The Use of the Collocation Algorithm for Estimating the Deformations of Soil-Shell Objects Made of Corrugated Sheets

Soil-shell objects made of corrugated metal sheets are characterized, among others, by the fact that bearings are not used in them.^[21] Therefore, the support conditions of the shells are inherently unidentified, which means that the freedom of rotation is not fully preserved (hence, the moments), but at the same time, displacements are still possible. In practice, a design (calculation) scheme is strived to be met, that is, the obtaining of an articulated and non-sliding support in the building under construction. The paper considers the construction phase of the soil-steel structure, which is when the largest internal forces and displacements are created in the shell. However, the geometric diversity of this type of structure often requires an individual approach to the discussed problem.^[25] Additionally, the soil-structure interaction of a buried structure is affected by: the material, size and stiffness; by the construction method; by the type and placement of the backfill material; and by the external loading.^[3,5,6,15] Time-dependent analysis is described in [14]. Many works concern the problem of adjusting the soil-steel interaction.^[1,25]

During the construction phase of a soil-shell structure, deformation of the shell is a random phenomenon. Soil impact resulting from different levels of ground^{[17,19,20,22]} on both sides of the shell, and also the compaction of backfill, are usually not symmetrical.^[23] Therefore, the function of the interaction of soil on the shell can be varied. Using the measuring base that is located on the shell gives the opportunity to observe the internal forces and displacements of the corrugated metal sheet of an object,^[9,13] as well as the interaction between the backfill and the shell.^[10,11]

To monitor shell structures with the largest spans, strain gauges and surveying techniques are used simultaneously.^[13] In the case of measuring geodetic displacements, internal forces are determined as an additional result. In turn, internal forces, and hence additional displacements, are determined on the basis of measuring unit deformations.^[9,11]. The paper shows the use of both measurement techniques in order to validate the collocation algorithm.^[11,12] The solution uses the finite difference method (FDM)^[4,7,16,18] when applied to beam systems. Therefore, an approximate solution is obtained from the FDM results, and thus, the differential coefficient η, which indicates a deviation from the exact FEM result, is used in the paper. The use of the collocation method^[2,11] with differential correction allows shell deformations in soil-shell structures, as well as in other construction models such as beam systems, to be correctly mapped.

The main advantage of the collocation algorithm and the differential solution is the ability to analyse a selected fragment of an object's structure, for example, one beam, without taking into account the boundary conditions in its contact with the rest of the structure. The loads of such a bar may not be precisely defined – they are coded in the bending moment diagram (in the change of the curvature), as is the case in the examples presented in the paper.

The paper uses the relation of bending moments in point i that are calculated on the basis of the deflection line in the vicinity of this point using a differential approach^[4,16] with the form of: (1)Mi(FDM)=−E⋅κi=−Ec2(wi−1−2wi+wi+1),M_i (FDM) = - {E} \cdot \kappa _i = {{ - {\bf {\it E}}} \over {c^2 }}\left( {w_{i - 1} - 2w_i + w_{i + 1} } \right), where: EI is the bending stiffness of the bar, while c is the distance between points in the mesh of an element. Formula (1) is an equivalent to the differential moment dependence and the deflection function, as given below: (2)M=−EId2wdx2.M = - EI{{d^2 w} \over {dx^2 }}.Equation (2) is an exact approach to the relationship of moments and the deflection in a beam.

An approximate solution is obtained from Formula (1). Table 1 summarizes the results that were obtained from the calculation of three types of loads applied to a scheme of a simply supported beam. When it is assumed in the scheme that the beam is divided into two parts, c = L/2 is obtained. The end points are the supports with numbers 0 and 4, and therefore, w₀ = w₄ = 0. With such assumptions, a bending moment in middle point 2 is obtained from the following dependence (3)M2=η⋅M2(FDM)=η4EIL22w2.M_2 = \eta \cdot M_2 (FDM) = \eta {{4EI} \over {L^2 }}2w_2 . In Formula (3), M₂ is an exact value obtained from Formula (2), while moment M(FDM) is calculated from (1). From the proportions of these results, as in (3), the differential coefficients η are determined and given in Table 1.

Table 1 presents the results of calculations when there are four sections of mesh in the beam (as in the scheme), which is subjected to the previously analysed loads. In the case of calculations for middle point 2, as in (3), symmetry is assumed, and hence, w₁ = w₃ and the following equation: (4)M2=η⋅M2(FDM)=η−16EIL22(w1−w2).M_2 = \eta \cdot M_2 (FDM) = \eta {{ - 16EI} \over {L^2 }}2(w_1 - w_2 ). When the bending moment in the intermediate point 1 is analysed, the following formula is used: (5)M1=η⋅M1(FDM)=η−16EIL2(−2w1−w2),M_1 = \eta \cdot M_1 (FDM) = \eta {{ - 16EI} \over {L^2 }}( - 2w_i - w_2 ), This is due to the fact that w₀ = 0. The values of the differential coefficients given in the table show that they depend on the load case, and therefore, on the shape of the bending moment diagram and the density of the beam division mesh into sections c.

Figure 1 shows an example of a simply supported beam loaded with intermittent distributed force q along length b. The division of the beam into sections with lengths L = 6b = 18c was assumed in the calculations. With these assumptions, the following deflection functions are obtained: –

beam section 0 < x < b with the load (6)w=3qc2b22EIxL{17−18[(xL)3+(L−xL)3]+(3L−xL−2)2},w = {{3qc^2 b^2 } \over {2EI}}{x \over L}\left\{ {17 - 18\left[ {\left( {{x \over L}} \right)^3 + \left( {{{L - x} \over L}} \right)^3 } \right] + \left( {3{{L - x} \over L} - 2} \right)^2 } \right\},

Figure 1

Figure 2a shows a scheme of an arch with a shape of a half circle that has radius R. The bending stiffness of arch EI, and the compressive stiffness EA, are constant along its length. The symmetrical load of the distributed force p(s) with a radial direction also causes symmetrical deformation of the shell, which is determined by the function r(s). In the arch, it is determined by the maximum displacements: vertical r_k – the uplift of the crown, and horizontal r_o – the narrowing of a side wall. Figure 2b shows a symmetrical, in relation to the crown, distribution of internal forces along the length of the arch. In general, the effect on arch p(s) is associated with the distribution of axial force N(s) and the distribution of bending moment M(s) along the arch according to the following equation: (11)p(s)=NR+d2Mds2.p(s) = {N \over R} + {{d^2 M} \over {ds^2 }}. In the analysed case of the arch, displacements depend on the internal forces included in the Mohr equation, which has the form of: (12)r=∫M⋅MrEIds+∫N⋅NrEAds+∫κ⋅Mr⋅ds+∫ε⋅Nr⋅ds.r = \int {{{M \cdot M_r } \over {EI}}ds + } \int {{{N \cdot N_r } \over {EA}}ds + \int {\kappa \cdot M_r \cdot ds + } \int {\varepsilon \cdot N_r \cdot ds.} } In this formula, M(s) and N(s) are the internal forces resulting from the load p(s), while M_r(s) and N_r(s) are tracking functions of displacements with a radial direction r. The paper uses the assumption of a negligible influence of axial forces. This is justified in the case of considering the loads located in the lower part of the arch, as in Figure 2a.

Figure 2

The integral approach^[9] is replaced in the paper by a differential relation, as in (10), in the form: (13)Mi=−EI(d2rds2+rR2)=η⋅Mi(FDM)= η−EIc2[ri−1−2ri+(cR)2ri+ri+1].M_i = - EI\left( {{{d^2 r} \over {ds^2 }} + {r \over {R^2 }}} \right) = \eta \cdot M_i (FDM) = \eta {{ - EI} \over {c^2 }}\left[ {r_{i - 1} - 2r_i + \left( {{c \over R}} \right)^2 r_i + r_{i + 1} } \right]. In this case, c is the length of the arch – a segment of a circle with a radius R formed between the points of the regular mesh of the division into elements. In the analysed example, load p is a radial hydrostatic interaction.

Figure 3 presents diagrams of the analysed quantities that were obtained from the FEM solution. The following numbering of points was adopted: 0 – a support, and 15 – a crown, and therefore, a quarter of the circle was divided into 15 segments. From the comparison of bending moments M(s) and M_MRS(s), and with the use of displacements r(s), as in Equation (13), the function η(s) that is shown in Figure 4 is obtained. Therefore, in order to obtain the correct bending moments (FEM) from the differential Equation (13), it is necessary to correct the displacements with the value of differential coefficients.

Figure 3

Figure 4

FEM results in the form of bending moments and displacements – treated as exact values – were used in the previously presented calculation examples. On this basis, the differential coefficients η were determined for various construction and load schemes. The results from the examination of objects, the bending moments obtained from strain gauges, and the displacements obtained from geodetic readings are used in the case of the analyses considered in the paper.

The testing of soil-shell objects made of corrugated sheets, as in Figure 5a, involves the use of strain gauges in the points of the scheme given in Figure 5b. From these measurements, the curvature change is obtained while using the principle of plane cross-sections: (14)−κ=−ME=εg−εDf. - \kappa = {{ - M} \over E} = {{\varepsilon _g - \varepsilon _D } \over f}. and the deformation in the cross-sectional axis of the circumferential beam: (15)ε=12f[(f+1)εg+(f−t)εD].\\varepsilon = {1 \over {2f}}\left[ {(f + 1)\varepsilon _g + (f - t)\varepsilon _D } \right]. These quantities appear in Mohr Equation (12), which is used to calculate displacements r.^[10,24] In the paper, however, they are used to calculate the function of displacements r from relationship (13), as in the given formula: (16)−κiηc2=ri−1−2ri+(cR)2ri+ri+1.{{ - \kappa _i } \over \eta }c^2 = r_{i - 1} - 2r_i + \left( {{c \over R}} \right)^2 r_i + r_{i + 1} . Calculations of the displacements start from the collocation point and the assumption of the displacement r_i of that point. Based on the initially adopted value r_i+1 (adjacent point), the displacement in the adjacent point (i−1) is also obtained from: (17)ri−1=κηc2+[2ri−(cR)2]ri−ri+1.r_{i - 1} = {\kappa \over \eta }c^2 + \left[ {2r_i - \left( {{c \over R}} \right)^2 } \right]r_i - r_{i + 1} . Because the initially adopted value r_i+1 is arbitrary, the appropriate displacement in the second collocation point is not obtained. In order to achieve the compatibility of results in this point, angular transformation is necessary. This transformation is also used to determine the differential coefficient η in Formula (17). Therefore, the displacements at the collocation points obtained from Formula (17) are accurate, and at the remaining points of the mesh, they can be considered as close to the correct values. If these results are similar, it can be seen that both displacement functions – from the geodetic measurements (including collocation ones), and that obtained from (17) – are adjusted.

Figure 5

The difference between integral algorithm (12) and differential algorithm (17) is basic. In (12), there are displacement tracking functions M_r(s) and N_r(s), in which the considered static scheme is treated as a design assumption. The construction of the support of the built shells shows that it is not an articulated and non-sliding connection, which is visible in the examples of measurement results carried out on objects. However, in Formula (17), there are no boundary conditions for the beam. They are replaced by displacements at two collocation points, as in Figure 6. Their position (any) is determined on the basis of, for example, geodetic measurements. It is important that these points are characteristic for the deformation of the beam, for example, the displacement of the crown and the sidewall of the soil-shell structure, as in Figure 6.

Figure 6

The algorithm given in the paper is intended for the analysis of the deformation of a shell during the construction phase of a soil-shell object when using strain gauges and geodetic measurements. In such a case, there are large displacement values, and the influence of axial forces on the strain is small. Such measurements are performed when monitoring a facility during the laying of a backfill, and also during the tracking of changes in the initial phase of the exploitation.^[13]

The analysed example of the object in Ostroda^[9] is the largest structure in Poland. It has the geometric characteristics of a shell with the shape of circle fragments, and a radius of curvature R = 16.632 m (and corner R_n = 6.12 m), span L = 25,724 m, and shell height H = 9.11 m. The geometry of the corrugated metal sheet is expressed by its technical designation UC 500×237×9.65. It is the Ultra Cor type and has parameters: a = 500 mm – wave length, f = 237 mm – wave height, t = 9.65 mm – sheet thickness. During the construction and monitoring of the object, strain gauge measurements were taken in the points given in Figure 6, and geodetic measurements in the points that are distant by 2c.

The results of the analyses given in the paper relate to the construction phase, that is, laying of the backfill, where z_g determines its thickness, as in Figure 5a. Figure 7 shows a comparison of the radial displacement results obtained from geodetic measurements r(uw) and those calculated using Formula (17) – as r(FDM), where z_g= H = 9.2 m. They refer to the bending moments diagram, which is associated with the change of curvature, as in Formula (14), when EI = 9.92 MNm² and the bands have a width of a = 0.5 m. After the angular transformation used in Equation (17) for the selected backfill thicknesses z_g [m] (given in brackets), the following differential coefficients were obtained: η(6.3)=4/7;η(7.5)=4/7;η(9.2)=5/9;η(10.4)=20/41.\eta (6.3) = 4/7;\;\;\;\;\;\eta (7.5) = 4/7;\;\;\;\;\;\eta (9.2) = {\bf 5/9};\;\;\;\;\;\eta (10.4) = 20/41. These values (mean values according to Figure 6) take into account the change in soil interaction on the shell at various backfill levels (when z_g = H, the backfill reaches the level of the shell's crown). Therefore, the bending moments diagram, as well as the form of displacements, change, as in Figure 7. The η values are also influenced by the component r(N) from Equation (12).

Figure 7

The analysed example of the Shumal Bridge object in Dubai, or more precisely in Ras Al Khajmah in the United Arab Emirates,^[23] is the largest in the world. The geometrical characteristics of the shell are as follows: span L = 32.39 m, shell height H = 9.57 m, and radius of curvature R = 29.68 m (and corner R_n = 8.19 m). The type of corrugated metal sheet is defined by its dimensions as UC 500×237×12 [mm], which are marked in Figure 5b (and in the paper) as UC a×f×t. During the construction and monitoring of the object, strain gauge measurements, as well as geodetic measurements were carried out in the points given in Figure 6. The results of the analyses presented in the paper relate to the construction phase, where z_g determines the thickness, as in Figure 5.

Figure 8 presents the diagrams of the radial displacements that were obtained from geodetic measurements for the characteristic five levels of the backfill, which were designated as A, B, ... E. In the facility, various backfill thicknesses on both sides were used, and their values are given in Table 2 – due to the twin system of the shell in the object. For this reason, there was a lack of symmetry in the deformation of the circumferential band of the shell, which was not observed in the object in Ostroda.

Figure 8

Figure 9 shows a comparison of the radial displacement results obtained from the geodetic measurements r(uw) and from calculations using Formula (17), as r(FDM). They refer to the diagrams of the bending moments that are related to the curvature change, as in Formula (14), when EI = 11.4 MNm², and the width of the band a = 0.5 m. Table 2 summarizes the values calculated from the comparison of curvature changes in the shell's crown. The displacement of the diagrams given in Figure 9 may be due to the horizontal displacements of the shell, which are a result of the lack of symmetry in soil interactions on the left and right of the shell.

Figure 9

The design scheme of the shell of the reinforced stone tunnel in Jedlina is given in Figure 2. The shell has a shape of a half circle with R = H = L/2 = 3.5 m. The profile of the corrugated metal sheet is of MP type with 200×55×5.5 [ mm] – where MP a×f×t (wave length, high, sheet thickness). Figure 2 presents the static scheme and results obtained from solving the arc using FEM.

In the study of the analysed object, changes in the shape of the shell during its strengthening were measured using terrestrial laser scanning^[17] and the Riegl VZ-400i device. The result of geodetic measurements were the radial displacement fields of the shell, which were processed into transverse profiles. Figure 10 shows one of the deflection profiles along the circumferential band of the shell. The parameter that determines the position of the analysed point on the arc is coordinate s, which is calculated along the circumference of the circle from the shell's crown (s = 0). Positive values indicate the displacement of the shell into the object. In the displacement line, in the vicinity of the crown, the impact (weight) of the concrete mix being implemented from this area is visible.

Figure 10

The diagrams of the deflection and bending moments of the shell that are shown in Figure 10 significantly differ from the results obtained from the calculations of the object model when using FEM, as shown in Figure 3. For the calculation of the bending moments, as shown in Figure 10, radial displacements r(s) and differential analysis, as given in Formula (1), were adopted. The constant bending stiffness EI = 103 kNm² of the circumferential band with a sheet width of a = 0.2 m was used. Therefore, in the object, the shell's support is not an articulated and non-sliding node. These reasons are the cause for the significant discrepancies between the calculation results, obtained from FEM, and the values of measured displacements, such as the shell's deformations.

Figure 11 presents the results of the analysis of the impact of the mesh density of the FDM elements when adopting the algorithm given in Formula (17). Two collocation points were used in the calculations: one in the crown (s = 0), and the other when s = −3.5 m. In this case, the curvature change was assumed based on the bending moments from Figure 10 and the relationship κ = M/EI. Due to the fact that the moments were determined on the basis of displacements r from Figure 10, full compliance of the diagrams is obtained with a dense mesh of the division into sections. Therefore, the differences of the diagrams that are visible in Figure 11 are only caused by the influence of the mesh and parameter c that are given in the key of the diagram. The direct cause of the differences is the complex function of the curvature change. Therefore, when κ(s) is a function with a complex waveform, it is necessary to use a dense measuring mesh. The case discussed in this point is a classic example where a collocation solution and FDM can be used.

Figure 11

In order to measure the internal forces in a shell's corrugated metal sheet, strain gauge measuring techniques are usually used. Due to significant displacements of the flaccid shell, geodetic measurements are also used. The displacement estimation algorithm that is discussed in the paper is based on the use of strain gauge (or fibre optic) measurements. It is intended for bar systems in which the design support conditions are not met. Collocation points in such a solution are used to consider the bar without constraints, but with specific changes in curvature, as the reference system. For this purpose, the transformation from the computational to the measuring system, that is, collocation points, is used. A small participation of axial forces in the displacements is beneficial for the algorithm.

Three examples of analysing the corrugated sheet shell are given in the paper. Good mapping of the deformation with the use of the collocation algorithm was indicated. In the case of a dense mesh and with a regular distribution of measuring points, it is convenient to use a differential approach. The analysed examples show a significant discrepancy between the support conditions of the FEM calculation model and the actual static conditions in the object at the construction site. Such analyses are performed when monitoring an object during the laying of the backfill and the tracking of changes in the initial phase of construction. The deformation of the shell during the construction is a random phenomenon. Soil impact that results from its different levels on both sides of the shell, and also from the backfill compaction, is usually not symmetrical. Therefore, the function of soil interaction on the shell can be varied.

The presented calculation algorithm can be used in other models of structures, which are modelled as a bar system. The main advantage of the algorithm is the ability to analyse a selected fragment of the structure of the object, for example, one bar without taking into account the boundary conditions in its contact with the rest of the structure. The algorithm is sensitive to measurement errors. Therefore, when there are measurement inaccuracies, the results will not match – apart from in the case of the collocation points, and an unadjusted solution is then obtained.