A Bayesian-based approach for inversion of earth pressures on in-service underground structures

Abstract

This paper presents a Bayesian inversion approach to identify earth pressures on in-service underground structures based on structural deformations. Ill-conditioning and non-uniqueness of solutions are major issues for load inversion problems. Traditional approaches are mostly based on an optimization framework where a smooth solution is uniquely determined using regularization techniques. However, these approaches require tuning of regularization factors that may be subjective and difficult to implement for pressure inversion on in-service underground structures. By contrast, the presented approach is based on a Bayesian framework. Instead of regularization techniques and corresponding tuning procedure, only physically plausible bounds are required for specifying constraints. The complete posterior distribution of feasible solutions is obtained based on Bayes' rules. By inferring the potential pressures with the complete posterior distribution, a natural regularization advantage can be shown. Specifically, this advantage is demonstrated in detail by a series of comparative tests: (1) the Bayesian posterior mean exhibits an inherent quality to smooth out ill-conditioned features of inversion solutions; (2) satisfactory inference of the pressures can be made even in the presence of non-uniqueness. These properties are valuable when observed data is noisy or limited. A recorded field example is also presented to show effectiveness of this approach in practical engineering. Finally, deficiencies and potential extensions are discussed.

Appendices

Appendix A: Linear interpolating vector

This “Appendix” presents the commonly-used linear interpolating vector I_z(z):

$$ q(z) = {\mathbf{I}}_{{\mathbf{Z}}} (z){\mathbf{x}} $$

(A1)

where q contains n − 1 pieces of the linear functions on intervals [z_i, z_i+1] (i = 1,…,n − 1), and vector x contains n unknown nodal values (x₁,x₂,…,x_n)^T. Coordinate of the nodes is denoted by z = (z₁,z₂,…,z_n)^T, and spacing between the nodes by Δs_i = z_i+1 − z_i.

The linear interpolant for q at z can be written as:

$$ q(z) = \sum\limits_{i = 1}^{n - 1} {a_{i} x_{i} + b_{i} x_{i + 1} } $$

(A2)

where,

$$ a_{i} = \left\{ {\begin{array}{*{20}c} {\frac{{z_{i + 1} - z}}{{\Delta s_{i} }}} & {z_{i} \le z \le z_{i + 1} } \\ 0 & {else} \\ \end{array} } \right., $$

(A3)

$$ b_{i} { = }\left\{ {\begin{array}{*{20}c} {\frac{{z - z_{i} }}{{\Delta s_{i} }}} & {z_{i} \le z \le z_{i + 1} } \\ 0 & {else} \\ \end{array} } \right., $$

(A4)

For mathematical convenience, matrix is adopted, and thus:

$$ {\mathbf{I}}_{{\mathbf{Z}}} (z) = (1, \ldots ,1)_{1 \times (n - 1)} \left[ {\begin{array}{*{20}l} {a_{1} } \hfill & {b_{1} } \hfill & {} \hfill & {} \hfill & {} \hfill \\ {} \hfill & {a_{2} } \hfill & {b_{2} } \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & \ddots \hfill & {} \hfill & {} \hfill \\ {} \hfill & {} \hfill & {} \hfill & {a_{n - 1} } \hfill & {b_{n - 1} } \hfill \\ \end{array} } \right]_{(n - 1) \times n} $$

(A5)

Appendix B: “Beam on elastic foundation” model

The partial differential governing functions of “beam on elastic foundation” model can be described as

$$ EI(z)\frac{{d^{4} y}}{{d{\kern 1pt} z^{4} }} + k(z)y = q(z) $$

(B1)

where EI is the flexural rigidity of the beam that may vary with depth z, y represents the deflection function of the beam. k(z)y is the reaction of the foundation, and k is the foundation stiffness. q is the pressure field determined by load parameters x (Eq. A1).

Discretization of (B1) using finite element method:

$$ {\mathbf{d}}^{\prime} = {\mathbf{K}}^{ - 1} {\mathbf{f}}\left( {{\mathbf{I}}_{{\mathbf{z}}} (z){\mathbf{x}}} \right) $$

(B2)

where d′ is the predicted deformation vector under pressure field q. f is a vector-valued function where f (I_z(z)x) is equivalent to q (i.e., I_z(z)x) with the transformation rules of virtual work. K is the global stiffness matrix that is assembled by element stiffness matrix k^e, consisting two parts:

$$ {\mathbf{k}}^{e} {\kern 1pt} = {\mathbf{k}}_{b} + {\mathbf{k}}_{f} $$

where k_b represents beam stiffness matrix. k_f is closely resembles the beam mass matrix due to the term k(z)y in (B1). Derivation of them has been presented in Griffiths [8]:

$$ \left. {\begin{array}{*{20}c} {k_{b(ij)} = \int_{{z_{e} }}^{{z_{e} + L}} {EI(z)\frac{{d^{2} {\kern 1pt} N_{i} }}{{d{\kern 1pt} z^{2} }}\frac{{d^{2} {\kern 1pt} N_{j} }}{{d{\kern 1pt} z^{2} }}} dz} \\ {k_{f(ij)} = \int_{{z_{e} }}^{{z_{e} + L}} {k(z)N_{i} N_{j} } dz} \\ \end{array} } \right\}i,j = 1,2,3,4 $$

(B3)

where L is the length of an individual beam element, z_e is the coordinate of element e, ξ = z–z_e, and

$$ \begin{array}{*{20}c} {N_{1} = \frac{2}{{L^{3} }}\xi^{3} - \frac{3}{{L^{2} }}\xi^{2} + 1,} & {N_{2} = \frac{1}{{L^{2} }}\xi^{3} - \frac{2}{L}\xi^{2} + \xi } \\ {N_{3} = - \frac{2}{{L^{3} }}\xi^{3} + \frac{3}{{L^{2} }}\xi^{2} ,} & {N_{4} = \frac{1}{{L^{2} }}\xi^{3} - \frac{1}{L}\xi^{2} ,} \\ \end{array} $$

(B4)

f is also assembled by element forces f^e(I_z(z)x) that is equivalent to the pressures q with the transformation of virtual work equation:

$$ {\mathbf{f}}^{e} \left( {{\mathbf{I}}_{{\mathbf{z}}} (z){\mathbf{x}}} \right) = \left[ {\begin{array}{*{20}c} 1 & 0 & { - \frac{3}{{L^{2} }}} & {\frac{2}{{L^{3} }}} \\ 0 & 1 & { - \frac{2}{L}} & {\frac{1}{{L^{2} }}} \\ 0 & 0 & {\frac{3}{{L^{2} }}} & { - \frac{2}{{L^{2} }}} \\ 0 & 0 & { - \frac{1}{L}} & {\frac{1}{{L^{2} }}} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {F_{p0} } \\ {F_{p1} } \\ {F_{p2} } \\ {F_{p3} } \\ \end{array} } \right\} $$

(B5)

where,

$$ \begin{array}{*{20}c} {F_{p0} = \int_{{z_{e} }}^{{z_{e} + L}} {q(z)} dz = \int_{{z_{e} }}^{{z_{e} + L}} {{\mathbf{I}}_{{\mathbf{z}}} (z)} {\mathbf{x}}dz} & {F_{p1} = \int_{{z_{e} }}^{{z_{e} + L}} {q(z)\xi } dz = \int_{{z_{e} }}^{{z_{e} + L}} {{\mathbf{I}}_{{\mathbf{z}}} (z){\mathbf{x}}\xi } dz} \\ {F_{p2} = \int_{{z_{e} }}^{{z_{e} + L}} {q(z)\xi^{2} } dz = \int_{{z_{e} }}^{{z_{e} + L}} {{\mathbf{I}}_{{\mathbf{z}}} (z){\mathbf{x}}} \xi^{2} dz} & {F_{p3} = \int_{{z_{e} }}^{{z_{e} + L}} {q(z)\xi^{3} } dz = \int_{{z_{e} }}^{{z_{e} + L}} {{\mathbf{I}}_{{\mathbf{z}}} (z){\mathbf{x}}\xi^{3} } dz} \\ \end{array} $$

(B6)