STIFFNESS AND STRENGTH OF STRUCTURAL LAYERS FROM COHESIONLESS MATERIAL

The deformation modulus and permissible stress are two independent parameters that depict the carrying capacity of foundations, including earthworks and ballast layer. Nevertheless, while designing the track superstructure or controlling its state, they are considered separate to each other, even though they are terms of the same measure. The scientific problem is due to the practical necessity of unified building rules and standards. The carrying capacity of earthworks and foundations is regulated with standards based both on deformation and on stress criteria, which are not related to each other. This plays particularly important role for railway ballast layer, as an intermediate between the solids and soil. The objective of the present research is to estimate the relationship between deformation modulus and the strength of ballast layer. An overview of modern approaches according to the relation between the stiffness, deformation modulus, elasticity and strength of soils and crushed stone is done. The strength of ballast layer is considered depending on the experimental test: the direct shear test, compressive strength in the uniaxial or biaxial stress state. Load transfer model in crushed stone is proposed. The load transfer angle and cone of loading distribution are determined based on the load transfer and compressive strength models. The relation between deformation modulus and strength is derived from two simple laboratory experiments with cohesionless ballast material. The experiment tests have shown that the ballast stiffness as well as its strength are influenced with the support stress. The measurement of elastic and residual settlements for the different support stress values enables to determine the relation. It can be potentially used for the development of methods for the ballast compaction control, unification of construction norms. The research result should be considered as an approach for unification of two different ways to reflect the carrying capacity of ballast layer.


Introduction
The ballast bed is, simply mechanically considered, cohesionless crushed stone. However, it is the most difficult element of the railway superstructure to understand and predict. Its settlement behavior determines the length of the tamping intervals and thus a significant part of the life cycle costs of the railway superstructure. The settlement behavior results from the relationship between load and load capacity. Conventionally, the load can be expressed as the mean contact stress between sleeper and ballast, and the load capacity as the corresponding permissible contact stress (Esveld, 2001). This approach derives from the mechanics of materials, which classifies a structure element as stable if the existing stress does not exceed the permissible stress (strength). The similar approach is presented within the phenomenological modelling  of ballast layer settlements with pressure dependent intensity of irreversible deformations. Soil mechanics deals with the carrying capacity of crushed stone. Here, the ground mechanical state is classified as permissible if the existing stress (load) does not exceed some permissible stress value, that depends on the corresponding deformation modulus (Fendrich and Fengler, 2013). This approach, at first sight, seems strange, because the permissible stress and the elasticity properties of materials are interrelated. Such an approach is not assumed in the mechanics of materials. The main reason of the discrepancy consists in fact that on the one hand carrying capacity of soils is not constant, but depends on many factors, such as its stress state, compaction (i.e. density), etc. On the other hand, the measurement of soil deformation modulus is the simple practical and reliable non-destructive method to control the quality of earthworks. The different approaches in mechanics of materials and soil mechanics arise from the different structural behavior of solids and crushed stone. This paper demonstrates the fundamental relationship between deformation modulus and strength as two alternative ways to quantify the carrying capacity of layers from crushed stone using two simple experiments. The scientific problem is current, and is considered in many recent studies. Authors Rybacki, et al. (2015) have studied the strength and static Young's modulus of shale rocks based experimental triaxial compression tests. The relation of these parameters to the concept of brittleness is proved and a good, nearly linear, correlation between compressive strength and static Young's modulus was found. The relationship between the small deformation and strength properties of undisturbed gravelly soils was evaluated within the studies of Enomoto, et al. (2013). The particularity of measurement was the cyclic triaxial compression tests together with dynamic measurements of wave propagation using a pair of accelerometers. The difference between the moduli measured statically and dynamically relating the grain size were discussed. Nabochenko, et al.  (Sysyn, et al. 2019b). The relation is used as the basic principle of the strength determination of sedimentary rocks with the measurement of its elastic properties (Umrao, et al., 2018). The data of P-wave velocity together with 45 datasets of other geomechanical properties are used to predict rocks strength with application of artificial neural network and fuzzy logic methods. The experimental investigation of railway ballast stiffness together with its settlement during 1 million loading cycles was presented by McDowell, et al. (2007). The ballast stiffness, that was measured with non-destructive methodthe falling weight deflectometerhas been incremented more than twice after all load cycles. The main problem of the measurement is that it was done for box test without free ballast slopes. The vertical stiffness of the track, including the stiffness of ballast layer, is considered by many authors (Droździel, et al. 2010, Grossoni, et al. 2018) as a parameter that characterize the ability of the track to resist to residual deformations.

Stability of crushed stone ballast layer 2.1. Direct shear test
Crushed stone, like the ballast bed, has already been destroyed in terms of mechanics of materials. The soil mechanics describes the strength behavior of the crushed rock in a very clear way on the basis of the direct shear test (Fig. 1). The crushed rock located in a horizontally displaceable test facility is subjected to a horizontal force H or shear stress. At the moment when a horizontal movement of the experimental device begins, the shear strength is reached. However, in contrast to the solid bodies, the strength is not a constant, but changes depending on the vertical force V or the normal stress σ, which acts on the test device. The relationship between strength and normal stress is linear. Two material parameters occur, which are called cohesion c and friction angle φ: (1) Cohesion is a measurable value of the adhesive power between the grains. E.g., concrete is a solid that results from the bonding of crushed stone with cement. Thus, its strength is determined mainly by the cohesion. Gravel or sand are typical crushed stones without cohesion. Their strength is simplified described with the relation: = • tan .
(2) Figure 2 illustrates the angle of friction based on Coulomb's friction law (Göbel, 2013), which formulates the permissible shear force T as a function of the normal force V and the coefficient of friction μ. The resulting force R is inclined by the friction angle φ relative to the vertical force. A coefficient comparison shows that friction coefficient and friction angle are in the relation μ = tan(Φ).
If one applies over depending on the intersection angle φ, one obtains the so-called MOHR stress circle (Fig. 4). It is characterized by the fact that the smallest normal stress is zero and the largest normal stress is . (4) The cutting angle can also be experimentally observated on fractured specimens in the case of a uniaxial stress state (Fig. 5). It is clear from (4) that uniaxial compressive strength exists only in the presence of a cohesion c. However, this does not occur in the ballast bed, so that the railway ballast bed has no uniaxial compressive strength.

Compressive strength in the biaxial stress state
On the basis of the relations of the biaxial stress state (which can be analogously transferred to the actual three-axis stress state), it can be shown that a compressive strength of the crushed stone occurs only in the presence of a horizontal support stress . For the derivation it is assumed that the vertical normal stress and the horizontal stress are the main stress components (Fig. 6). If one applies over σ depending on the cutting angle φ, one obtains the so-called MOHR stress circle for the biaxial stress state (Fig. 7). In an analogous approach as in the uniaxial stress state, the compressive strength results in additional dependence on the horizontal support stress (Fig.7 above). The compressive strength is identical to the vertical principal stress : Setting the horizontal support stress to zero results in the uniaxial stress state, that was already derived in (4). In contrast to the uniaxial stress state, materials with no cohesion (c = 0) have strength in the biaxial stress state as long as a support stress acts (Fig.7 above):

Elasticity of the crushed stone
The linear elasticity of a material, including the crushed stone, is determined by the elastic modulus E [N/mm²]. A horizontal layer of crushed stone is assumed with a height h on a flat, rigid surface; its elastic properties should be calculated, i.e. after its vertical elastic settlement s as a function of the applied vertical stress , which is lower than the strength . The relation of both variables gives the bedding modulus C [N/mm³] of ballast bed: = .
The elastic settlement produces a negative deformation (compression) in the vertical direction: HOOKE's law combines stress, deformation and modulus of elasticity: By a coefficient comparison of (8) and (10), the modulus of bedding can also be expressed as a function of the elastic modulus: Expressing the elastic properties in the form of the vertical elastic settlement as a function of the vertical force , the support point stiffness cz [N/mm] can be received: The vertical stress is related to the vertical force on the contact surface A (in railway construction, the sleeper area); in turn, HOOKE's law also applies: Thus, the support stiffness can also be expressed as a function of the modulus of elasticity:

Load transfer in crushed stone
The load transfer model answers the question of the spatial distribution of the stress components, which determine the location (sliding line) and the beginning of the flow. The following considerations refer, for simplicity, to the case of a single load acting on an elastic space. Boussinesq (Bobe, 1983) has analytically deduced the stress components for this. According to Fig. 8, the vertical single load V produces in the half-space a vertical normal stress , the horizontal stresses and ℎ (which correspond to the horizontal stress in (7)) and the shear stress . The ratio of the shear stress and the normal stress follows from this: A coefficient comparison of (16) with the material law from the direct shear test according to (2) shows that the angle is identical to the friction angle φ of the crushed stone. It follows that the load transfer occurs within a load transfer cone at the edge of which the yield point of the material is reached (Fig.  9). The Boussinesq's solution refers to the half-space and explains the load transfer within a load transfer cone. In the case of railroad ballast, the dimension of the half-plane cannot be considered as infinite. Its free edges are limited with a slope angle (Fig. 10). This raises the question in what relation are the angle of the load transfer cone φ and the ballast slope angle β. • sin = • cos • tan .
According to (17), the shape and load transfer in the railway ballast follows from the identity of the slope angle and the angle of friction β = φ (Fig. 11).

Experimental determination of deformation modulus and strength
A frame filled with sand under a vertical loading of a square stamp is used as physical model of ballast layer (Fig. 12). The push force and the stamp movement s are measured. The load consists of cyclical and monotonous increasing parts. Fig. 13 shows the vertical stress converted to the stamp surface. Fig. 13. Vertical stress as a function of time A second experiment is made with the same loading, which differs from the first one in that the frame has been removed (Fig. 14).  To explain the order of magnitude, the average contact stress between threshold and ballast in a conventional railway superstructure should be estimated at this point. An axle load of 200 kN generates 2 vertical interpolation forces of 50 kN, taking into account the load distribution under the rail. This corresponds to a vertical threshold load of 100 kN. In a conventional sleeper geometry (length 2500 mm, width 250 mm) and the assumption of a uniformly distributed contact stress results a contact stress of = 0.16 N/mm². In general, the contact stress distribution between the threshold and gravel is irregular and therefore quickly reaches twice as big value, i.e. the strength limit that has been reached in the experiment without frame. Fig. 17 shows the derivative of The deformation modulus , can be interpreted as the mean elastic modulus E between the points MINi and MAXi. Fig. 18 shows the comparison of the current deformation modulus , and the current strength , . . There is an obvious correlation, which is due to the following linear relationship can be described:

Fig. 18. Deformation modulus and strength
Each new load cycle according to Fig. 13 results in a compression of the crushed stone. The compression corresponds to the formation of a hydrostatic stress state in the loaded volume below the stamp, i.e. an increase in both the vertical stress and the support stress . The support stress could be increased in the experiment without frame only as far as allow the supporting frictional forces of the outer borders. On the other hand, in the test with frame, the support is caused not only the activated frictional forces of the outer borders, but also the horizontal pressure of the frame. According to equation (7), the strength with frame is larger. The permissible stress of consolidated surface protection layers and subgrade is given as a function of the dynamic modulus of elasticity (Lichtberger 2003, p. 210): The deformation modulus, that is determined according to Figures 17-18 corresponds to a static deformation modulus. The dynamic elastic modulus is usually around a factor of 1.2 to 2.5 above the static deformation modulus . From a coefficient comparison of (18) and (19), it follows that the safety factor of the condition expressed by equation (19) is 1.4 to 2.9. It should again be noted that the strength according to (18) or the permissible load according to (19), in contrast to solids is not a constant, but depends on the compression, so the size of the existing horizontal support stress. The load capacity, however, depends on the boundary conditions, i.e. from the ability to carry support stresses. This also determines the direction of future work.

Conclusions and subsequent studies
The influence of the horizontal support stress on the strength of crushed rock layer is considered. The support stress is built up by the formation of a hydrostatic stress state in the area of the loaded volume. This explains the increase in the deformation modulus with the current vertical stress. This process cased with consolidation. It is finished when slip planes in the crushed rock appear. It is associated with a drastic increase in settlement and is therefore characterized in practice as a loss of carrying capacity. Between strength and deformation modulus -in contrast to solids -there is a close correlation with crushed rock. As a result of this correlation, it becomes possible to predict the strength of the crushed stones based on their deformation modulus. The advantage of this approach is obvious: the measurements of deformation modulus require minimal effort and are non-destructive. It is shown that the size of the reached compression and thus the size of the reached strength or the reached deformation modulus depends on the horizontal storage conditions of the crushed rock. The relation found between strength and deformation of ballast layer has a very high practical significance. It can be used for the development of nondestructive methods for the ballast compaction control, unification of construction norms, as well as the strategic maintenance planning and improvement of railway infrastructure (Wieczorek, et al. 2018).