In this article, the composite plate foundation is mainly comprised of: (1) the upper four independent concrete foundations, (2) the protection plate at the bottom, and (3) a sliding layer of suitable thickness, which is usually composed of sand and gravel and situated in between the concrete foundations and the protection plate (refer to Figs. 1-3).

As it has been discussed earlier, the existing methods of elastic thin plate (i.e., four free edges rested on an elastic foundation) are not suitable for the composite plate foundation in mining areas owing to the deformation of the surface-soil complex. Therefore, to calculate the bending moment of a protection plate under the influence of mining subsidence, the plate has been assumed as a beam with a rectangular cross-section, which is rested on an elastic soil, by considering the following assumptions:

1. The soil in the mining areas can be considered as a perfect elastic and the reaction is proportional to its compressional deformation.
2. Only pressure (no tension) is considered on the contact surface between the soil and the foundation subsidence.
3. The plate foundation subsides by having a full contact to the soil, i.e., there is no space between them.
4. It has been assumed that the composite plate foundation is always in an elastic state without considering the damages to its foundation.
5. It has been assumed that the subsidence basin is moving along the vertical direction of the transmission line by considering the force and the deformation under the linear load.
6. As it is almost possible that the surface deformation, the maximum wind load and the ice load are coexisting at the same time, therefore, only the gravity of the tower and its associated lines are considered.
7. By considering the sliding layer between the independent foundation and the protection plate, it has been assumed that the vertical pressure (F_NV) on the top surface of the plate caused by the tower and its lines is equivalent to load q₁ that is uniformly distributed along its longitudinal direction and its distribution length is B (i.e., the width of independent foundation).

After analyzing the moment of the protection plate along the moving direction of the mining basin, the moment of these sections along the vertical direction can be calculated by using the same method.

It has been assumed that the width of the equivalent beam is one-half times to that of the plate’s transverse size. The simple model of composite plate foundation has been shown in Fig. (4) by considering the gravity only (i.e., the deformation of the ground surface is not considered) for onward simple process of estimation. In this article, we do not consider the protruding parts of structure for a 500 mm long section at its outer end, as it has been shown in Fig. (1).

Fig. (4). A simple model of a composite foundation.

The q_JC and q_KZ can be obtained as follows (refer to Fig. 4):

(1)

(2)

When the subsidence leads to a continuous and limited displacement of the ground surface, the interactive model of the composite plate foundation and the subsidence ground as well as the corresponding coordinate system have been shown in Fig. (5). In Fig. (5), s is the distance between maximum subsidence point and left boundary point of the protecting plate. The length of plate along moving direction of the subsidence area is named as l. The width of the single concrete foundation bottom edge is named b.

Fig. (5). The displacement and deformation of the protection plate just before and after its subsidence.

In Fig. (5), it has been assumed that the origin of the coordinate system is located on the lower left end of the plate. In accordance to the classic theory of an expected subsidence, the subsidence curve of the moving basin after the horizontal seam is fully mined [18-20] and it can be assumed as follows:

(3)

Here, L is the horizontal distance between the point of maximum subsidence and the boundary point.

In accordance to the load distribution (refer to Fig. 4), the plate has been divided into the three different sections for its solution. The cross-sectional displacement of the plate is such that x∋[0, b], [b, l-b] and [l-b, l] are denoted as y₁(x), y₂(x) and y₃(x) respectively.

In those three sections, the differential equation of the displacement of the plate for an elastic foundation can be expressed as follows (refer to Eqs. (4)-(9)).

(1) For x∈[0, b],

(4)

The general solution of y₁ is as follows:

(5)

(2) For x∈[b, l-b],

(6)

The general solution for y₂ is as follows:

(7)

(3) For x∈[l-b, l],

(8)

The general solution for y₃ is as follows:

(9)

where,;

To get solution of the twelve undetermined constants from A₁ to A₁₂ as described in the Eqs. (5), (7) and (9), the following boundary conditions need to be considered.

For x =0, at the left end of the protection plate,

(10)

(11)

For x =b, at the inward flange of the left independent foundation,

(12)

(13)

(14)

(15)

For x =l-b, at the inward flange of the right independent foundation,

(16)

(17)

(18)

(19)

For x =l, at the right end of the plate,

(20)

(21)

Thereafter, the following equations can be obtained after substituting the Eqs. (5), (7), and (9) into the Eqs. (10)-(21) of boundary conditions:

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

Next, the solutions of A₁-A₁₂ can be obtained by using Eqs. (22)-(33), whereas their expressions are too complex for estimation purposes. Keeping this in view, the specific parameters including α, l, b, m, s, q₁, q₂, γ_m1, γ_m2, d and k may first be substituted into Eqs. (22)-(33), and, thereafter, the method of elimination can be used by either invoking through programming languages (MATLAB) or other numerical analysis software for determining the aforesaid undetermined constants. Next, the expression of deformation of the plate in each section y₁(x), y₂(x) and y₃(x) can be obtained by substituting the undetermined constants into the Eqs. (5), (7) and (9). Finally, the bending moment at the different cross-sectional points and the counterforce at the bottom of the protection plate can be obtained in accordance with Eqs. (34) and (35) respectively.

(34)

(35)

A composite protection plate, of a transmission tower in mining areas, has been chosen to verify the feasibility of the model and its proposed method as described earlier. The maximum settlement of the surface in the area is assumed as: w₀ = 2m, the half of the subsidence length of the basin is assumed as: L = 200m, the length of the protection plate of the transmission tower is assumed as: l = 20 m, and the thickness is assumed as: H = 0. m. The side length of the upper independent foundation is approximately taken as: b = 5 m and the embedment depth of the foundation is assumed as: d = 3 m. The equivalent load of the upper tower is considered as: q₁ = 5kN/m. The weighted average unit weight of the foundation and the soil is taken as: γ_m1 = 20kN/m³. The unit weight of the filling soil is γ_m2 = 17kN/m³. The elastic modulus of base concrete is considered as: E = 2.55×10¹⁰Pa. The bedding value is approximately taken as: k = 3×10⁴kN/m³. The values of parameter α, m are 0.3575 and 0.1592, respectively. The major results have been shown as follows:

For the purposes of analysis and discussion of the force and deformation conditions of the foundation at different positions in the subsidence areas, the plate has been divided into the three segments along the moving direction of mining subsidence area. Each segment has further subdivided into the four small sections. The arrangement and the serial numbers of the control cross-sections have been shown in Fig. (6). The flexure deformation and the bending moment of the nine control points have been assessed. The distances of the big board from the point of maximum subsidence are 0, L/8, 2L/8, 3L/8, 4L/8, 5L/8, 6L/8, 7L/8 and L. The actual assessment in the case of s=L is the right end of the complex plate and it coincides with the boundary of the subsidence area. The actual assessment in the case of s=0 is the left end of the big plate and it coincides with the maximum subsidence point. Therefore, their corresponding values are 0, 15, 40, 65, 90, 115, 140, 165 and 180 m respectively.

Fig. (6). A schematic diagram of the control sections of the protecting plate.

The aforesaid protection plate has been designed and analyzed by the proposed model. The corresponding undetermined constants of different values of s have been obtained (refer to Table 1).

To verify the correctness of the proposed solution, the following result of s at zero is taken as an example. The undetermined constants have been substituted in Eqs. (5), (7) and (9). The functions of the flexure deformation for the protection plate have been obtained, as these have been shown in Eqs. (36)-(38).

(36)

(37)

(38)

Thereafter, by considering the unit width approximately equal to 1 m, the bottom reaction of the base integral sum of a big plate and the resultant force of the bottom reaction of the plate have been obtained by using the integral method for the bottom reaction of the plate (refer to Eq. (39)).

(39)

where, , and can respectively be estimated by using the Eqs. (40)-(42).

(40)

(41)

(42)

The sum of the basal reaction can be obtained by using the known parameters including α, l, b, m, s, q₁, q₂, γ_m1, γ_m2, d and k by substituting into Eqs. (40)-(42), as it has been shown in Eq. (43).

(43)

The resultant force of the upper load at each increment of 1 m width has been shown in Eq. (44).

(44)

The ratio between the basal reaction sum and the resultant of the upper load is shown as follows:

(45)

This shows that the earlier described solution can satisfy the conditions of the static equilibrium, and, the method proposed in this article is accurate and reliable.

Fig. (7) shows that the relevant changes in the vertical displacement at each cross-section of the protection plate for different positions in subsidence areas. The different s-values of the plate have been calculated based on the ratio between the difference of the vertical displacement at center of an independent foundation and the horizontal distance (between the center points), as it has been shown in Fig. (8).

Fig. (7). The vertical displacement of a cross-section by considering its different positions.

Fig. (8). A relation between the global tilt of a protection plate and its position.

Figs. (7 and 8) show that the whole vertical displacement of subsidence for the protection plate has been occurred due to the gravity as well as the interaction between the upper edge of a uniformly distributed load (when located at the bottom of the basin, s = 0 m) and the edge of the basin (s =180 m), where tilting is negligibly small. The ever increasing tilting happens, when the location changes from the bottom of the basin to its position at s=90 m. The slope deformation is large, when s = 90 m. It is consistent with the maximum tilting of the surface that happens at the central position of the basin, i.e., from its edge location towards its bottom.

Fig. (9) shows that the distribution of the bending moment of the protection plate at each of its cross-section controls it, when it is at different positions of the subsidence basin.

Fig. (9). The distribution of a bending moment at control sections.

Fig. (9) shows that the deformation process of the composite plate foundation experiences from the negative curvature to the positive curvature deformation process, in the whole process of the dynamic deformation. The section of the plate in the vicinity of the inside edge of an independent foundation may be taken to be the control section to design in negative curvature phase and the middle section of big plate should be taken to be the control section to design in negative curvature phase. The plate section can be assumed as a control section for the design purposes during the phase of a positive curvature. The designed internal force can be assumed for the worst scenario of a negative curvature phase during the dynamic deformation of the surface. The bending moment of the plate changes between its positive and negative extremes during the deformation of the surface. Hence, the structure of the foundation may be adopted as follows: for its upper and lower edges, the double reinforced layout scheme with a symmetrical reinforced pattern.

Fig. (10) shows the distribution of the contact pressure between the bottom side of the plate foundation and the soil, when s is ranged between 0 and 180 m.

Fig. (10). The distribution of a contact pressure between the plate foundation and its corresponding soil.

Fig. (10) further shows that the plate is situated in the curvature area, when s is less than 90 m. The distribution of the contact pressure is similar to a U-shape. The contact pressure at both ends is large, whereas it is minimum at central bottom of the plate. The plate is situated at the largest positive curvature position, when s = 40 m. The maximum contact pressure is 88.6 kN/m², whereas the minimum contact pressure is 54.4 kN/m². The distribution of the contact pressure is M-shaped, when s is more than 90 m. The largest pressure is located at the position of an independent foundation and most probably in the central position of an independent foundation. The contact pressure at the end and center of the plate is small. For example, when s=140 m, the maximum pressure is 65.9 kN/m², whereas the minimum is only 53.9 kN/m². Fig. (10) also shows the change of contact pressure at the central position of the plate and it is very small and it has nothing to do with s.