Effects of Heterogeneity and Nonlinearity on Uplift Characteristics of Shallow Horizontal Anchor Plates

Abstract

Based on the upper bound theorem of limit analysis and variational principle, the uplift resistance of shallow horizontal anchor plates in heterogeneous soil and three-dimensional failure surface of overlying soil are investigated. The results calculated by the theoretical approach are compared with results of the existing solutions and numerical results obtained by software PLAXIS 3D to verify the validity of the variational analysis. Furthermore, the influence of the heterogeneity and nonlinearity on the uplift characteristics of shallow horizontal anchor plates is discussed. The results show that: (i) Increase in the uplift resistance and expansion of failure range are observed with decreasing nonlinear coefficient m and variation coefficient k_t of soil uniaxial tensile strength as well as increasing variation coefficient k_c of soil initial cohesion. (ii) Heterogeneity and nonlinearity jointly determine the shape of failure surface. This study enriches the analytical method of anchor plates and has great significance for stability and safety of anchor plates.

Appendix 1: Three-Dimensional Failure Mechanism of Circular Anchor Plate

1.1 Modelling

When the circular anchor plate is subjected to vertical uplift load, failure mechanism is presented as a three-dimensional axisymmetric failure surface, as shown in Fig. 

Fig. 15

Three-dimensional failure mechanism of shallow horizontal circular anchor plate

15b. The rupture surface is constructed through rotating a curve around the z-axis. The curve function in the xoz plane is expressed as x(z), as shown in Fig. 15a. The buried depth and radius of the circular anchor plate are H and r, respectively; the surcharge pressure is q; the uplift load is P_u and ν is the velocity in the kinematically admissible velocity field.

1.2 Theoretical Derivation of the Uplift Resistance for Circular Anchor Plate in Single-Layer Heterogeneous Soil

1.2.1 Calculation of Internal Energy Dissipation Power

The volume dV and the rupture surface area dS corresponding to the unit thickness dz of the overlying heterogeneous soil for the circular anchor plate are calculated respectively as follows:

$${\text{d}}V = {\uppi }x^{2} \left( z \right){\text{d}}z$$

(33)

$${\text{d}}S = 2{\uppi }x\left( z \right)\sqrt {x^{\prime}\left( z \right)^{2} + 1} \cdot {\text{d}}z$$

(34)

The internal energy dissipation power is written as:

$$D = \int_{0}^{H} {2{\uppi }x\left( z \right)\sqrt {x^{\prime}\left( z \right)^{2} + 1} \cdot \left( {\tau \cos \varphi - \sigma_{n} \sin \varphi } \right)} {\text{d}}z \cdot v$$

(35)

1.2.2 Calculation of Power by External Force

The power of gravity can be obtained using Eq. (36):

$$W_{\gamma } = \int_{0}^{H} {\gamma \left( z \right)} \cdot \left( { - v} \right){\text{d}}V = \int_{0}^{H} {{\uppi }\gamma \left( z \right)x^{2} \left( z \right)} {\text{d}}z \cdot \left( { - v} \right)$$

(36)

The power of the uplift load is calculated as follows:

$$W_{{p_{u} }} = P_{u} \cdot v$$

(37)

The power of the surcharge load can be expressed as follows:

$$W_{q} = {\uppi }U^{2} q \cdot \left( { - v} \right)$$

(38)

1.2.3 Calculation of the Uplift Bearing Capacity

Based on the virtual power theory, we can get:

$$D = W_{\gamma } + W_{{p_{u} }} + W_{q}$$

(39)

Similarly

$$\tan \varphi _{t} = x^{\prime } \left( z \right)$$

(40)

The expression of ultimate uplift capacity P_u can be obtained by substituting Eqs. (35) –(38) and (40) into Eq. (39):

$$P_{u} = \int_{0}^{H} {\left\{ {\left. {2{\uppi }x\left( z \right) \cdot \left[ {\tau - \sigma_{n} x^{\prime}\left( z \right)} \right] + {\uppi }\gamma \left( z \right)x^{2} \left( z \right)} \right\}} \right.} {\text{d}}z + {\uppi }U^{2} q$$

(41)

Note

$$\xi \left( {z,x,x^{\prime},\tau ,\sigma_{n} } \right) = 2{\uppi }x\left( z \right) \cdot \left[ {\tau - \sigma_{n} x^{\prime}\left( z \right)} \right] + {\uppi }\gamma \left( z \right)x^{2} \left( z \right)$$

(42)

$$Q = {\uppi }U^{2} q$$

(43)

Substituting Eqs. (15) and (42) into Eq. (16), the following equations can be obtained:

$$x^{\prime}\left( z \right)= \tan \varphi _{t} = \frac{{c_{0} \left( z \right)}}{{m\sigma _{t} \left( z \right)}}\left[ {1 + \sigma _{n} /\sigma _{t} \left( z \right)} \right]^{{\left( {1 - m} \right)/m}}$$

(44)

$$\sigma_{n}^{\prime } { = }\frac{{{\text{d}}\sigma_{n} }}{{{\text{d}}z}} = - \gamma \left( z \right) - \tau /x\left( z \right)$$

(45)

The boundary conditions are:

$$\begin{gathered} x\left( {z = 0} \right) = r \hfill \\ x\left( {z = H} \right) = U \hfill \\ \end{gathered}$$

(46)

Similarly

$$\left. {\left( {\frac{\partial \xi }{{\partial x^{\prime}\left( z \right)}} + \frac{\partial Q}{{\partial U}}} \right)} \right|_{z = H} = 0$$

(47)

1.3 Theoretical Derivation of the Uplift Resistance for Circular Anchor Plate in Multi-layer Heterogeneous Soil

The failure mechanism of the horizontal shallow circular anchor plate in multi-layer heterogeneous soil is constructed by following principles: the soil is divided into n layers along the depth direction. The radius of the circular anchor plate is r; the buried depth is H; the surcharge load is q; and the uplift load is P_u. The thickness of the i-th layer is h_i = H_i-H_i-1, and the function expression of the soil rupture surface within the range of H_i-1 ~ H_i in the xoz plane is x_i(z), and in the x direction, the failure range at the soil interface H_i is U_i. For the i-th heterogeneous soil layer, the variation of soil parameters also satisfies Eq. (22).

As a result, the expression of the uplift bearing capacity for the circular anchor plate in multi-layer heterogeneous soil is obtained using Eq. (48):

$$P_{u} = \left( {\sum\limits_{i = 1}^{n} {\int_{{H_{i - 1} }}^{{H_{i} }} {\xi_{i} } {\text{d}}z} + Q} \right){ 1} \le i \le n$$

(48)

where i represents the i-th layer of soil, and H₀ = 0, H_n = H.

Note

$$\xi_{i} \left( {z,x_{i} ,x^{\prime}_{i} ,\tau_{i} ,\sigma_{ni} } \right){ = }2{\uppi }x_{i} \left( z \right) \cdot \left[ {\tau_{i} - \sigma_{ni} x^{\prime}_{i} \left( z \right)} \right] + \gamma_{i} \left( z \right){\uppi }x_{i} \left( z \right)^{2}$$

(49)

$$Q = {\uppi }U_{n}^{2} q$$

(50)

By substituting Eqs. (26) and (49) into Eq. (27), the following equations can be obtained:

$$x^{\prime}_{i} = \tan \varphi_{ti} = \frac{{c_{0i} \left( z \right)}}{{m_{i} \sigma_{ti} \left( z \right)}}\left[ {1 + \sigma_{ni} /\sigma_{ti} \left( z \right)} \right]^{{\left( {1 - m_{i} } \right)/m_{i} }}$$

(51)

$$\sigma_{ni}^{\prime } = \frac{{{\text{d}}\sigma_{ni} }}{{{\text{d}}z}} = - \gamma_{i} \left( z \right) - \tau_{i} /x_{i} \left( z \right)$$

(52)

Displacement boundary conditions:

$$\left\{ \begin{gathered} x_{1} \left( {z = 0} \right) = r \hfill \\ x_{i} \left( {z = H_{i} } \right) = x_{i + 1} \left( {z = H_{i} } \right) = U_{i} \left( {1 \le i \le n - 1} \right) \hfill \\ x_{n} \left( {z = H_{n} } \right) = U_{n} \hfill \\ \end{gathered} \right.$$

(53)

Similarly

$$\left. {\left( {\frac{{\partial \xi_{n} }}{{\partial x^{\prime}_{n} \left( z \right)}} + \frac{\partial Q}{{\partial U}}} \right)} \right|_{{z = H_{n} }} = 0$$

(54)

Another variation transversality condition can be expressed as follows:

$$\left. {\frac{{\partial \xi_{i} }}{{\partial x_{i}^{\prime } \left( z \right)}}} \right|_{{z = H_{i} - 0}} = \left. {\frac{{\partial \xi_{i + 1} }}{{\partial x^{\prime}_{i + 1} \left( z \right)}}} \right|_{{z = H_{i} + 0}} \Rightarrow \left. {\sigma_{ni} } \right|_{{z = H_{i} - 0}} = \left. {\sigma_{{n\left( {i + 1} \right)}} } \right|_{{z = H_{i} + 0}}$$

(55)

Based on the above conditions and Runge–Kutta method, the uplift resistance of circular anchor plate can be obtained.