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Jump to:   Page 1 ... Page 2 ... Page 3 ... Page 4 ... Translation in English; ... Part 2

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FLEXIBLE-PILE ABUTMENTS FOR ROADWAY BRIDGES Part I – Theoretical Principles & Design Methods (Translation from Bulgarian) author: Ass. Prof. D. M. Shapiro, Ph.D. State Institute for Road Research and Design (USSR) – Voronezh branch (text and figures prepared by Mario Behar; text review – G. Hristov & D. M. Shapiro)           The improvement in cost-effectiveness, and of the structural and technological principles of design of abutments, by creating design schemes that reflect the real mechanism of force interaction between the soil medium and the structure, is a compelling task for the theory of bridge construction. To this objective was dedicated the scientific and technological cooperation between the design organizations State institute for Road Research and Design (Voronezh branch) – USSR, and ”RoadDesign” – People’s Republic of Bulgaria (PRB), and the Road Research Institute – Sofia (subcontractor for the experimental part of the work). This cooperation was concluded in 1985. The widespread use of flexible-pile abutments is typical for the bridge engineering practice of our two countries. The standard designs and the existing production facilities are oriented primarily toward this type of abutments. In the same time, the most extensively used design scheme [for abutments] – as a retaining wall, is practically unsuitable for design of flexible-pile abutments, which do not retain the earth medium, rather, they are within a stable such medium.           The joint research and field tests enabled us to experimentally give proof of and refine the offered earlier design methods for single-line drilled-pile and column-pile (with classic flat-base support) flexible abutments [1-3]. A scheme of the complex of such methods as recommended by the Soviet State institute for Road Research and Design and ”RoadDesign” institute is presented on fig. 1. The groups and types of limit states presented in the scheme correspond to the general formulations in the COMECON (Committee for Economical Cooperation [of the ex-communist bloc]) standard 384-76. The control calculations provide more specific approach in accordance with the structures, and forms of destruction and deformation, specific for the different categories considered. The design schemes for flexible-pile abutments are based on two models for the behavior of the soil medium, that imply: the first one – linear relationship between the acting forces and the displacements (linearly-deformable medium), and the second – limit state of balance along a given sliding surface.           The first design scheme is based on assumptions characteristic for the linear-contact problem. The tests performed [4] confirmed the correctness of the mathematical model of the method of local elastic deformations with distribution foundation-bed factor Cz according to the equation: Eq. (1) … (see the Bulgarian original) where k = 2000 kN/m4 is the proportionality factor, and z – the coordinate along the height of the pile (column-pile), measured down from the fixing point at the cap-beam that unites several piles. The method of initial parameters is being used for these calculations, as well as the principles and the basic equations presented in [5], according to which the stressed state of a single-line drilled-pile (column-pile) abutment is considered using a system of equations, whose bending parameters (horizontal displacement yz, rotation φz, moment Mz, shear Qz) corresponding to coordinate z, are determined according to the initial parameters yo, φo, Mo and Qo.           In the design of flexible-pile abutments using the above described scheme, the following effects are taken into consideration:           - permanent and temporary loads applied to the cap-beam – represented by their resultants at its lower-edge level: horizontal H and vertical P components, and moment M (fig. 2a & fig. 2b);           - uneven settlement (rotation) of the basement of the column-pile abutment as a result of the unbalanced loading from the front sloped portion of the earthwork and the backfill behind the abutment, as well as from the [horizontal] earth pressure applied to the abutment (fig. 2c) – for the drilled-pile abutments this effect is not considered;           - horizontal displacements of the bed-soil line, related to the different weight of the front sloped portion of the earthwork and the backfill, and to the deformation of the fill-soil-body itself under its own weight.           The loads (P, H, M) applied to the cap-beam, are composed by those coming from the superstructure of the bridge and by those from the transition plates (their dead load and temporary loads): the horizontal force coming from the superstructure and determined by considering the interaction with the other supports [6]; the active (Coulomb) earth pressure of the soil behind the backwall; the dead load of the cap-beam. The load safety factors are assumed according to the recommendations of SNiP 2.05.03-84 (USSR). The initial parameters for design of the single-line flexible-pile abutments are being determined using the following equations [1]: Eqs. (2) … (see the Bulgarian original) Eqs. (3) … (see the Bulgarian original) Eqs. (4) … (see the Bulgarian original)           Equations (2) are common for drilled-pile and column-pile abutments, (3) are good only for drilled-pile abutments, and (4) – only for column-pile abutments. The notations α, A1, A2 … D3, D4 are assumed in accordance with [5], EI is the bending stiffness of the drilled-piles (column-piles), n – the number of piles.           The uneven settlement of the flat foundation of the column-pile abutment generates rotation of the foundation by an angle φ"h (see fig. 2c), determined using the following equation [7]: Eq. (5) … (see the Bulgarian original) where ωs is the angle of general inclination of the flat surface below the foundation as a result of the dead load of the front sloped portion of the earthwork and the backfill, e – eccentricity of the force P with respect to the centroid of foundation plate; Mφ – parameter, a moment in the foundation bed as a result of the resistance of the soil medium against the horizontal movement of the column-pile for a unit rotation of the foundation, determined according to [1, 7]; G – angle of rotation of the foundation produced by the unit moment at the foundation bed surface, as deter-mined according to equation (10) – addendum 2, SNiP 2.02.01-83; t – thickness of the foundation plate.           The initial parameters for calculation of the bending of the column-piles as a secondary effect, related to the rotation of the foundation to an angle φ"h , are determined using the following equations: Eqs. (6) … (see the Bulgarian original)           As the special investigation showed, the horizontal displacements of the soil medium produce insignificant forces in the drilled-piles and column-piles, which might be neglected. This effect must be taken into consideration in the calculations based on limit states of group two when determining the horizontal displacements of the cap-beam using the approximate formula: Eq. (7) … (see the Bulgarian original) where γo is the specific weight of the fill, HH – the height of the approach fill, EH – the modulus of deformation for the backfill and the front sloped portion of the earthwork; A and B – tabulated parameters, determined depending on the type of and the ratio between the deformation moduli of the fill and the base earth medium; h – height of the drilled-pile (column-pile); δMQ – parameter, equal to the horizontal displacement of the upper end of the drilled-pile (column-pile) produced by the unit moment [5] – for drilled-piles … (see formula in the Bulgarian original), and for column-piles … (see formula in the Bulgarian original).           The design scheme of flexible-pile abutment as an anti-landslide structure [2] is based on the principles of the grapho-analytical method of the theory of limit balance. The sliding surface is represented by a trough-like outline with cross-section along the broken line ABCD (fig. 3) with angle of the side-slopes β = arccotg n = const. This assumption allows for simplification of the calculation, making the latter suitable for computer, and bringing the assumed form of the sliding surface closer in similarity to the one observed. The design procedure takes into account the [restraining] support action of the flexible-pile abutment with respect to the moving earth medium (the cut-off body), and by analogy with the method of G. M. Shaguliantz [8] the reverse-algorithm problem is being solved: for given safety factor with respect to the structure γn, the overall horizontal component E of the moving and restraining forces acting along the assumed surface is being determined.           The basic equation for this design process looks like this: Eqs. (8) … (see the Bulgarian original) where ΔEi is the increment of the active horizontal forces within the limits of the i-th segment; ΔIi – the resultant of the filtering pressure forces within the limits of the i-th segment; … (see formulae in the Bulgarian original) – tangent (moving or restraining) components and normal components along the sliding surface due to the weights Gci and Gδi of the middle and side sections of the i-th segment respectively; Sci, Sδi – area of the sliding surfaces in the limits of the middle and side sections of the i-th segment; Io – average slope of the depression curve of the dropping level of underground waters; Vwi – volume of the submerged portion of the i-th segment; αi – angle of slope of the mid-section of the i-th segment with respect to the horizontal line; φi, ci – internal friction angle and cohesion in the section through the axis of the i-th segment; φδi, cδi – similar parameters for the side sections of the i-th segment (average values); N – total number of segments, in which the “cut-off” body is being divided.           The sign plus (minus) in equations (8) corresponds to the moving (restraining) forces Tci and Tδi. In this case the safety factor γn is assumed to be equal to 1.4 for the moving, and γn = 1.0 for the restraining forces. The form of the sliding surface (center, radius of the axial section, and slope angle of the side sections, see fig. 3) is determined by choosing the most unfavorable one, with a criterion – the resultant E to be reaching its maximum. This method can be used to the same extent to check the stability of the abutment together with the front sloped portion of the earthwork and the backfill against deep or local landslide movement. The distribution diagram of the horizontal load on the flexible-pile abutment along its height is assumed to have a triangular shape with a maximum ordinate at the sliding surface (see fig. 3) and uniform distribution along the width of the back surface.           Main characteristic features of the design of flexible-pile abutments are the use of the method of the equivalent system and the method of checking of the stability against prolonged deformations.           The method of the equivalent system [9, 10] is developed with the aim to correctly consider the weight of the front sloped portion of the earthwork, the backfill and approach fill in the calculations for the fill of the foundation of the abutment. The method consists in the following (fig. 4): to the foundation is being added the weightless layer z; the acting load is being replaced with semi-infinite strip load with intensity po and strip width B. Eqs. (9) … (see the Bulgarian original) where m is the relative arithmetic-average value of the horizontal dimension of the sloped portions of the soil-fill in front of the abutment, γo – specific weight of the fill, HH – the height of the soil-fill, b – width of the fill at its upper end. For this design method, the stress components that are determined using ready equations from the theory of elasticity, correspond to the test data and engineering expectations. The overall pressure q put into effect by the foundation of the column-pile is determined as follows: Eq. (10) … (see the Bulgarian original) where σh = αz . po – stress due to the po load (αz – tabulated factor, reflecting the ratio between the vertical normal stress in given point and the intensity of the strip loading [10]); p1 – additional pressure, transferred to the abutment foundation; and the earth pressure γ'.h (γ' – specific weight of the base layer, h – depth of penetration of the foundation within the base layer).           The check-up for stability of weak layers in the foundation medium for long-term plastic deformations is based on the assumption for admissibility of the active loading with the condition, that the “plastic domains” in the base layer obtained using the theory of elasticity (where Mohr-Coulomb strength conditions are not complied with) do not surpass certain dimensions*. The check-up under consideration consists in finding the factor Kd.abut., which must not surpass 1.0. Eqs. (11) … (see the Bulgarian original) where HH.saf is the safe height of the earth-fill; φ and c – angle of internal friction and cohesion of the base layer under consideration; z1 – depth level of the investigated layer at a [vertical] distance r = 3-5 m from its upper boundary (fig. 4b); β1 – factor that depends of the form, the 3D-dimensions of the active load, the depth level, and the angle of internal friction of the base layer under investigation [12].           The institutes in cooperation have developed “Recommendations for the design of single-line flexible drilled-pile and column-pile (with flat foundation) abutments for roadway bridges”. In the Voronezh branch of the Soviet state institute for Road Research and Design the included in these “Recommendations” methods for design of flexible-pile abutments are in use since 1973 [13, 14]. For the time until 1986, 53 bridges (102 abutments) have been designed. The comparative techno-economical calculations show, that the refined design of abutments (compared with the SN 200-62 norms and their replacing equivalent sections of SNiP 2.05.03-84) brings the possibility to save for an abutment, in the average: in construction cost – 3-5 thousand rubles or 20-30%; in reinforcement expenses – 2.5-4.0 [metric] tones; in amount of concrete – 8-15 m3.           In the same time, in weak unstable base layers and significant heights of the earth-fill, the use of the recommended design methods might also not lead to cost reductions, and even can bring to an increase in expenses. The cost-increase in these cases should be considered as legitimate due to the increased reliability of the structures. ___________________________ * Similar assumption is used for the design of weak foundations of road base layers [11].            REFERENCES:            1. Shelyapin R.S., D.M. Shapiro, Improvement of the design methods for bridge flexible-pile abutments as a way to reduce the construction costs, “Transport Construction” Magazine, 1974, issue # 6.            2. Shapiro D.M., Design of the flexible-pile abutment as an anti-landslide structure, “Transport Construction” Magazine, 1980, issue # 11.            3. Shapiro D.M., Improvement of the design and construction of bridge flexible-pile abutments, “Highways” Magazine, 1983, issue # 11, pp. 18-20.            4. Shelyapin R.S., D.M. Shapiro, Tests of flexible-pile abutment models, “Transport Construction” Magazine, 1972, issue # 8.            5. Zavriev K.S., G.S.Shpiro, Design of deep bridge foundations, Moscow, “Transport” Editorial, 1970.            6. Grinberg E.I., D.M. Shapiro, About the design of roadway bridges, “Transport Construction” Magazine, 1978, issue # 10.            7. Shelyapin R.S., D.M. Shapiro, Contact problem for bending of a bar in a moving surrounding medium, “College News”, Section “Civil Engineering and Architecture”, 1971, issue # 10.            8. Shehuniantz, G.M., Railroads, 2nd Edition, Moscow, Editorial “Transport”, 1969.            9. Shelyapin R.S., D.M. Shapiro, Improvement of the design methods for bridge flexible-pile abutments, “Highways” Magazine, 1971, issue # 10.            10. Shapiro D.M., Design of the foundations of flexible-pile bridge abutment, “Highways” Mag., 1974, issue # 9.            11. Evgeniev, E.I., V.D. Kazanovskiy, The lower earth layers of roadways on weak soils, Moscow, Editorial “Transport”, 1976.            12. Shapiro D.M., Refining the design of foundations for flexible-pile bridge abutments, Investigation of regional structures foundations, Voronezh, Editorial “Voronezh State University”, 1984.            13. Design recommendation for single-line drilled-pile and column-pile highway bridge abutments as elastic supports in linearly-deforming medium – Soviet State Institute for Road Research and Design, Moscow, 1978.            14. Design recommendation for highway bridge flexible-pile abutments – Soviet State Institute for Road Research and Design, Moscow, 1982. Back to Top

Jump to:   Page 1 ... Page 2 ... Page 3 ... Page 4 ... Translation in English; ... Part 1

Jump to:   Page 1 ... Page 2 ... Page 3 ... Page 4 ... Translation in English; ... Part 1

Jump to:   Page 1 ... Page 2 ... Page 3 ... Page 4 ... Translation in English; ... Part 1

Jump to:   Page 1 ... Page 2 ... Page 3 ... Page 4 ... Translation in English; ... Part 1

FLEXIBLE-PILE ABUTMENTS FOR ROADWAY BRIDGES Part II - Results from the research and field tests of some [bridge] structures* (Translation from Bulgarian) Ass. Prof. D. M. Shapiro, Ph.D. (USSR), eng. Georgui Hristov, eng. Mario Behar, eng. M. A. Zaharov (USSR) (text and figures prepared by Mario Behar; text review with some additions - D.M.Shapiro ... and G.Hristov?)           Introduction           According to the program for scientific and technical cooperation, joint research and field tests were performed on existing structures and structures under construction in the People’s Republic of Bulgaria (PRB). The goal of this work was: experimental check-up of the design methods developed for single-line flexible-pile abutment systems [1-3]; and confirmation of conclusions driven from earlier experimental research work [4, 5], using measuring equipment on significant number of existing structures with different age, and field tests on structures under construction.           1. Field investigations on existing structures           The investigation of these structures was performed using measuring equipment, and consisted of:           - definition of: deformations in the steel-rubber (elastomeric) bridge bearings that are re-sult of longitudinal horizontal displacements; and relative horizontal displacements of the upper plates of steel roll-bearings with respect to the lower ones (fig. 1);           - measurement of the gap opening of the bridge joints.           The total number of investigated structures was seven (20 abutments). The type of the abutments and their dimensions are presented in table 1. All of the observed abutments were single-line-pile (-column) abutments, with big cross-section (80 x 80 cm, 90 x 90 cm, d = 120-138 cm), positioned 6 m at center, enabling this way for bulldozers and compaction machines to go in between. The front sloped portion of the earthwork and the backfill have been erected simultaneously with the approach fill using the same soil material (incl. semi-permeable one). The deposit, leveling and layer-by-layer compaction of the soil have been performed mechanically. Concrete for the drilled piles has been poured, and cap-beams for the two types of abutments (pile-cast-in-place, and pre-cast column-piles) have been either poured, or mounted, after the soil material has been put in place completely.           The age of the structures at the beginning of the research process was between 15 and 52 months.           The measurements were performed using micrometer, and were repeated 3 to 7 times for different yearly and day-night temperatures. The measurements of deformations (displacements) in the bearings were averaged at each abutment for each measurement, and then compared with the theoretical results. The opening of the expansion joints (the gap between the backwall and the faces of the beams) was used to control the state of these joints and their ability to follow the temperature movements of the superstructure. The observations were performed during an 18-month period (from September 1983, till March 1985).           The comparative calculations were made, assuming design scheme for the flexible-pile-abutments as an elastic support in linearly-deforming medium [1] – in compliance with the Recommendations [3]. For the comparison of measured deformations and such computed based on theory, the theoretical displacements a of the roll-bearings (for structures #1 & #2), as well as the longitudinal forces X in the superstructures (for the rest of the structures), were determined. In the case of the latter group of structures, the obtained forces were compared with the total force due to the observed deformation of the steel-rubber (elastomeric) bearings. The parameters a and X were obtained taking into consideration the following effects:           - moments due to vertical dead loads applied on the abutment (support reactions for the superstructure and the transition plate [on the outer face of the abutment], as well as weight of the cap-beam);           - horizontal earth pressure on the backwall;           - horizontal displacements of the soil medium around the abutment;           - superstructure deformations (due to changes in temperature, shrinkage and creep of concrete, elongation and shortening of the upper and lower chords of the beams during bending);           - rotation (heeling) of the foundations of pre-cast [concrete] pile-abutments;           On figs. 2a & 2b are presented samples of comparison of the measurements performed for structures #1 and #6 with the theoretical graphs for the relationships a = f1(t°) and X = f2(t°). On the diagrams the average displacements and deformations are presented by points, united each to the other with straight lines, reflecting the sequence (in time) in which the results have been obtained.           Because the “temperatures of closing of the system” (the temperatures at which the expansion joints have been initially fixed to the superstructure and the abutment) were unknown, it was assumed that said temperatures should be in the range of 0 to 30°C, which encompasses 80-90% of the average day-night temperatures during the year. That’s why the theoretical graphs only relatively determine the limits within which the deformations (displacements) of the bearings can be positioned at different temperatures. It should be noted that the temperature of closing of the system is a relative quantity, because the beam positioning and the assembly of the “plastic pin points” of the thermally continuous superstructures (or of the “hidden cap-beams” in frame systems) is being done along a period of time, during which the air temperature cannot be constant.           The performed measurements proved the normal work of all of the investigated structural elements. The data in fig. 2, as well as in similar diagrams, made according to the processed results of measurements on other structures, attest to the correspondence between the horizontal movements in the bearings and the seasonal changes in temperature. Unidirectional horizontal movements related to the plastic deformations of the soil medium, or any other irreversible long-term processes, were not recorded.           The same comparative calculations were used for all of the investigated abutments, and the results of these calculations in a satisfactory manner agreed with the facts at hand. This enables us to conclude that the assumed design scheme is correct. Confirmation of such a conclusion is the comparison in table 1 of the theoretically obtained parameters δXp that represent the force in the steel-rubber bearings for temperature change of 1°C, and their experimental analogues δXe = Σ(Xi+1 – Xi)/Σ(t°i+1 – t°i) – the arithmetically averaged ratio between the differences in deformation forces Xi+1, Xi and temperatures t°i+1, t°i , where i, i+1 are the indexes reflecting consecutive measurements. In the majority of cases (12 out of 14) an approximate equivalence between δXp and δXe, or correlation δXp < δXe, were obtained. These results show, that the real stiffness of the abutments is equal or greater than the design stiffness.           2. Field investigations during construction           These were performed on structure #8 (frame viaduct with spans 16 + 2 x 24 + 16 m), and structure #9 (beam viaduct, “simple-span” type with spans of 3 x 20 m). The fill around the abutments, as well as the approach fill, were done with sand-clay mixed with gravel. The earth works were performed mechanically, while complying with the requirements for quality of soil compaction.           In one of the column-piles (dimensions 80 x 80 cm) at two of the abutments of each structure were installed string stress-strain gauges of type 587, with base 200 mm (made in Czechoslovakia). The way they were installed is presented on fig. 3a & fig. 4a. Later, during the construction, with their help were followed the strains produced by the loads imposed to the column-piles of the abutments. The investigation itself included the following experiments:           - testing of a column-pile with horizontal force, applied at its upper end, after covering around the abutment with soil up to the bottom edge of the cap-beam; simultaneously, data from the string gauges was collected, and the horizontal displacement yo was followed;           - data collection from the string gauges in the process of construction work: At structure #8 data was collected during a 13-month period, starting with the erection of the front sloped portion and the backfill, until the very end of the construction; at structure #9 data was collected only during the erection of the front sloped portion and the backfill up to the bottom edge of the cap-beam.           In processing the strain data collected during the column-pile loading, average values of the bending moments were determined using the equation: Eq. (1) … (see the Bulgarian original) where ε1 and ε2 are the strains measured in the elongated and shortened zones of the cross section; EI – the bending stiffness of the column-pile (the E-modulus was obtained in a test – submitting the column-pile to predetermined unit load; for structure #9); ν = 0.34 m – the distance from the centerline to the position of the strain gauges.           In fig. 3 & fig. 4 are presented comparisons of bending moments – the ones obtained based on the measuring data, with those based on theory and obtained according to [1, 3]. Additionally, a comparison is presented of the theoretically obtained horizontal displacements of the upper end of the column-piles under horizontal force applied at this level with the ones measured in the field.           For the comparative calculations reflecting the behavior of structure #8 under loads applied during construction, the following quantities were also considered:           - the rotation of the foundation of the column-pile, resulting from uneven settlement under the front sloped portion of the soil-fill and the backfill;           - the moment on the column-pile due to the eccentrically applied resultant support reaction from the superstructure and the transition plate [on the outer face of the abutment];           - the temperature, the elastic and plastic deformations, and the shrinkage of the concrete in the superstructure.           The design calculations were repeated twice: for factored and un-factored loads, with safety factors (for loading) assumed according to SNiP 2.05.03-84 [Soviet norms].           The comparative calculation for structure #9 were performed only for factored loads due to the uneven settlement of the foundation.           The form and signs of all compared graphs (theoretical and experimental) practically coincided. The theoretically obtained values for the horizontal displacements yo (fig. 3g & fig. 4g) exceeded to some extent the values of the ones measured. For values for yo around 10 mm, the measured values exceeded the theoretical ones by 20-25%. This percentage had a tendency toward zero for yo between 15 mm and 20 mm. This correspondence between theoretical and experimental values confirms in the best way the assumed mathematical model.           The theoretical graphs for bending moments Mz due to the horizontal load applied at the upper end of the column-pile, are similar to those obtained during field-tests for structure #8. For structure #9 the theoretical values of Mz were greater than those experimentally obtained by 20-40%.           For structure #8 the bending moments in the column-pile, determined from data obtained during construction, exceeded to some extent the corresponding theoretical values based on factored [normative] loads. Certain reserve with respect to the test results was obtained after additionally factoring said normative loads by the load reliability coefficient. For structure #8 the comparison of similar tests with the theoretically obtained parameters showed good correspondence between the compared quantities.           Conclusion           As a result of the research carried out, the following basic ideas in the design method for flexible-pile abutments as elastic supports in a linearly-deforming medium were experimentally confirmed: the definition of the force interaction between the structural elements of the abutment and the soil in compliance with the solution for the linear media-contact problem; the use for these design calculations of the method of local elastic deformations with distribution foundation-bed factor according to the expression Cz = K.z (K = 2000 kN/m4 [4]), and of the expressions and empirical parameters of the method as presented in [6]; the influence on the stress-strain state of the flexible-pile abutment by the uneven settlement of the foundation bed under the different weights of the sloped front portion of the fill and of the backfill. The method used to determine the bending of the column-piles according to [1] was also confirmed. The obtained results open the way for extensive application of the design models [1, 2, 3], which provide for improved cost-effectiveness and reliability of the design solutions for flexible-pile abutments. Future experimental work, related to the perfection of the design methods for flexible-pile abutments and the scope of their application, would be able to provide new knowledge in this area. ___________________________ * The research, field tests, and processing of the data were done by the Bulgarian part of the team in the Road Research Institute - Sofia. After that the results and the comparative calculations were summarized by the Soviet part of the team.            REFERENCES:            1. Shelyapin R.S., D.M. Shapiro, Improvement of the design methods for bridge flexible-pile abutments as a way to reduce the construction costs, “Transport Construction” Magazine, 1974, issue # 6, pp. 43-44.            2. Shapiro D.M., Improvement of the design and construction of bridge flexible-pile abutments, “Highways” Magazine, 1983, issue # 11, pp. 18-20.            3. Recommendation for the design of highway bridge flexible-pile abutments - Soviet State Institute for Road Research and Design, Moscow, 1982.            4. Shelyapin R.S., D.M. Shapiro, Tests of flexible-pile-abutment models, “Transport Construction” Magazine, 1972, issue # 8, pp. 46-47.            5. Grinberg E.I., D.M. Shapiro, V.P. Ribchinskiy, Results from field-tests of flexible-pile abutments, “Transport Construction” Magazine, 1978, issue # 1, pp. 41-42.            6. Zavriev K.S., G.S.Shpiro, Design of deep bridge foundations, Moscow, “Transport” Editorial, 1970. Back to Top home ... other publications