The multi-layered ring under parabolic distribution of radial stresses combined with uniform internal and external pressure

A recently introduced solution for the stressand displacementfields, developed in a multi-layered circular ring, composed of a finite number of linearly elastic concentric layers, subjected to a parabolic distribution of radial stresses, is here extended to encompass a more general loading scheme, closer to actual conditions. The loading scheme includes, besides the parabolic radial stresses, a combination of uniform pressures acting along the outerand innermost boundaries of the layered ring. The analytic solution of the problem is achieved by adopting Savin’s pioneering approach for an infinite plate with a hole strengthened by rings. Taking advantage of the results provided by the analytic solution, a numerical model, simulating the configuration of a threelayered ring (quite commonly encountered in practical applications) is validated. The numerical model is then used for a parametric analysis enlightening some crucial aspects of the overall response of the ring.


INTRODUCTION
n a recently published paper [1] the stress-and displacement-fields developed in a circular multi-layered ring were considered both analytically and numerically.The ring was composed by a finite number of concentric layers made of different linearly elastic materials and it was loaded by a parabolic distribution of radial stresses acting along two finite arcs of the outer boundary of the outer-most layer.In general, the configuration of multi-layered rings is common in quite a few engineering applications of increased interest, ranging from Biomechanics (human aorta) to Fluid Mechanics (insulated pipes).Such composite rings are usually loaded either by uniformly distributed internal or external pressure or by a combination of them.In the study by Markides et al. [1] a more complicated loading scheme was considered consisting of radial stresses acting along two finite arcs of the external boundary of the outer-most layer (antisymmetric with respect to the geometric centre of the ring) distributed according to a parabolic law.Such a loading scheme appears in case a circular ring is compressed between either plane or curved jaws (as it is for example the jaws of the device introduced by the International Society for Rock Mechanics (ISRM) [2] for the standardized implementation of the Brazilian-disc test).The solution by Markides et al. [1] is here generalized to confront a more general loading scheme including, besides the parabolic radial stresses, a combination of uniform pressures acting along the outer-and inner-most boundaries of the I layered ring.For the analytic solution of the problem, the procedure proposed by Savin [3] for an infinite plate with a hole strengthened by rings is adopted as it was done also in ref. [1].The specific procedure is based on the complex potentials technique introduced by Muskhelishvili [4].The analytic solution achieved provides full-field closed-form expressions for both the stresses and the displacements.However, the specific expressions, in spite of their "elegance" and their numerous advantages (related mainly to their generality and the fact that they are expandable to rings that are made up of any number of concentric layers), are somehow "lengthy" and cumbersome.As a result, detailed parametric analyses of the (quite a few) factors influencing the final outcomes become rather difficult.To overcome the above mentioned difficulty, the problem is revisited, also, numerically using the Finite Element method and the commercially available software ANSYS.For the validation of the numerical model, the data obtained from the analytic solution for the case of a ring made up of three concentric layers are used.Then the validated numerical model is used for a parametric study of the role of some critical parameters (i.e., the elastic modulus and thickness of the intermediate layer and also the length of the arcs loaded by the parabolic distribution of radial stresses) on the overall stressand displacement-fields developed in the ring.

THEORETICAL CONSIDERATIONS
The problem and the basic assumptions onsider a multi-layered hollow cylinder of length w consisting of n concentric hollow cylinders of different thicknesses, perfectly joined together without any gaps at all.The cylinders are assumed to be made of homogeneous, isotropic and linearly elastic materials.The multi-layered cylinder as a whole is in equilibrium under the simultaneous action of three different kinds of loading: inner and outer uniform pressure all along its inner and outer lateral surfaces and a parabolic pressure acting along two finite parts of its outer periphery, antisymmetric with respect to the section's center.All three types of loading act in the body's normal cross-section and remain constant along its length.Ignoring its weight, stresses and displacements are to be determined at any point of the multi-layered cylinder.Clearly, the as above described configuration corresponds to a 1 st fundamental problem of plane linear elasticity for the body's crosssection, i.e., the multi-layered circular ring.In this context, Muskhelishvili's method of complex potentials [4] will be employed for the analytic solution.

Mathematical formulation
Under the above assumptions, the multi-layered ring is considered lying in the z=x+iy=re iθ complex plane, Fig. 1.The origin of the Cartesian reference coincides with the center of the ring.The n constituent concentric rings are numbered in such a way so that they are encountered in the order 1, 2,…, n as one moves from the origin towards the outer perimeter of the multi-layered ring.The arbitrary ring is denoted by the index j (1≤j≤n) and its boundaries are Lj and Lj+1 corresponding to the radii r=Rj and r=Rj+1 (Rj+1> Rj), respectively.In general, it holds that (R2-R1)≠(R3-R2)≠…(Rn+1-Rn).Adjacent constituent rings are perfectly joined together along their common interfaces.The inner boundary of the innermost ring, L 1 , for r=R 1 , is subjected to a uniform pressure of magnitude pI>0.The outer boundary of the outermost ring, Ln+1, for r=Rn+1, is under the simultaneous action of a uniform pressure of magnitude pE>0 exerted all over Ln+1, and a parabolic pressure of magnitude p(θ)>0 acting along two finite arcs of L n+1 , antisymmetric with respect to the ring's center.Each one of these arcs has a length equal to 2ωoRn+1, where ωo corresponds to the half loaded rim.Particularly, p(θ) is here considered equal to: where ϕ o is the arbitrary angle formed by the axis of symmetry of the parabolic pressure and x-axis (measured from x-axis in the anticlockwise direction) and F (F>0) is the resultant force due to p(θ).Obviously, for the as above described loading conditions, the multi-layered ring as a whole and each constituent j-ring separately, are in equilibrium and for that configuration the stress-and displacement-fields are to be determined for each j-ring.
According to Muskhelishvili, the latter can be implemented by obtaining on each j-ring two analytic functions of the complex variable z, the complex potentials φ j (z) and ψ j (z), in terms of which stresses and displacements are expressed as [4]: Figure 1: The configuration of the problem and definition of symbols.
In Eqs.(2)(3)(4) prime and double prime denote first and second derivative, bar denotes the complex conjugate and  is the real part; in addition, μ j denotes the shear modulus and κ j is the Muskhelishvili's constant, equal to 3-4ν j or (3-ν j )/(1+ν j ), for plane strain or plane stress respectively, with ν j indicating the Poisson's ratio.
Regarding φj(z) and ψj(z), their analytic character on each j-circular ring permits representing them by Laurent series as: Therefore the problem reduces to the determination of the coefficients ( ) j k a and ( ) j k b , a task that will be achieved by ful- filling the boundary conditions of the problem.Before implementing this step, however, it is to be mentioned that it is not possible to obtain directly all ( ) j k a and ( ) j k b ; namely, for a 1 st fundamental problem φj(z) is determined apart from an arbitrary complex constant and an arbitrary real constant C appearing in a term of the form iCz. In this context, ( ) 0 j a and the ima- a remain arbitrary.In the particular problem studied, both are set equal to zero for convenience.What is more, ψ j (z) is obtained apart from an arbitrary complex constant so ( ) 0 j b remains arbitrary.Actually, it can be seen that for the present problem only one out of all ( ) 0 j b can be arbitrarily fixed while the remaining ones are subjected to determination.Here it is assumed that  (1) 0

b
It is only under these simplifying assumptions that the remaining coefficients ( ) j k a and ( ) j k b can be determined with the aid of the boundary conditions of the problem.These conditions are classified into two categories: i.The boundary values of stresses on the multi-layered ring's boundaries as a whole, reading as: ii.The values of stresses and displacements along the interfaces Lj+1, j=1,…,n-1, between the j-and (j+1)-ring, due to their firmly joining, reading as: where  (1)   1 is the real part of (1)   1 a .
For the 13n-1 unknowns appearing in Eqs.(15-18), i.e., for   exists a number of 13n-1 equations, as expected and thus the problem has a solution.In other words, Eqs.(15), ( 16), ( 17) and (18) comprise 2n, 3n-1, 4n and 4n equations, respectively, i.e., a total sum of 13n-1.Thus, ( ) j k a and ( ) j k b are indeed obtainable from Eqs.(15-18), and in this context φj(z) and ψj(z) can be recapitulated as follows: ( 2) 5,7,9,... 5,7,9,... ( ) , 1, 2,..., provide the solution in the case external uniform and parabolic pressure are zero and the multi-layered ring is only under the action of internal uniform pressure p I .Particularly, in all last three cases (d, e, f), i.e., when only uniform pressure over one or both the two boundaries (internal and external) of the multi-layered ring is present, the complex potentials are significantly simplified to:  In order to demonstrate the capabilities of the solution introduced, the distribution of the radial and tangential stresses along the diameter corresponding to the symmetry axis of the parabolic load, is plotted in Fig. 2, for three combinations of the constituent loading schemes, i.e., parabolic distribution of radial stresses and uniform internal pressure, parabolic distribution of radial stresses and uniform external pressure, and finally all three schemes simultaneously.The discontinuity of the distribution of transverse stress at the interfaces of the rings is the main characteristic of the plots.As it was expected, this discontinuity is stronger for the case of simultaneous action of the parabolic distribution and external uniform pressure.

NUMERICAL ANALYSIS -APPLICATION FOR n=3 CONSTITUENT RINGS
s a next step, a parametric numerical analysis was decided to be carried out, in order to explore the influence of various factors on the displacement-and stress-fields developed in a multi-layered ring.The ring with three layers under plane strain conditions was decided to be considered, due to its increased practical interest.The principal advantage of the numerical scheme is that it models the "multi-layered ring -jaw" complex (as an integrated elastic system) rather than the isolated ring.Before the main numerical study, the model was properly validated as it is described in next section.

Construction and validation of the reference numerical model
For the proper validation of the numerical model advantage was, initially, taken of the analytic solution for the most complex loading case, i.e., the partial parabolic pressure.For symmetry reasons (materials and geometry) only a quarter of the ringjaw complex (0 ο ≤θ≤90 ο ) was constructed with the aid of ANSYS software.The geometry of the reference model is shown in Fig. 3.For the radii of the layers it holds that Rj=25, 30, 34, 50 mm, for 1≤j≤4.All three interfaces (i.e., between the constituent rings and between the jaw and the outer ring) were taken into account.The model was meshed using the PLANE182 element.Contact elements (CONTA172 and TARGE169) were used to model the interfaces.The interfaces between the successive rings were considered as mutually bonded without separation.A very small (tending to zero) value was assigned to the coefficient of friction at the interface between the jaw and the external surface of the outer ring, in order to minimize (eliminate) friction stresses.This assumption was dictated by the fact that the analytic solution, described in the previous section, considers an isolated ring under exclusively radial stresses without any friction stresses at all.The analysis performed was linearly elastic.The moduli of elasticity and the Poisson's ratios assigned to each one of the rings were Ej=30, 10, 3 GPa and νj=0.25,0.30, 0.35, respectively, for 1≤j≤3.The jaw was considered to be made of steel (E jaw =210 GPa and ν jaw =0.30).
Figure 3: The reference numerical model and detailed views of the finally adopted mesh.
Due to the model's symmetry the following boundary conditions were adopted: i.The motion along the y-axis (loading axis) of all the nodes of the lower line of the ring was restricted.

A
ii.All the nodes of the left (vertical) line of the ring as well as the nodes of both sides of the jaw were restricted of shifting horizontally (i.e., normally to the loading axis).iii.A uniform vertical displacement on the nodes of the upper line of the jaw was imposed.The optimum element size for the meshing of the model was chosen based on a proper convergence analysis.The variation of the polar stresses developed along "Path 0", i.e., the outer periphery of the outermost ring, and especially around the position where the major differences are observed is plotted in Fig. 4a.The embedded graph corresponds to the stresses along the whole length of "Path 0".In addition, the total stress applied on point A (see Fig. 3) is plotted in Fig. 4b.Both graphs indicate that convergence is satisfactory for about 20000 elements.In order to obtain more accurate results (and given that the CPU-time consumption is not prohibitive), a number of about 38000 elements was chosen for the final meshing.Before proceeding to the parametric study, the numerical model was used to check the assumption of the analytic solution concerning the parabolic distribution of radial stresses along the loaded arc of the ring.This was considered absolutely necessary because the specific assumption is based on the solution of a different problem, i.e., that of the "compact disc -jaw" contact problem [5] and therefore adopting a similar distribution for the "ring -jaw" problem appears perhaps somehow arbitrary.In this context, the vertical displacement of the upper node (point A in Fig. 3) of the ring (which is, in fact, the only point of the ring in contact with the jaw before the onset of loading) was determined according to the analytic solution described in previous section, assuming now that ωo is not arbitrarily prescribed (as it was considered in Eqs.( 1)), but it is rather provided, together with Pc, by the relevant compact disc-jaw contact problem [5].Then a uniform displacement was applied at the steel jaw (simulating in fact the load exerted by the loading frame during the laboratory experiments with the device suggested by ISRM for the standardized Brazilian-disc test), the magnitude of which was determined by demanding that it should result to a vertical displacement of point A equal to the respective one provided by the analytic solution.The results of the numerical analysis for the distribution of pressure along the common contact arc, developed at the ring-jaw interface are plotted in Fig. 5, in conjunction with the parabolic pressure applied on the ring during the theoretical solution.It is seen from this figure that the two distributions are in relatively good agreement, despite some differences concerning their amplitude (equal to about 140 MPa for the analytic solution and 122 MPa for the numerical one) and also their flatness (the numerically obtained distribution appears somehow flatter).These differences should be expected, given that the parabolic distribution is an approximation of the actual "cyclic" distribution [5,6] (developed during the contact of two elastic cylindrical bodies), which was adopted in an attempt to achieve closed-form analytic expressions for the distribution of stresses a displacements all over the disc [7,8].(It is here recalled, that for the actual "cyclic" distribution analytic expressions for the stress field cannot be straightforwardly obtained [5]).
Figure 5: The applied pressure applied on the ring based on both the analytical and the numerical analysis.
Besides the above mentioned slight differences, it is quite astonishing that the analytic and the numerical approaches provide almost identical results for the "total" pressure applied on the ring indicating the static equivalency of the two solutions.Even more astonishing is the fact that the length of the contact arcs obtained by the two solutions (numerical and analytic) is again almost identical: The contact arc 2ω ο determined from the numerical model equals to 24.67 o while the respective one of the analytical solution is equal to 24.59 o .The same excellent agreement was observed, also, for the variation of displacements and stresses along strategic paths of the model, as it can be seen, for example, in Fig. 6, in which analytical versus numerical results concerning the vertical displacement (Fig. 6a) and the polar stresses (Fig. 6b) are plotted for "Path 1", i.e., along the diameter corresponding to the axis of loading symmetry.The differences observed are quite marginal.For example, for the displacements, the difference does not exceed 4% (Fig. 6a) while for the polar stresses the differences are almost tending to zero (Fig. 6b).As an additional validation step, results of the numerical analysis were compared against the respective ones analytically obtained, also, for some combinations of the three loading schemes considered (parabolic pressure, external uniform pressure and internal uniform pressure).For this specific step the jaw was removed and the multi-layered ring was modeled as an isolated elastic body (for convenience).The following loading schemes were imposed on the ring: Case 1: Pressure was applied on the nodes of the outer perimeter of the outermost ring along a contact length 2ωο which was assumed equal to 20 o .The parabolic pressure variation corresponds to a statically equivalent force equal to 5 kN applied by the loading frame.Case 2: Internal pressure equal to 10 MPa was applied on all nodes along the inner perimeter of the three-layered ring.Case 3: External pressure equal to 10 MPa was applied on all nodes along the outer perimeter of the three-layered ring.The normal polar stresses (both radial and transverse) obtained by the numerical model for all six loading cases mentioned above are presented in Fig. 7 in juxtaposition to the respective analytical results.The plots are realized, as in Fig. 6, along the diameter corresponding to the symmetry axis of the loading distribution.It is clear that the correlation between analytic and numerical results is quite satisfactory, for both stresses and for all loading schemes without any exception.Recapitulating the data of the validation procedure, it can be definitely concluded that the numerical model approaches the analytic solution in an excellent manner.Therefore, it is safe for the model to be used for a thorough parametric analysis of the role of the (quite a few) factors, which are expected to crucially affect the overall mechanical response of the multi-layered ring, in case it is subjected to various combinations of the three constituent loading schemes considered (internal and external pressure and parabolic variation of radial stresses).Moreover, the results of the present section, besides validating the numerical model, indicated that the actual stress distribution along the contact arc at the ring-jaw interface is rather insensitive to whether the disc squeezed between the ISRM jaws is compact or hollow, i.e., ring (either it is homogeneous or it consists of successive layers with different mechanical properties), at least for low to moderate values of the externally applied load and provided relatively thick-walled rings are considered.

Parametric numerical analysis
The as above exhaustively validated numerical model is now used for a parametric analysis.In this direction, a series of new models was constructed for the most complicated loading constituent, i.e., the parabolic pressure.Three parameters were taken into account in the present study, the influence of which is expected to be more crucial.These parameters are: b.The thickness of the intermediate ring, t 2 =R 3 -R 2 , keeping the ring's outer diameter constant, i.e., increasing t 2 results to decrease of the thickness of the outermost ring, t 3 =R 4 -R 3 .c.The contact length, or equivalently the length of the loaded arc, 2ωo, assuming that the overall externally imposed load is kept constant.In this case the jaw was not taken into account.Concerning the distribution of the pressure applied along each contact arc, it was determined by taking advantage of Eqs.(1).The numerical values assigned to each one of the three parameters studied are summarized in Tab. 1.In addition, the variation of the applied pressure for the values considered for the length of the half contact arc ω o is plotted in Fig. 8.As it is expected, by decreasing the length of the loaded arc (while keeping the overall load constant) the amplitude of the distribution increases, leading to increased stress concentration in the immediate vicinity of the loaded arc, which in turn increases the possibility for premature failure in this region of the ring.This aspect should be carefully taken into account in case the external layer of the ring is made of relatively brittle materials. Parameter

RESULTS AND DISCUSSION
The influence of the elastic modulus of the intermediate ring, E 2 he displacement along the loading axis (i.e., y-axis) developed all over the ring's area is presented in Fig. 9a for the reference model.Its variation along y-axis for various E 2 values is exhibited in Fig. 9b.It is interesting to observe that, although the values of E 2 vary within a relatively wide range (from -50% to +100% of the respective value of the reference model), the displacements along y-axis are more or less insensitive to this variation.The maximum difference does not exceed 3% and it is observed for E 2 =20 GPa.On the contrary, the stresses developed in the ring are much more sensitive to the changes of E 2 .To enlighten this point (and considering that the polar reference system is much more convenient for the specific application), the variation of the transverse stress (σ θ ) along the loaded diameter is plotted in Fig. 10 for all E 2 values studied.It is seen that the influence of E 2 is very strong, especially at the interface points between the constituent rings.As it is perhaps expected, the "stress jump" T is amplified as the difference of the modulus of elasticity of ring 2 from the elastic modulus of the adjacent ring increases.Therefore, the variation of σθ from ring 1 (innermost) to ring 2 (intermediate) is smoother when E2=20 GPa while from ring 2 to ring 3 (outermost) the variation is smoother when E 2 =5 GPa.It is also noted that σ θ is negative all along rings 2, 3 (along the specific path) for all E2 values studied, while it changes sign somewhere along ring 1.The variation of the polar stresses developed along the interfaces is quite interesting.For the interface between rings 1, 2 and more specifically along the outer periphery of ring 1 (Fig. 11a), σ θ is affected more severely.When the value of E 2 is between the elastic moduli of the two surrounding rings, the variation of σθ is again very smooth with small variations.For E2<10 GPa the transverse stress becomes larger as θ increases while the opposite is true for E2>10 GPa.It is also seen from the same figure that the values of σ rθ developed along the interface are of the same order of magnitude with σ θ .As the modulus of elasticity of ring 2 increases, σrθ also increases, attaining its maximum value when θ~65 o .The radial stress also attains its maximum value when θ is equal to about 65 o , although a slight "move" to a lower value of θ is observed as E 2 increases.It is also to be noted that the changes of σ r are not significant all along the outer periphery of ring 1.The variation of stresses along the inner periphery of ring 3 (interface between rings 2 and 3) is plotted in Fig. 11b.It is clear that the variation of both radial and shear stresses is qualitatively similar to the respective one along the outer periphery of ring 1 but their magnitudes are lower.The radial stress attains its maximum value when θ is equal to about 60 o and the differences are mainly detected when θ varies from about 75 o to 90 o .Concerning the shear stress, it attains its maximum value when θ~70 o .In addition, it seems that the exact position of the maximum shear stress "moves" slightly to larger θ values with increasing E 2 .On the other hand, the variation of the transverse stress is monotonous all along the specific path, contrary to what it was observed along the outer periphery of ring 1.

The influence of the thickness of the intermediate ring, t 2
The effect of the thickness of the intermediate ring on the variation of stresses along the outer periphery of ring 1 (Fig. 12a) is very similar to the one presented in Fig. 11a (where the parameter studied was the modulus of elasticity of ring 2) both qualitatively and quantitatively.For t 2 =4 mm (reference model) the variation of σ θ is again smoother.For t 2 <4 mm σ θ increases with increasing θ while for t2>4 mm σθ increases with decreasing θ.The shear stress, on the other hand, increases continuously with increasing t2 and its maximum value is again detected around θ~65 o independently of the values of t2 considered here.Increase of t 2 leads also to increase of the radial stress developed.Again its maximum value seems to "move" slightly towards larger θ-value as t2 increases.
The variation of stresses along the inner periphery of ring 3 is exhibited in Fig. 12b and it is similar to the respective one in Fig. 11b.However, and despite this qualitative similarity, the thickness of the intermediate ring affects more the stress field along the inner periphery of the outermost ring 3. The differences are generally larger, especially around θ=0 o and θ=90 o .This is true for both radial and transverse stresses.On the contrary, the shear stress attains its maximum value at θ~65 o when t2=1 mm while as t2 increases the maximum shear stress appears at slightly larger values of θ.Another interesting point is that the contact length between the outermost ring and the jaw, as well as the pressure applied along it, are only slightly influenced by the thickness of the intermediate ring, as it can be seen in Fig. 13.The lower t 2 is, the smaller the contact length is (i.e., for t2=1 mm, 2ωo=23.34o while for t2=6 mm, 2ωo=24.67o ) and the lower the maximum pressure applied on the ring is (keeping the externally applied displacement on the jaw constant).

The role of the contact length, 2ωo
Keeping the "total" pressure applied constant and at the same time narrowing the contact length ω o , results to interesting changes of the polar stresses developed along the interfaces.It is recalled here that the specific parameter is studied for an isolated ring rather than the ring-jaw complex.Therefore the contact length for the reference model is here set equal to that obtained during the validation procedure, which was equal to 2ω ο =24.60 ο .

Combining Eqs.( 3 -
10) and comparing terms of e ikθ of the same k-order, in the expressions resulting, the following equations are obtained involving ( ) condition of uniform pressure on L 1 : ' terms are missing since are all zero.Before concluding it should be mentioned that: (a) by setting pI=0 in the first of Eqs.(15), the present formulae [Eqs.(15-20)]provide the solution in case internal uniform pressure is zero and the multi-layered ring is only under the action of external uniform pressure p E and external parabolic pressure p(θ) (along two finite arcs of its outer periphery, antisymmetric with respect to x-axis), (b) by setting pE=0 in the second of Eqs.(15), the present formulae [Eqs.(15-20)]provide the solution in case external uniform pressure is zero and the multi-layered ring is only under the action of internal uniform pressure p I and external parabolic pressure p(θ), (c) by setting pI=0 in the first of Eqs.(15) and pE=0 in the second of Eqs.(15), the present formulae [Eqs.(15-20)]provide the solution in the case the multi-layered ring is only under the action of external parabolic pressure p(θ), (d) by setting P c =0 in Eqs.(15, 17 and 18), the present formulae [Eqs.(15-20)]provide the solution in the case external parabolic pressure is zero and the multi-layered ring is only under the action of internal uniform pressure pI and external uniform pressure p E , (e) by setting P c =0 in Eqs.(15, 17 and 18) and p I =0 in the first of Eqs.(15), the present formulae [Eqs.(15-20)]provide the solution in the case internal uniform and external parabolic pressure are zero and the multi-layered ring is only under the action of externally imposed pressure p E and (f) by setting P c =0 in Eqs.(15, 17 and 18) and p E =0 in the second of Eqs.(15), the present formulae [Eqs.(15-20)]

b
are now just provided by only Eqs.(15), for P c = 0, and for: p I ≠ 0, p E ≠ 0 in case (d), p I = 0, p E ≠ 0 in case (e) and pI ≠ 0, pE = 0 in case (f).

Figure 4 :
Figure 4: (a) The variation of the polar stresses developed along "Path 0" (the embedded graph corresponds to the whole length of "Path 0") and (b) the total stress applied on point A (see Fig.3) for various numbers of elements.

Figure 6 :
Figure 6: Analytical vs. numerical results for the (a) vertical displacement and (b) the polar stresses along "Path 1".

Figure 7 :
Figure 7: Analytical vs. numerical results for (a) simple and (b) complex loading schemes.Blue lines correspond to radial stress while red ones correspond to transverse stress.
modulus of elasticity of the intermediate ring, E2.

Figure 8 :
Figure 8: The variation of the pressure applied on the ring for various ωo-values.

Figure 9 :
Figure 9: (a) The distribution of the u y component of the displacement field in [m].(b) The variation of u y along "Path 1" for various values of the elastic modulus of the intermediate ring (E 2 ).The colours of the lines of the graph are in agreement to the colour code of the embedded figure.

Figure 10 :
Figure 10: The transverse stress along "Path 1" for various values of the elastic modules, E 2 , of the intermediate ring.The colours of the lines of the graph are in agreement to the colour code of the embedded figure.

Figure 11 :
Figure 11: (a) The variation of the polar stresses along "Path 2" (outer periphery of the innermost ring) for various E 2 -values.(b) The variation of the polar stresses along "Path 3" (inner periphery of the outermost ring) for various E 2 -values.

Figure 12 :
Figure 12: The variation of the polar stresses developed along (a) "Path 2" -outer periphery of innermost ring and (b) "Path 3" -inner periphery of outermost ring for various values assigned to the thickness of the intermediate ring (t 2 ).

Figure 13 :
Figure 13: The variation of the total pressure applied on the ring for various t 2 -values.

Figure 14 :
Figure 14: The variation of the polar stresses developed along (a) "Path 2" -outer periphery of innermost ring and (b) "Path 3" -inner periphery of outermost ring for various values assigned to the half contact length (ωo).

Table 1 :
The numerical values assigned to each parameter.