Theoretical Calculation of RCC Layer Thickness Based on Equivalent Elastic Model and Numerical Study on Dam Deformation

Abstract

The layer-by-layer construction process of roller-compacted concrete (RCC) leads to the formation of layer zones, resulting in the transversely isotropic constitutive behavior of the RCC dam. Crucially, the thickness of the layer zone exerts a non-negligible influence on the overall deformation behavior of the RCC dam structure. An analytical expression for the thickness of the layer zone is derived based on the equivalent elastic modulus theory, and the influence of concrete with the same and different properties on the thickness of the layer zone is investigated. Numerical analysis is employed in conjunction with the RCC dam project to quantitatively assess the influence of layer thickness on the displacement of the dam structure. The numerical results demonstrate that considering the thickness of the layer zone leads to a substantial increase in both horizontal and vertical displacements of the dam. Furthermore, the range of extreme values of layer thickness is established using the variational principle of elasticity mechanics. When the elastic modulus ratio of RCC with layer zones to bulk concrete β is 0.85–0.99, a reasonable thickness of the layer can be obtained. When the weakening coefficient k and β are close to 1, the thickness of the layer is close to 10 cm, which is close in nature to the RCC dam.

1. Introduction

The layer-by-layer rolling construction technique of roller-compacted concrete (RCC) leads to the formation of weak layer zones [1,2,3,4,5]. The mechanical properties of layer zone and bulk concrete are quite different, which affects the properties of RCC, such as compressive strength, tensile strength, shear strength and impermeability. Due to the layer zone structure, the dam body exhibits transversely isotropic characteristics and bidirectional elastic anisotropy. The thickness of layer zones affects the elastic modulus of RCC units, and consequently, affects the deformation of the entire dam structure [6,7,8]. Therefore, theoretical research on the thickness of layer zones is essential.

Based on the series and parallel models of composite materials, Gu et al. [9] proposed a calculation method for determining the thickness, instantaneous elastic modulus, and delayed elastic modulus of the layer zone, based on the gradient characteristics of the RCC layer zone. Wei et al. [10] established a rheological analysis model for the layer zone based on the series–parallel concept in composite material mechanics. Zhou et al. [11,12] regarded the RCC layer as a transversely isotropic body and proposed the theoretical solution for the equivalent deformation parameters of the layer zone. A sensitivity analysis of the main calculation parameters of the deformation of an RCC dam was carried out. It was found that the elastic modulus of the body and the elastic modulus of the layer zone are the most sensitive to the deformation of the dam.

Liu et al. [13] converted the layered structure into the equivalent continuum model of a transversely isotropic body for calculation, in accordance with the energy-equivalent continuum modulus. Liu et al. [14] converted the layered structure into an equivalent continuum model of a transversely isotropic body for calculation, based on the principle of displacement equivalence, and the results showed that the elastic parameters are independent of the thickness of the layer zone. Li et al. [15] established an analysis model for the elastic calculation parameters of typical rolled layers by considering the spatial gradient characteristics of RCC and combining the series–parallel models of composite material mechanics, and solved the elastic gradient parameters of RCC dams by using the particle swarm optimization algorithm. Peng et al. [16,17] assumed that the dam body satisfies the constitutive relation of transversely isotropic media, derived the range of layer zone thickness by establishing a thin-layer calculation model based on the principle of minimum potential energy and the principle of minimum complementary energy, and inversed the horizontal and vertical elastic modulus of the dam body, as well as the elastic modulus and thickness of layer zones in combination with finite element calculation. Liu et al. [18] established a discrete element model for the double-layer rolling and shear resistance tests of RCC, and analyzed the meso-mechanism of RCC layer zone formation during construction. The results showed that rolling parameters exert a certain influence on the embedment value, compaction characteristics, layer stress, and layer bonding quality of RCC. The reduction of rolling thickness is conducive to improving the layer bonding quality, and rolling thickness is the most sensitive parameter affecting the layer bonding quality. Azizmohammadi M [2] and Bayqra S H et al. [19] analyzed the influence of the water–cement ratio, time interval for concrete, and cushion mortar on the strength and permeability of RCC through an experimental study. The results showed that with the increase in age, the bond strength of joints decreases and the permeability increases. By increasing the cement content of the cushion mortar, the shear and tensile strength were improved, and the permeability coefficient was significantly reduced. Xu et al. [6] proposed a moisture-based prediction method for RCC dam interlayer strength, achieving over 95% accuracy by monitoring the critical 10–20 mm zone, effectively replacing traditional time-based control. Shen et al. [7] investigated the influence of layer interval time on the microstructure and macro-mechanical properties of overlay transition zones in RCC dams, establishing relationships between microcrack characteristics and failure surfaces to provide experimental references for dam construction. Shen et al. [8] conducted in situ tests to investigate the shear properties of RCC dam interlayers under various conditions. Recent advances in roller-compacted concrete (RCC) dam research have integrated moisture-based interlayer strength prediction models with equivalent elastic frameworks to optimize layer thickness, demonstrating that environmental factors critically influence bonding quality and that optimal thickness can be determined. Through the analysis of multiple engineering test data, Jiang [20] found that even if the RCC with layer zones was poured at allowed intervals, the influence of layer zones on the performance reduction of RCC cannot be ignored. The safety assessment of the RCC dam structure must consider the influence of layer zones. The above researchers have considered the influencing factors of the layer zone thickness, but did not give a specific formula for calculating the thickness of the layer zone.

This paper intends to use a combination of theoretical and numerical analysis methods to study and discuss the thickness of the RCC layer zone based on the equivalent elastic model. Combined with the actual RCC dam project, the numerical analysis of the dam deformation considering the influence of the layer zone is carried out to provide practical guidance for the design and construction of the RCC dam project.

2. Calculation Theory of RCC Equivalent Elastic Modulus

2.1. Constitutive Relation of RCC with Layer Zones

The layer is caused by the thin-layer construction of RCC. In fact, the layer is not a surface, but a layer zone with a certain thickness. The layer zone connects the upper and lower concrete to form a roller-compacted layer unit. The rolling unit containing the layer zone is shown in Figure 1, where b is the thickness of the layer zone and B is the thickness of the rolling layer, generally 30 cm.

The construction technology for the layer-by-layer rolling of RCC engineering results in the formation of layer zone structures, and the dam body shows the characteristics of transverse isotropy. The transversely isotropic body is a special case of the orthotropic body, which is reduced from 21 independent elastic parameters to five independent parameters. According to Hooke‘s theorem, the relationship between strain and stress is shown in Equation (1) [21,22,23].

$$\left\{\begin{array}{l}{\epsilon }_x\\ {\epsilon }_y\\ {\epsilon }_z\\ {\gamma }_{xy}\\ {\gamma }_{yz}\\ {\gamma }_{zx}\end{array}\right\}=\left[\left.\begin{array}{cccccc}{\displaystyle \frac{1}{E}}& -{\displaystyle \frac{\mu }{E}}& -{\displaystyle \frac{\mu \prime }{E\prime }}& 0& 0& 0\\ -{\displaystyle \frac{\mu }{E}}& {\displaystyle \frac{1}{E}}& -{\displaystyle \frac{\mu \prime }{E\prime }}& 0& 0& 0\\ -{\displaystyle \frac{\mu \prime }{E\prime }}& -{\displaystyle \frac{\mu \prime }{E\prime }}& {\displaystyle \frac{1}{E\prime }}& 0& 0& 0\\ 0& 0& 0& {\displaystyle \frac{1}{G}}& 0& 0\\ 0& 0& 0& 0& {\displaystyle \frac{1}{G\prime }}& 0\\ 0& 0& 0& 0& 0& {\displaystyle \frac{1}{G\prime }}\end{array}\right]\right.\left\{\begin{array}{l}{\sigma }_x\\ {\sigma }_y\\ {\sigma }_z\\ {\tau }_{xy}\\ {\tau }_{yz}\\ {\tau }_{zx}\end{array}\right\}$$

(1)

where E, G, and µ are the elastic modulus, shear modulus, and Poisson’s ratio perpendicular to the layer zones plane, respectively. E′, G′, and µ′ are the elastic modulus, shear modulus, and Poisson’s ratio parallel to the layer zones plane, respectively.

2.2. Calculation Theory of Equivalent Elastic Modulus

2.2.1. Equivalent Elastic Modulus of Springs

The equivalent elastic modulus is obtained by using the decomposition stiffness method to decompose a complex structure into several structures. The total stiffness of the structure is composed of the stiffness of each substructure to form an equivalent structure. The equivalent structure should ensure that the total thickness of the original structure is the same as the total stiffness [24,25,26,27,28]. The equivalent stiffness is calculated according to the method for parallel and series connection of springs. The series and parallel models of springs are shown in Figure 2.

The spring stiffness refers to the force required for a spring to produce a unit elongation. K₁ and K₂ refer to the stiffness of Spring 1 and Spring 2, respectively. When two springs are connected in series, the equivalent stiffness of the series structure is assumed to be K, and the total deformation of the two springs under the action of force P is S. The equivalent stiffness K of the springs in series can be derived as follows:

$${\displaystyle \frac{1}{K}}={\displaystyle \frac{1}{K_1}}+{\displaystyle \frac{1}{K_2}}$$

(2)

When two springs are connected in parallel, the two springs produce the same deformation S under the action of force P. Assuming that the equivalent stiffness in parallel is K, the formula for the equivalent stiffness of springs in parallel can be derived as follows:

K = K₁ + K₂

(3)

2.2.2. Equivalent Elastic Modulus of Concrete

The calculation method for the spring equivalent stiffness is also applicable to the calculation of the equivalent elastic modulus of concrete. It is assumed that the thickness and width of the two concrete blocks are the same, the thickness is l_t, the width is l_b, the length of the two concrete blocks is l_a1 and l_a2, and the unit deformation length is $\mathrm{\Delta }l$. When the two concrete blocks are connected in series, as shown in Figure 3a, the equivalent stiffness K of the two concrete blocks is given in Equation (4).

$$K={\displaystyle \frac{E_{ec}{l}_b{l}_t\mathrm{\Delta }l}{l_{a_1}+l_{a_2}}}$$

(4)

In the formula, E₁ and E₂ are the elastic modulus of Concrete Block 1 and Concrete Block 2, respectively; l_a1 and l_a2 are the lengths of Concrete Block 1 and Concrete Block 2, respectively; and E_ec is the equivalent elastic modulus of the series structure.

The formula for the equivalent elastic modulus of concrete structures in series can be derived from Equations (2) and (4):

$${\displaystyle \frac{l_{a_1}}{E_1}}+{\displaystyle \frac{l_{a_2}}{E_2}}={\displaystyle \frac{l_{a_1}+l_{a_2}}{E_{ec}}}$$

(5)

When two concrete blocks are connected in parallel, as shown in Figure 3b, the equivalent stiffness K of the two concrete blocks is shown in Equation (6).

$$K={\displaystyle \frac{E_{eb}\left(l_{a_1}+l_{a_2}\right)l_t\mathrm{\Delta }l}{l_b}}$$

(6)

where E_eb is the equivalent elastic modulus of the parallel structure.

The formula for the equivalent elastic modulus of concrete structures in parallel can be derived from Equations (3) and (6):

$$E_1{l}_{a_1}+E_2{l}_{a_2}=E_{eb}\left(l_{a_1}+l_{a_2}\right)$$

(7)

2.2.3. The Elastic Modulus of the RCC Layer Zone

The layer structure of RCC makes the dam body present transversely isotropic properties. The difference in elastic modulus between the direction parallel to the layer zones and that perpendicular to them affects the strength and stability of the RCC [29,30]. According to the calculation method for the equivalent elastic modulus of concrete, the equivalent elastic modulus of the RCC unit with the layer zone is calculated, and its equivalent structure is shown in Figure 4. The RCC unit consists of three parts: the upper concrete, the lower concrete and the layer zone. The elastic modulus of the three parts is E₁, E₂, and E_b. The thickness of the RCC compacted layer is B, and the length of the layer zones is b. The length of the upper and lower concrete is the same as a.

Assuming that the equivalent elastic modulus in the direction perpendicular to the layer zone is E_V, and the equivalent structure in this direction is in series form, it can be obtained from Equation (5).

$${\displaystyle \frac{a}{E_1}}+{\displaystyle \frac{a}{E_2}}+{\displaystyle \frac{b}{E_b}}={\displaystyle \frac{B}{E_V}}$$

(8)

Let E₂ = nE₁, and the equivalent elastic modulus E_V perpendicular to the direction of the layer zone can be obtained as follows:

$$E_V={\displaystyle \frac{nB{E}_1{E}_b}{a\left(1+n\right)E_b+nb{E}_1}}$$

(9)

Assuming the equivalent elastic modulus parallel to the layer zone is E_H, and the equivalent structure in this direction is in parallel form, it can be obtained from Equation (7).

$$E_{1}a+E_{2}a+E_{b}b=B{E}_H$$

(10)

Let E₂ = nE₁, and the equivalent elastic modulus perpendicular to the layer zones can be obtained as follows:

$$E_H={\displaystyle \frac{a\left(1+n\right)E_1}{B}}+{\displaystyle \frac{b{E}_b}{B}}$$

(11)

From the Equations (9) and (11), the elastic modulus E_b formula of the layer zone can be obtained.

$$E_b={\displaystyle \frac{2nb{E}_1}{\left[2n{E}_1/E_V-(1+n)\right]+(1+n)b}}$$

(12)

3. The Analytical Formula for the Thickness of the Layer Zone

3.1. The Layer Thickness of the Concrete with the Same Properties

For the RCC materials in the same zone, the upper body and the lower body concrete are of the same nature. Assuming that the elastic modulus of the body RCC is E, the elastic modulus of the upper body and the lower body concrete is E₁ = E₂ = E, and the working condition is n = 1. The calculation model is shown in Figure 5.

Assuming that the elastic modulus ratio of the vertical and parallel layers of RCC is a constant λ, then

$$\lambda =E_V/E_H$$

(13)

Assuming that the elastic modulus ratio of RCC with layer zones to bulk concrete is β, then

E_RCC = E_V = βE

(14)

where E_RCC is the RCC elastic modulus with layer zones, and E is the RCC elastic modulus.

The rapid construction technology of the layer-by-layer rolling of RCC forms a layered structure, which makes the dam material show the characteristics of transverse isotropy and two-way anisotropic elastic modulus. It is pointed out in Reference [31] that the elastic modulus in the vertical direction of the hydrostatic roller-compacted concrete dam in the United States is only 0.5 times that in the parallel direction. The elastic modulus in the vertical direction of the Longtan RCC dam in China is 0.8 times that in the parallel direction. Therefore, the value range of parameter λ in this paper is set to 0.5~0.8. According to the experimental data in the literature [32,33], the statistical constant β ranges from 0.86 to 0.99, and the layer zone is a weak zone of concrete. Its value is related to the treatment measures and treatment effect of the concrete layer, and the worse the construction quality, the smaller the coefficient.

From Equations (8) and (9), the equivalent elastic modulus E_V of the vertical layer of RCC and the equivalent elastic modulus E_H of the parallel layer can be obtained:

$$E_H=\beta E/\lambda$$

(15)

According to the series relationship, the equivalent elastic modulus of the RCC unit with the layer zone is calculated, and the elastic modulus E_b of the layer zone can be obtained.

$$E_b={\displaystyle \frac{\beta Eb}{\left(1-\beta \right)B+\beta b}}$$

(16)

According to the parallel relationship, the equivalent elastic modulus of the RCC unit with the layer zone is calculated, and the analytical formula of the thickness b of the layer zone is obtained.

$$b={\displaystyle \frac{(\frac{\beta }{\lambda }-1)(1-\beta )B^2}{(2-\beta /\lambda )\beta B-1}}$$

(17)

where β is the ratio of the elastic modulus of concrete with layer to the elastic modulus of concrete body; b is the thickness of roller-compacted concrete layer with layers; and λ is the elastic modulus ratio of concrete in vertical and parallel planes.

3.2. Layer Thickness of Concrete with Different Properties

For the RCC materials in the transition zone of different regions, the upper body and the lower body concrete have different properties. It is assumed that the elastic modulus of the body roller-compacted concrete is E₁, and the lower body part is E₂. Let E₂ = nE₁, the working condition is n $\ne$ 1, and the calculation model is shown in Figure 6.

The equivalent structure without the layer zone is shown in Figure 6a. The equivalent elastic modulus is calculated according to the series and parallel structures, as follows:

$$E_y={\displaystyle \frac{2n}{1+n}}{E}_1$$

(18)

$$E_x={\displaystyle \frac{1+n}{2}}{E}_1$$

(19)

where E_y is the elastic modulus in the y-direction without the layer zones, and E_x is the elastic modulus in the x-direction without the layer zones.

The formula for the elastic modulus ratio λ between the y-direction and x-direction can be obtained as follows:

$$\lambda =E_y/E_x=4n/{(1+n)}^2$$

(20)

As shown in Figure 6b, for the RCC unit with the layer zone, the elastic modulus E_b of the layer zone in the equivalent structure can be obtained by calculating the elastic modulus according to the series equivalent structure.

$$E_b={\displaystyle \frac{2n{E}_{1}b}{(1+n)(B/\beta -B+b)}}$$

(21)

If n = 1, then Equation (21) is Equation (16).

Then, the equivalent elastic modulus is calculated according to the parallel structure, and the quadratic equation with only parameter b can be obtained.

$${(1-n)}^2{b}^2-{(1+n)}^{2}B(2-\beta -{\displaystyle \frac{1}{\beta }})b-{(1+n)}^2{B}^2(1-\beta )({\displaystyle \frac{1}{\beta }}-1)=0$$

(22)

By solving a quadratic equation of one variable, the analytical formula of the thickness b of the layer is obtained.

$$b={\displaystyle \frac{B(1+n)}{2{(1-n)}^2}}\left(\left(1+n\right)\left(2-\beta -{\displaystyle \frac{1}{\beta }}\right)+\sqrt{{(1+n)}^2{\left(2-\beta -{\displaystyle \frac{1}{\beta }}\right)}^2+4{(1-n)}^2\left(1-\beta \right)\left({\displaystyle \frac{1}{\beta }}-1\right)}\right) (n \ne 1)$$

(23)

3.3. Analytical Formula for Layer Zone Thickness

From Equations (17) and (23), the total analytical formula for the layer zone thickness b can be obtained as follows:

$$\left\{\begin{array}{l}b={\displaystyle \frac{(\frac{\beta }{\lambda }-1)(1-\beta )B^2}{\beta B(2-\beta /\lambda )-1}}(n=1)\\ b={\displaystyle \frac{B(1+n)}{2{(1-n)}^2}}\left(\left(1+n\right)\left(2-\beta -{\displaystyle \frac{1}{\beta }}\right)+\sqrt{{(1+n)}^2{\left(2-\beta -{\displaystyle \frac{1}{\beta }}\right)}^2+4{(1-n)}^2\left(1-\beta \right)\left({\displaystyle \frac{1}{\beta }}-1\right)}\right)(n\ne 1)\end{array}\right.$$

(24)

(1)

n = 1

When n = 1, the b value is related to the values of β and λ, and has nothing to do with the size of the concrete elastic modulus. The relationship between the b value and the values of β and λ is shown in

Table 1. The relationship between the layer zone thicknesses b and β, λ (n = 1).

β	λ = 0.52	λ = 0.53	λ = 0.55	λ = 0.60	λ = 0.70	λ = 0.75	λ = 0.76
	b/cm	b/cm	b/cm	b/cm	b/cm	b/cm	b/cm
0.86	10.39	8.89	6.92	4.01	1.52	0.88	0.77
0.90	10.49	8.79	6.49	3.60	1.41	0.87	0.79
0.94	9.86	7.76	5.32	2.73	1.06	0.68	0.62
0.98	6.66	4.45	2.60	1.17	0.43	0.28	0.26
0.99	4.38	2.67	1.46	0.62	0.23	0.15	0.14

.

It can be seen from Table 1 that when λ ∈ (0.52, 0.76), the larger the values of β and λ, the smaller the thickness b value of the layer zone is, and the closer the performance of the RCC is to the body concrete, which is in line with engineering practice. When n = 1, the relationship between the b value and the β value of the layer thickness is as shown in Figure 7.

As can be seen from Figure 7, the b value decreases with an increasing β value. Furthermore, the larger the value of λ, the smaller the value of b. As the value of β decreases, the rate of increase in the layer zones’ thickness b slows down. The larger the values of β and λ, the smaller the layer zones’ thickness b, which is consistent with the actual situation.

From Figure 7, it can be seen that the b value of the layer thickness decreases with the increase in the β value. The larger the values of β and λ are, the smaller the thickness b value of the layer zone is, which is consistent with the actual situation.

(2)

n ≠ 1

When n ≠ 1, the b value of the thickness of the layer zone is related to the value of β and n. With the change in β, the relationship between the b value and the n value of the layer thickness is shown in Figure 8.

From Figure 8, it can be seen that when the n value is closer to 1, the thickness of the layer zone is larger, and the thickness of the layer zone decreases more obviously with the increase in the β value. When the n value approaches infinity, the elastic modulus of the upper and lower body concrete differs excessively, and there will be deformation coordination problems. The layer thickness is close to 0; the layer cannot be formed, and the upper and lower bodies cannot be bonded.

4. A Numerical Study on Dam Deformation Considering the Layer Thickness

4.1. Dam Material Zoning Without Considering the Influence of the Layer Zones

The typical dam section of an RCC dam is a RCC gravity dam, with a crest width of 15 m, a crest elevation of 630 m, a maximum dam height of 215 m, a total crest length of 990 m, and a total of 39 dam sections. In this paper, the typical dam sections are used for the finite element calculation, and the dam section and material zoning are shown in Figure 9.

The parameters of the dam material, cushion and bedrock in each zone are shown in

Table 2. Material parameters.

Name	Type	Aggregate	Elastic Modulus/GPa	Design Value of Tensile Strength/MPa	Standard Value of Tensile Strength/MPa	Density (kg/m3)	Poisson’s Ratio
Zone 1	C15	Sandstone	10.0	0.91	1.27	2200	0.167
Zone 2	C20	Sandstone	12.8	1.10	1.54	2200	0.167
Zone 3	C25	Sandstone	14.0	1.27	1.78	2200	0.167
Cushion	C20	Limestone	25.5	1.10	1.54	2350	0.167
Bedrock	--	--	9.0	--	--	2500	0.220

.

4.2. Material Zoning Considering the Influence of the Layer Zones

RCC dam Zone 1, Zone 2 and Zone 3 are roller-compacted zones. The construction technology of layer-by-layer rolling is adopted. After considering the influence of the layer, countless rolling units are formed in each rolling zone. Due to the different properties of the upper and lower concrete, a transition zone is formed between the concretes with different properties, and the material zoning considering the influence of the layer zones is shown in Figure 10.

The height of each rolling unit is B (B = 30 cm), and it is assumed that the layer zones are formed in the middle of the roller-compacted unit, and the transition zone is considered as a roller-compacted unit. The calculation model of the rolling unit with the layer zone is shown in Figure 11.

According to the equivalent elastic modulus theory, the two-way elastic modulus of transversely isotropic RCC is calculated, that is, the elastic modulus of parallel and vertical layers. Each rolling zone is composed of numerous identical rolling units. The bidirectional elastic modulus of each rolling zone is equal to the bidirectional elastic modulus of the rolling unit. The transversely isotropic body constitutive relationship is used to consider the influence of the layer zone, so that the layer zone modeling can be ignored and the calculation efficiency can be improved. In this example, the ratio of the elastic modulus of concrete with layers to the elastic modulus of the concrete body β is 0.9. The parallel and vertical elastic modulus ratio λ of RCC is 0.6. The calculated layer zones’ thickness and two-way elastic modulus are shown in

Table 3. Layer zone thickness and elastic modulus.

Type	Elastic Modulus of Body Concrete/GPa		n = E2/E1	Layer Zone Thickness b/cm	Elastic Modulus of Layer Zone Eb/GPa	Ex /GPa	Ey /GPa
	Upper Body E1	Lower Body E2
Zone 1	10.0	10.0	1.00	3.60	0.87	8.90	4.41
Zone 2	12.8	12.8	1.00	3.60	1.11	11.40	5.65
Zone 3	14.0	14.0	1.00	3.60	1.21	12.47	6.18
1–2 transition zone	10.0	12.8	1.28	17.00	9.38	10.26	10.11
2–3 transition zone	12.8	14.0	1.09	26.00	11.85	12.06	12.04
3–4 transition zone	14.0	28.0	2.00	8.10	13.22	18.90	16.80
4–5 transition zone	28.0	9.0	0.32	5.60	8.51	16.65	12.26

.

It can be seen from Table 3 that when n = 1, the thickness B of the layer zones is independent of the elastic modulus. When n ≠ 1, the thickness B of the layer zones is related to the elastic modulus ratio n (n = E₂/E₁). Considering the influence of the layer zones, the dam material is regarded as a transversely isotropic material; the change in the elastic modulus of the vertical layer zones is obvious, and the elastic modulus of the layer zones’ concrete is lower than that of the bulk concrete.

4.3. Calculation Model and Working Load

4.3.1. Finite Element Calculation Model

The dam body is regarded as a plane strain problem for finite element calculation. The dam material is analyzed by the D-P model. The bottom and side of the dam foundation rock are constrained by normal constraints, and the other boundaries are free. The plane182 element in the ANSYS 14.0 program is used, and the finite element model is shown in Figure 12.

4.3.2. Calculation of Working Conditions and Loads

Considering the influence of the layer zone, the displacement and stress changes in the dam body with or without the influence of the layer zone are compared. The working conditions and loads of the finite element calculation are shown in

Table 4. Calculation of working conditions and loads of the finite element.

Working Condition	Upstream Water Level (m)	Downstream Water Level (m)	Load
			Deadweight	Hydrostatic Pressure	Uplift Pressure
No layer	627.00	463.11	√	√	√
Consideration layer	627.00	463.11	√	√	√

.

4.3.3. Dam Deformation Analysis

The horizontal displacement cloud diagram and vertical displacement cloud diagram of the dam body calculated by the finite element method under two working conditions, with or without considering the influence of the layer zones, are shown in Figure 13.

Six points are taken from the dam bottom to the dam crest on the downstream surface. The relationship between the dam displacement and the dam height is shown in Figure 14. Taking the height of the dam bottom as 0 m, it is analyzed that the displacement of the dam body changes with the increase in the height of the dam body, as shown in

Table 5. Calculation results of RCC dam displacement.

Point	Dam Height/m	Without Considering the Layer Zones		Considering the Layer Zones
		Horizontal Displacement/cm	Vertical Displacement/cm	Horizontal Displacement/cm	Vertical Displacement/cm
1	0	0.87	4.93	1.00	4.94
2	48.2	1.11	6.22	1.21	6.94
3	80.3	2.03	6.88	2.29	8.21
4	144.6	4.12	7.38	4.91	9.33
5	191.0	5.45	6.98	6.71	8.86
6	210.0	5.93	7.03	7.41	8.97

.

From Figure 14 and Table 5, it can be seen that in the displacement analysis of six points along the downstream surface of the dam body after considering the layer zone, the horizontal displacement and vertical displacement increase with the increase in the dam body position. The maximum horizontal displacement occurs at the sixth point of the dam crest, and the maximum horizontal displacement increases from 5.93 cm to 7.41 cm, an increase of 1.25 times. The maximum vertical displacement occurs at the fourth point, in the middle and upper part of the dam, which increases from 7.03 cm to 8.97 cm, an increase of 1.26 times. For the safety of the dam structure design, the influence of layer zone thickness should be considered.

5. Discussion on the Extreme Value of the Layer Zone Thickness

5.1. Rolling-Unit Model of Layer Zone

In order to facilitate the study on the extreme thickness of the layer zone, the rolling unit of a transversely isotropic body with a length of 1 m, a width of 1 m, and a height of B is taken as the calculation model, and the rolling-unit model with the layer zone is shown in Figure 15.

It is assumed that the allowable stress is applied on the rolling unit, and the allowable strain is generated under the condition of continuous deformation and displacement boundary conditions. Assuming that the shear stress is zero, the stress and strain relationship of the transversely isotropic body is shown in Equation (25) [34,35,36,37].

$$\left\{\begin{array}{l}{\sigma }_x\\ {\sigma }_y\\ {\sigma }_z\end{array}\right\}=\left[\begin{array}{ccc}{D}_{11}& D_{12}& D_{13}\\ D_{12}& D_{11}& D_{13}\\ D_{13}& D_{13}& D_{33}\end{array}\right]\left\{\begin{array}{l}{\epsilon }_x\\ {\epsilon }_y\\ {\epsilon }_z\end{array}\right\}$$

(25)

where stiffness matrix D are $D_{11}=E(E\prime -\mathrm{E}\mu {\prime }^2)/(1+\mu )\left[(1-\mu )E\prime -2E\mu {\prime }^2\right]$; $D_{13}=E\prime \mathrm{E}\mu \prime /\left[(1-\mu )E\prime -2E\mu {\prime }^2\right]$; $D_{12}=E(\mu E\prime +\mathrm{E}\mu {\prime }^2)/(1+\mu )\left[(1-\mu )E\prime -2E\mu {\prime }^2\right]$; and $D_{33}=E{\prime }^2(1-\mu )/\left[(1-\mu )E\prime -2E\mu {\prime }^2\right]$. E and µ are the elastic modulus and Poisson’s ratio perpendicular to the layer surface, respectively. E′ and µ′ are the elastic modulus and Poisson’s ratio parallel to the layer zone.

5.2. Layer Thickness Extreme Value

The variational method of elasticity lists the energy equation, according to the energy principle, and solves the functional extremum problem. The calculation result is an approximate solution. The principle of minimum potential energy means that under the action of a given external force, all possible displacements satisfying the equilibrium conditions make the total potential energy of the elastic system take the minimum value. The principle of minimum complementary energy means that under the action of external force, the stress satisfying the deformation coordination condition makes the total complementary energy of the elastic system take the minimum value. In this section, the minimum potential energy principle and the minimum complementary energy principle are used to calculate the extreme range of the thickness of the layer zone [16].

5.2.1. Maximum Value

Suppose that the upper-body concrete, lower-body concrete, and layer-zones concrete in the layer zone elements are isotropic, the thickness of the rolling unit is B, E₁ is the elastic modulus of the upper body of the roller-compacted concrete, and a is the thickness of the upper body and lower body of the roller-compacted concrete. E₂ is the elastic modulus of the lower part of the RCC, and b is the thickness of the RCC layer zone; E_b is the rolling-layer elastic modulus; E₂ = nE₁; and E_v is the equivalent modulus. It is assumed that the allowable strain field of the rolling unit with zero shear stress is:

$$\left\{\begin{array}{l}\left[{\epsilon }_y\right]={\epsilon }_y=\epsilon \\ \left[{\epsilon }_x\right]=\left[{\epsilon }_z\right]=\mu \epsilon \end{array}\right.$$

(26)

Let ε₁ = $m_1\epsilon$, ε₂ = $m_2\epsilon$, ε₃ = $m_3\epsilon$, and m₁ = nm₂, and from aε₁ + aε₂ + bε₃ = Bε, we can get Equation (27).

$$\left\{\begin{array}{l}{\epsilon }_1=n{m}_2\epsilon \\ {\epsilon }_2=m_2\epsilon \\ {\epsilon }_3={\displaystyle \frac{B-a{m}_2(1+n)}{b}}\end{array}\right.$$

(27)

where m₁, m, and m₃ are undetermined coefficients, and subscripts 1, 2, and 3 represent upper-body concrete, lower-body concrete, and layer-zone concrete, respectively.

The strain energy U_ε of the element body in the layer zones is:

$$U_{\epsilon }={\displaystyle \frac{1}{2}}\left({\sigma }_{1x}{\epsilon }_{1x}+{\sigma }_{1y}{\epsilon }_{1y}+{\sigma }_{1z}{\epsilon }_{1z}\right)V_1+{\displaystyle \frac{1}{2}}\left({\sigma }_{2x}{\epsilon }_{2x}+{\sigma }_{2y}{\epsilon }_{2y}+{\sigma }_{2z}{\epsilon }_{2z}\right)V_2+{\displaystyle \frac{1}{2}}\left({\sigma }_{3x}{\epsilon }_{3x}+{\sigma }_{3y}{\epsilon }_{3y}+{\sigma }_{3z}{\epsilon }_{3z}\right)V_3$$

(28)

where V₁, V₂, and V₃ are the volumes of the upper-body concrete, the lower-body concrete, and the layer-zone concrete, respectively.

The stress expression obtained from Equation (25) is:

$$\left\{\begin{array}{l}{\sigma }_x=D_{11}{\epsilon }_x+D_{12}{\epsilon }_y+D_{13}{\epsilon }_z\\ {\sigma }_y=D_{12}{\epsilon }_x+D_{11}{\epsilon }_y+D_{13}{\epsilon }_z\\ {\sigma }_z=D_{13}{\epsilon }_x+D_{13}{\epsilon }_y+D_{33}{\epsilon }_z\end{array}\right.$$

(29)

From Equations (28) and (29), Equation (30) can be obtained.

$$U_{\epsilon }={\displaystyle \frac{{\epsilon }^{2}V}{2B}}\left[\begin{array}{l}2a\left[\left(D_{11}^1+D_{12}^1+D_{11}^2+D_{12}^2\right)+2b\left(D_{11}^3+D_{13}^3\right)\right]{\mu }^2+4a\left[n{D}_{13}^1+D_{13}^2-D_{13}^3\left(1+n\right)\right]\mu m_2\\ -{\displaystyle \frac{2Ba}{b}}\left(1+n\right)D_{33}^3{m}_2+a\left(n^2{D}_{33}^1+D_{33}^2+{\displaystyle \frac{D_{33}^{3}a}{b}}{\left(1+n\right)}^2\right)m_2^2+4D_{13}^{3}B\mu +{\displaystyle \frac{D_{33}^3{B}^2}{b}}\end{array}\right]$$

(30)

The undetermined coefficients m₂ and µ can be obtained from the functional extremum ${\displaystyle \frac{\mathit{\partial }{U}_{\epsilon }}{\partial \mu }}=0$ and ${\displaystyle \frac{\partial U_{\epsilon }}{\partial k_2}}=0$. By substituting into Equation (30), the strain energy extremum U_εmin can be obtained as:

$$U_{\epsilon \min }={\displaystyle \frac{{\epsilon }^{2}V}{2B}}f(b)$$

(31)

According to the overall research and analysis, the strain energy in the real state is:

$$U={\displaystyle \frac{E_V{\epsilon }^{2}V}{2}}$$

(32)

According to the minimum potential energy theorem:

$$U\le U_{\epsilon \min }$$

(33)

Substituting Equations (31) and (32) into Equation (33), Equation (34) can be obtained:

$$f(b)\ge B{E}_V$$

(34)

When the upper-body concrete, lower-body concrete, and layer-zone concrete are connected in series, and when b = b_max, $f{(b)}_{\min }=B{E}_V$, and the maximum value of layer zones’ thickness is obtained by substituting into Equation (9):

$$b_{\max }={\displaystyle \frac{\left(1+n\right)E_1-2E_V}{\left(1+n\right)E_1-2E_b}}B$$

(35)

5.2.2. Minimum Value

It is assumed that the allowable stress field of the element body with layer zones is in a unidirectional pure stress state:

$$\left\{\begin{array}{l}\left[{\sigma }_y\right]={\sigma }_y=\sigma \\ \left[{\sigma }_x\right]=\left[{\sigma }_z\right]=\left[{\tau }_{xz}\right]=\left[{\tau }_{xy}\right]=\left[{\tau }_{yz}\right]=0\end{array}\right.$$

(36)

The residual energy U_σ corresponding to the allowable stress site is:

$$U_{\sigma }={\displaystyle \frac{{\sigma }^{2}V}{2B}}\left({\displaystyle \frac{a}{E_1}}+{\displaystyle \frac{a}{E_2}}+{\displaystyle \frac{b}{E_b}}\right)$$

(37)

The complementary energy U corresponding to the real stress field is:

$$U={\displaystyle \frac{{\sigma }^{2}V}{2E_V}}$$

(38)

According to the principle of minimum potential energy:

$$U_{\sigma }\ge U$$

(39)

Substituting Equations (37) and (38) into Equation (39), we can get:

$$b\ge {\displaystyle \frac{E_b\left[2n{E}_1-\left(1+n\right)E_V\right]}{E_V\left[2n{E}_1-\left(1+n\right)E_b\right]}}B$$

(40)

According to Equation (40), the minimum value of the thickness of the layer zones is obtained when taking the equal sign:

$$b_{\min }={\displaystyle \frac{E_b\left[2n{E}_1-\left(1+n\right)E_V\right]}{E_V\left[2n{E}_1-\left(1+n\right)E_b\right]}}B$$

(41)

From Equations (35) and (41), the formula for the thickness extreme range of the layer zones is obtained:

$${\displaystyle \frac{E_b\left[2n{E}_1-\left(1+n\right)E_V\right]}{E_V\left[2n{E}_1-\left(1+n\right)E_b\right]}}B\le b\le {\displaystyle \frac{\left(1+n\right)E_1-2E_V}{\left(1+n\right)E_1-2E_b}}B$$

(42)

5.2.3. Discussion

We discuss the value range of the thickness of the layer zone. Assuming n = 1, B = 30 cm, the elastic modulus of the body concrete E₁ = 10 GPa, $E_V=\beta E_1$, and the elastic modulus of the layer zones is E_b = k E₁ GPa. Equation (42) is included to obtain:

$$b_{\min }={\displaystyle \frac{k(1-\beta )}{\beta (1-k)}}B , b_{\max }={\displaystyle \frac{1-\beta }{1-k}}B$$

(43)

where k is the weakening coefficient of the layer relative to the elastic modulus of the matrix concrete. The extreme value of the layer thickness is discussed when k and β are 0.10–0.99, as shown in Figure 16. The reasonable range of layer zone thickness is indicated by the dashed-line box, specifically where the elastic modulus of the layer zone is at least 0.5 times that of the main body. The layer zone is a weak zone; the strength is too low to be conducive to the safety of the engineering structure, and the minimum b value is less than B/3. The reasonable range of the layer thickness is shown in

Table 6. Reasonable range of layer thickness b.

β	k	b/cm	β	k	b/cm	β	k	b/cm
0.85	0.50	5.30–9.00	0.95	0.50	1.58–3.00	0.99	0.60	0.45–0.75
0.85	0.60	7.94–11.25	0.95	0.60	2.37–3.75	0.99	0.70	0.71–1.00
0.90	0.50	3.33–6.00	0.95	0.70	3.68–5.00	0.99	0.80	1.21–1.50
0.90	0.60	5.00–7.50	0.95	0.80	6.32–7.50	0.99	0.90	2.73–3.00
0.90	0.70	7.78–10.00	0.99	0.50	0.30–0.60	0.99	0.95	5.76–6.00

.

It can be seen from Figure 16 and Table 6 that the reasonable value of layer thickness β is 0.85–0.99, which is consistent with the statistical results of the experimental data in References [32,33]. The larger the k value is, the thicker the layer thickness is, and the closer the layer strength is to the body strength. When k and β are close to 1, the thickness of the layer zone is close to 10 cm, which is close to the nature of the body.

6. Conclusions

In order to consider the influence of layer zones, this paper studies the calculation method for layer zone thickness and the influence of layer zones on dam deformation. The main conclusions are as follows:

(1)

Based on the equivalent elastic model, the analytical formula of the thickness of the layer zone is derived by introducing the three parameters of β, λ and n, according to the two working conditions of concrete with the same property (n = 1) and concrete with different properties (n ≠ 1). When n = 1, the thickness b value of the layer zone decreases with the increase in λ and β values, which is consistent with the actual situation. When n ≠ 1, the closer the value of n is to 1, the β value is larger, and the thickness of the layer zone is smaller. When the n value approaches infinity, the difference between the elastic modulus of the upper and lower concrete is too large, and the problem of deformation coordination will occur. The upper and lower bodies cannot be bonded.

(2)

A finite element numerical analysis is carried out with an example. The results show that the horizontal displacement and vertical displacement of the dam increase significantly after considering the influence of the layer zones. In order to ensure the safety of the dam structure design, the influence of the thickness of the layer zones should be considered.

(3)

The extreme range in the thickness of the layer zone is discussed by using the variational theory in elasticity. When β is 0.85–0.99, a reasonable thickness of the layer is obtained, which is consistent with the statistical results of the experimental data in References [19,20]. The larger the k value is, the thicker the layer thickness is, and the closer the layer strength is to the body strength. When k and β are close to 1, the thickness of the layer is close to 10 cm, which is close to the nature of the body.

This study on the layer thickness of heterogeneous concrete is a valuable research question. In practical engineering, there may even be situations where the elastic modulus ratio of the two types of concrete is greater than 2. How to control the thickness of the layer zone, improve the performance of the layer zone, and solve the problem of deformation coordination is an urgent engineering problem that needs to be solved, and this is also a future research direction.