2        0.8327      0.0604
Fig. 1: load c     1        0.8361      2.3146      0.2226
                   2        0.8361      0.0614
Fig. 2: load a     1        0.4578      0.0382      0.2349
                   2        0.2316      0.0859
Fig. 2: load b     1        0.8876      3.0844        --
                   2        0.1099      0.0592
Fig. 2: load c     1        1.1461      3.0381      0.2394
                   2        0.2529      0.0507
Fig. 3: load a     1        0.5003      0.0340      0.1952
                   2        0.3483      0.0841
Fig. 3: load b     1        0.4268      2.3385        --
                   2        0.4268      0.0886
Fig. 3: load c     1        0.8558      1.7498      0.2148
                   2        0.4152      0.0846

Table 3. Results of the mode of deformation research for scheme in
Fig. 4

Model number:    Layer        Max         Max     Deflection,
load             number   compressive   tensile       mm
                            stress,     stress,

                              MPa

Fig. 4: load a     1        0.2264      0.0744      0.0535
                   2        0.0784      0.0909
Fig. 4: load b     1        0.6071      2.4933        --
                   2        0.6071      0.0754
Fig. 4: load c     1        0.6830      2.4603      0.0566
                   2        0.6830      0.0118

Table 4. Results of the mode of deformation research for schemes
in Figs 6-9

Model number     Layer        Max         Max     Deflection,
load             number   compressive   tensile       mm
                            stress      stress

                              MPa

Fig. 6: load a     1        0.5677      0.0625      0.2058
                   2        0.3287      0.0405
Fig. 6: load b     1        0.8311      4.6493        --
                   2        1.1160      0.2933
Fig. 6: load c     1        1.4566      4.3901      0.1815
                   2        1.3417      0.0916
Fig. 7: load a     1        0.4495      0.0442      0.2329
                   2        0.2637      0.0694
                   3        0.3603      0.0158
Fig. 7: load b     1        1.9450      3.8694        --
                   2        0.5743      0.0111
                   3        1.9450      5.5593
Fig. 7: load c     1        2.0001      3.8794      0.3409
                   2        0.6542      0.0110
                   3        2.0001      5.5582
Fig. 9: load a     1        0.4321      0.0575      0.2354
                   2        0.3000      0.0491
                   3        0.3779      0.0163
Fig. 9: load b     1        0.9367      4.2712        --
                   2        2.9491      0.0984
                   3        2.9491      5.3770
Fig. 9: load c     1        1.2849      4.2766      0.3792
                   2        2.9503      0.0983
                   3        2.9503      5.3676

Table 5. Results of the mode of deformation research for schemes
in Figs 10 and 11

Model number:     Layer        Max         Max     Deflection,
load              number   compressive   tensile       mm
                             stress,     stress,

                               MPa

Fig. 10: load a     1        0.3725        --        0.1956
                    2        0.1833      0.0485
                    3        0.2826      0.0143
                    4        0.5183      0.0392
Fig. 10: load b     1        1.0737      0.0660        --
                    2        0.0532      0.0145
                    3        1.1288      0.0169
                    4        1.1288      2.9483
Fig. 10: load c     1        1.0771      0.0613      0.2048
                    2        0.2178      0.0145
                    3        1.1212      0.0168
                    4        1.1212      2.9528
Fig. 11: load a     1        0.5635        --        0.1929
                    2        0.1829      0.0654
                    3        0.4273      0.0148      0.1929
                    4        0.5594      0.0244
                    5        0.7216        --
Fig. 11: load b     1        0.7599      0.0319        --
                    2        0.0519      0.0165
                    3        1.0406      0.0159        --
                    4        1.0406      3.0874
                    5        0.3052      0.0630
Fig. 11: load c     1        0.7326        --        0.1985
                    2        0.1999      0.0353
                    3        1.0361      0.0098
                    4        1.0361      3.0967
                    5        0.6926        --

Table 6. Results of the mode of deformation research for
schemes in Figs 12 and 13

Model number,     Layer        Max         Max     Deflection,
load              number   compressive   tensile       mm
                             stress,     stress,

                               MPa

Fig. 12: load a     1        0.2864      0.7329      0.0797
                    2        0.0717      0.0397
                    3        2.4020      0.7329
                    4        2.4020      0.2555
                    5        2.3812      1.1252
Fig. 12: load b     1        0.1950      0.0727        --
                    2        0.0458      0.0295
                    3        0.2538      0.2407
                    4        0.4474      2.4000
                    5        0.2085      0.0727
Fig. 12: load c     1        0.3169      0.6360      0.0835
                    2        0.0962      0.0294
                    3        2.3292      0.6360
                    4        2.3292      2.4056
                    5        2.3011      0.9173
Fig. 13: load a     1        0.2747      0.0461      0.1974
                    2        0.2161      0.0414
                    3        0.5833      0.0315
Fig. 13: load b     1        1.9538      2.5399        --
                    2        0.7943      0.1496
                    3        1.9802      4.9736
Fig. 13: load c     1        2.0500      2.5664      0.1479
                    2        0.8881      0.1465
                    3        2.1536      4.9894