Accumulative Bayesian detection of displacement constants of a hybrid indeterminate box girder with variable scale gradient theory

Abstract

With general Bayesian theory, the accumulative Bayesian objective function of displacement constants of a hybrid indeterminate box girder was found. The gradient matrix of accumulative Bayesian objective function to displacement constants and the calculative covariance matrix were both derived. The finite curvilinear strip controlling equation of a pinned box girder was derived and the hybrid indeterminate problem of a continuous curvilinear box girder with diaphragm was solved based on agglomeration theory. Combined with one-dimensional (1D) Fibonacci automatic search scheme of optimal step length, the variable scale gradient theory was utilized to research the stochastic detection of displacement constants of the hybrid indeterminate curvilinear box girder. Then the detection steps of displacement constants of the hybrid indeterminate curvilinear box girder were presented in detail and the detection procedure was developed. Through some classic examples, it is achieved that the accumulative Bayesian detection of displacement constants of the hybrid indeterminate curvilinear box girder has perfect numerical stability and convergence, which demonstrates that the derived detection model is correct and reliable. The stochastic performances of displacement constants and structural responses are simultaneously deliberated in an accumulative Bayesian objective function, which proves to have high computational efficiency. The variable scale gradient method incessantly changes the spatial matrix scale to engender new search directions during the iterative processes, which makes the derived accumulative Bayesian detection of the displacement constants more efficient.

The box girder is generally applied in civil engineering and especially the curvilinear box girder often appears in structural engineering.^1,2 The superior mechanical characteristics including larger torsion stiffness and lighter weight naturally exist in this kind of structure.^3,4 The research results of mechanical analysis of the box girder have usually emerged with the improvement of the construction technology.^5,6 The layered shell element based on the supposition of the ignorance of the transversal shearing stress is deduced in Ye et al. ⁷ and the analysis of ultimate loads of a prestressed concrete multi-T girder is accomplished. Then in the study of Zhang, ⁸ the degraded solid element is applied for the box girder, and with the failure criteria of concrete material, the mechanical performances after cracking are studied in detail. Compared with the layered shell element, the finite strip element has fewer nodes and elements in structural analysis, the mechanical responses of the box girder are completed in Zhang et al., ⁹ and, with agglomeration theory, the hybrid indeterminate curvilinear box girder is resolved. ¹⁰ Before the analysis of the box girder, the systematic constants including elastic modulus and Poisson’s ratio must be mastered, otherwise the prearrangement cannot be put into practice.^11,12 However, it is sometimes quite difficult to grasp displacement constants exactly. The displacement constants are often presupposed by engineering experiments and even sometimes subjectively given by engineering experiences, which cannot take the influence of stochastic factors into consideration or cannot accurately reflect the complex practical circumstance.^13–15 How to determine the displacement constants efficiently is worth researching. During the available achievements, the Kalman filtering theory which has the advantages of self-revision and auto-optimization has been applied in the detection of displacement constants, which shows that if the initial constants are improper, the Kalman adaptation matrix might be divergent. ¹⁶ The Powell direct optimization method has been successfully used in the constant detection of the box girder. ¹⁷ In the Powell iterative processes, the improvement of the systematic constants only depends on the revision of the objective function without any search direction, which consequentially leads to lower computational efficiency. ¹⁸ In Zhang et al., ¹⁹ a new search method for optimal step length is employed to improve the computational efficiency, but it cannot alter the inherent disadvantage of the Powell direct optimization method. And fortunately the gradient search methods can conquer the defect just disserted to a certain extent.

Thus, the accumulative Bayesian objective function of displacement constants of the hybrid indeterminate curvilinear box girder is derived and the finite curvilinear strip element for the box structure is obtained. In order to find the detection analytical model of displacement constants of the hybrid indeterminate curvilinear box girder, the variable scale gradient theory is combined with. And through analysis of some classic examples, several laws about accumulative Bayesian detection of displacement constants are recognized and achieved.

Accumulative Bayesian objective function of displacement constants of a hybrid indeterminate box girder

Although the Bayesian statistical method is quite different from the classical statistical methods, the parameters estimated by the referred methods are identical under the condition of large samples. In the case of small samples, the Bayesian statistical method can make full use of all kinds of information and the results are more reliable. The characteristic of the Bayesian statistical method is that it can make full use of existing information, such as general information, empirical information, and sample information, and base statistical inference on posterior distribution. This not only reduces the statistical error caused by the small sample size, but can also be inferred without data samples.^20–23 During the process of the variable scale gradient detection of displacement constants of the hybrid indeterminate curvilinear box girder, the displacement constants are totally considered as stochastic variables, which are recorded as the stochastic vector $\mathbf{E}=[\begin{array}{cccc}{E}_1\hfill & E_2\hfill & \cdots \hfill & E_m\hfill \end{array}{]}^T$ (where m is vector E’s dimension) to take the detection of poly constants. From general Bayesian statistical theory,^24–26 it can be attained as

$$f(\mathbf{E}|{\mathbf{W}}^{*})=\frac{f({\mathbf{W}}^{*}|\mathbf{E})f(\mathbf{E})}{f({\mathbf{W}}^{*})}$$

(1)

where $f(\mathbf{E})$ is the priori distribution function, $f({\mathbf{W}}^{*}|\mathbf{E})$ is the conditional distribution function of the systematic response, $f({\mathbf{W}}^{*})$ is the systematic response distribution function, and $f(\mathbf{E}|{\mathbf{W}}^{*})$ is the posterior distribution function. It is supposed that displacement constants $\mathbf{E}$ are conformed to normal distribution, and then the expression of the priori distribution function $f(\mathbf{E})$ is

$$f(\mathbf{E})=(2\mathit{\pi }{)}^{-\frac{m}{2}}{\left|{\mathbf{C}}_{\mathbf{E}}\right|}^{-1}\exp [-\frac{1}{2}{(\mathbf{E}-{\mathbf{E}}_0)}^T{\mathbf{C}}_{\mathbf{E}}^{-1}(\mathbf{E}-{\mathbf{E}}_0)]$$

(2)

where ${\mathbf{E}}_0$ and ${\mathbf{C}}_{\mathbf{E}}$ are, respectively, the expectation vector and covariance matrix of displacement constants $\mathbf{E}$ .

In practical engineering, the displacements at the measuring nodes must be measured many times and the measured displacement data ${\mathbf{W}}_i^{*}$ of each time are all samples of ${\mathbf{W}}^{*}$ . If the general Bayesian objective function is derived to identify constants, there are many assignments repeated.^27–30 Thus, the accumulative Bayesian objective function of displacement constants of the hybrid indeterminate curvilinear box girder is attained. The joint density function of ${\mathbf{W}}_i^{*}$ is $\underset{i=1}{\overset{n}{\Pi }}f({\mathbf{W}}_i^{*}|\mathbf{E})$ , where n is the tested times of displacement responses. From maximum likelihood theory, it can be obtained

$$f({\mathbf{W}}^{\ast }|\mathbf{E})={(2\mathit{\pi })}^{-\frac{mn}{2}}\prod _{i=1}^n{\left|{\mathbf{C}}_{{\mathbf{W}}_i^{\ast }}\right|}^{-1}\times \exp \left[-\frac{1}{2}\sum _{i=1}^n{({\mathbf{W}}_i^{\ast }-{\mathbf{W}}_i)}^T{\mathbf{C}}_{{\mathbf{W}}_i^{\ast }}^{-1}({\mathbf{W}}_i^{\ast }-{\mathbf{W}}_i)\right]$$

(3)

where ${\mathbf{W}}_i$ is the displacement response vector from the computational result, which is evidently the implicit function of displacement constants $\mathbf{E}$ . Substituting equations (2) and (3) into equation (1), the accumulative Bayesian objective function J can be derived as

$$J=\sum _{i=1}^n{({\mathbf{W}}_i^{\ast }-{\mathbf{W}}_i)}^T{\mathbf{C}}_{{\mathbf{W}}_i^{\ast }}^{-1}({\mathbf{W}}_i^{\ast }-{\mathbf{W}}_i)+{(\mathbf{E}-{\mathbf{E}}_0)}^T{\mathbf{C}}_{\mathbf{E}}^{-1}(\mathbf{E}-{\mathbf{E}}_0)$$

(4)

To obtain the detection result of displacement constants with variable scale gradient theory, from equation (4) the gradient matrix of accumulative objective function J to displacement constants $\mathbf{E}$ is expressed as

$$\frac{\partial J}{\partial \mathbf{E}}=\sum _{i=1}^{n}2{\left(\frac{\partial {\mathbf{W}}_i}{\partial \mathbf{E}}\right)}^T{\mathbf{C}}_{{\mathbf{W}}_i^{*}}^{-1}({\mathbf{W}}_i-{\mathbf{W}}_i^{*})+2{\mathbf{C}}_{\mathbf{E}}^{-1}(\mathbf{E}-{\mathbf{E}}_0)$$

(5)

Expanding ${\mathbf{W}}_i(\mathbf{E})$ in the form of Taylor formula at the expectation point and keeping only the first two items, it can be derived as

$${\mathbf{W}}_i(\mathbf{E})={\mathbf{W}}_i(\overline{\mathbf{E}})+{\mathbf{G}}_i(\overline{\mathbf{E}})(\mathbf{E}-\overline{\mathbf{E}})$$

(6)

where ${\mathbf{G}}_i(\overline{\mathbf{E}})={\partial {\mathbf{W}}_i/\partial \mathbf{E}|}_{\mathbf{E}=\overline{\mathbf{E}}}$ , which is the essential matrix of displacement response to displacement constants. Substituting equation (6) into equation (5), it can be obtained

$$\frac{\partial J}{\partial \mathbf{E}}=\sum _{i=1}^{n}2{\mathbf{G}}_i^T{\mathbf{C}}_{{\mathbf{W}}_i^{\ast }}^{-1}({\overline{\mathbf{W}}}_i+{\mathbf{G}}_i\mathbf{E}-{\mathbf{G}}_i\overline{\mathbf{E}}-{\mathbf{W}}_i^{*})+2{\mathbf{C}}_{\mathbf{E}}^{-1}(\mathbf{E}-{\mathbf{E}}_0)$$

(7)

where ${\overline{\mathbf{W}}}_i={\mathbf{W}}_i(\overline{\mathbf{E}})$ . When the derived objective function J reaches the minimum value, equation (7) becomes equal to zero. Then it can be derived

$$\begin{array}{c}\multicolumn{1}{c}{[\sum _{i=1}^n{\mathbf{G}}_i^T{\mathbf{C}}_{{\mathbf{W}}_i^{\ast }}^{-1}{\mathbf{G}}_i+{\mathbf{C}}_{\mathbf{E}}^{-1}]\mathbf{E}=\sum _{i=1}^n{\mathbf{G}}_i^T{\mathbf{C}}_{{\mathbf{W}}_i^{\ast }}^{-1}}\\ \multicolumn{1}{c}{({\mathbf{W}}_i^{*}-{\overline{\mathbf{W}}}_i+{\mathbf{G}}_i\overline{\mathbf{E}})+{\mathbf{C}}_{\mathbf{E}}^{-1}{\mathbf{E}}_0}\end{array}$$

(8)

Assuming $\mathbf{H}=\sum _{i=1}^n{\mathbf{G}}_i^T{\mathbf{C}}_{{\mathbf{W}}_i^{\ast }}^{-1}{\mathbf{G}}_i+{\mathbf{C}}_{\mathbf{E}}^{-1}$ and $\mathbf{M}={\mathbf{H}}^{-1}$ $[{\mathbf{G}}_1^T{\mathbf{C}}_{{\mathbf{W}}_1^{*}}^{-1},{\mathbf{G}}_2^T{\mathbf{C}}_{{\mathbf{W}}_2^{*}}^{-1},\cdots ,{\mathbf{G}}_n^T{\mathbf{C}}_{{\mathbf{W}}_n^{*}}^{-1}]$ , from equation (8) the detection value $\hat{\mathbf{E}}$ of displacement constants $\mathbf{E}$ of the hybrid indeterminate box girder can be written as

$$\hat{\mathbf{E}}=(\mathbf{I}-\mathbf{MG}){\mathbf{E}}_0+\mathbf{M}{\mathbf{W}}^{*}-\mathbf{M}(\overline{\mathbf{W}}-\mathbf{G}\overline{\mathbf{E}})$$

(9)

where $\mathbf{I}$ is a unit matrix and ${\mathbf{W}}^{*}=[{\mathbf{W}}_1^{*},{\mathbf{W}}_2^{*},\cdots ,{\mathbf{W}}_n^{*}{]}^T$ , where ${\mathbf{W}}_i^{*}$ is the measured vector of displacement responses of the ith time; $\overline{\mathbf{W}}=[{\overline{\mathbf{W}}}_1,{\overline{\mathbf{W}}}_2,\cdots ,{\overline{\mathbf{W}}}_n{]}^T$ , where ${\overline{\mathbf{W}}}_i$ is the computational vector of displacement responses of the ith time at the expectation point $\overline{\mathbf{E}}$ ; $\mathbf{G}=[{\mathbf{G}}_1,{\mathbf{G}}_2,\cdots ,{\mathbf{G}}_n{]}^T$ , where ${\mathbf{G}}_i$ is the sensitivity matrix of the ith time. Supposing that the priori information ${\mathbf{E}}_0$ of displacement constants $\mathbf{E}$ of the hybrid indeterminate curvilinear box girder has no relation with the measured displacement responses ${\mathbf{W}}^{*}$ , from equation (9) the covariance matrix of $\hat{\mathbf{E}}$ can be written as follows

$${\mathbf{C}}_{\hat{\mathbf{E}}}=[\mathbf{I}-\mathbf{MG}]{\mathbf{C}}_{\mathbf{E}}{[\mathbf{I}-\mathbf{MG}]}^T+\mathbf{M}{\mathbf{C}}_{{\mathbf{W}}^{*}}{\mathbf{M}}^T$$

(10)

where ${\mathbf{C}}_{{\mathbf{W}}^{*}}=\mathrm{diag}({\mathbf{C}}_{{\mathbf{W}}_1^{*}},{\mathbf{C}}_{{\mathbf{W}}_2^{*}},\cdots ,{\mathbf{C}}_{{\mathbf{W}}_n^{*}})$ and ${\mathbf{C}}_{{\mathbf{W}}_i^{*}}$ is the covariance matrix of the measured displacement responses of the ith time. Using the non-singularity property of ${\mathbf{C}}_{\mathbf{E}}$ and ${\mathbf{C}}_{{\mathbf{W}}^{*}}$ , equation (10) can be written as

$${\mathbf{C}}_{\hat{\mathbf{E}}}={[{\mathbf{C}}_{\mathbf{E}}^{-1}+{\mathbf{G}}^T{\mathbf{C}}_{{\mathbf{W}}^{*}}^{-1}\mathbf{G}]}^{-1}$$

(11)

Equation (11) can be derived as

$${\mathbf{C}}_{\hat{\mathbf{E}}}={[{\mathbf{C}}_{\mathbf{E}}^{-1}+\sum _{i=1}^n{\mathbf{G}}_i^T{\mathbf{C}}_{{\mathbf{W}}_i^{*}}^{-1}{\mathbf{G}}_i]}^{-1}$$

(12)

The mechanical solution of the hybrid indeterminate curvilinear box girder

Finite curvilinear strip element theory

A finite curvilinear strip element is suitable for mechanical research in the determinate girder shown in Figure 1. The different kinds of $\varphi$ called the conic angle are utilized to reveal different relations between strain and deformation with Novozhilov theory. ¹⁰ The relations are listed as

$$\begin{array}{c}{\epsilon }_x=\frac{\partial u}{\partial x},{\epsilon }_{\theta }=\frac{1}{r}\frac{\partial v}{\partial \theta }+\frac{w\cos \varphi +u\sin \varphi }{r},{\gamma }_{x\theta }=\frac{1}{r}\frac{\partial u}{\partial \theta }\hfill \\ \multicolumn{1}{c}{+\frac{\partial v}{\partial x}-\frac{v\sin \varphi }{r}}\\ {\mathrm{X}}_x=-\frac{{\partial }^{2}w}{\partial x^2},{\mathrm{X}}_{\theta }=-\frac{1}{r^2}\frac{{\partial }^{2}w}{\partial {\theta }^2}+\frac{\cos \varphi }{r^2}\frac{\partial v}{\partial \theta }-\frac{\sin \varphi }{r}\frac{\partial w}{\partial x}\hfill \\ {\mathrm{X}}_{x\theta }=2(-\frac{1}{r}\frac{{\partial }^{2}w}{\partial x\partial \theta }+\frac{\sin \varphi }{r^2}\frac{\partial w}{\partial \theta }+\frac{\cos \varphi }{r}\frac{\partial v}{\partial x}-\frac{\sin \varphi \cos \varphi }{r^2}v)\hfill \end{array}$$

(13)

where the x-axis is the longitudinal direction in the initial section, the y-axis is the tangential direction in the initial section, and the z-axis is vertical to the plane which includes the x-axis and the y-axis; r is the radius variable; ${\theta }_y$ is the angular displacement rotated around the y-axis; u, v, and w are, respectively, the displacements along the x-, y-, and z-axis; ${\epsilon }_x$ and ${\epsilon }_{\theta }$ are, respectively, the strain along the x-axis and the θ-axis; ${\gamma }_{x\theta }$ is the shearing strain in the plane which includes the x-axis and the θ-axis; ${\mathrm{X}}_x$ and ${\mathrm{X}}_{\theta }$ are, respectively, the bending curvature along the x-axis and the θ-axis; and ${\mathrm{X}}_{x\theta }$ is the planar torsion curvature in the plane which includes the x-axis and the θ-axis.

Figure 1.

The determinate pinned curvilinear box.

As for the pinned curvilinear box structure, the displacement interposition function of the finite curvilinear strip element shown in Figure 2 is

$$\mathbf{f}=\sum _{m=1}^r{\mathbf{f}}_m=\sum _{m=1}^r{\mathbf{N}}_m{\mathbf{\delta }}_m$$

(14)

where $\mathbf{f}=[u,v,w{]}^T$ and ${\mathbf{f}}_m=[u_m,v_m,w_m{]}^T$ are, respectively, the deformation vector and the mth deformation vector in the whole coordinate system; ${\delta }_m$ is the peak value deformation vector of the nodal line in the finite curvilinear strip element method; ${\mathbf{N}}_m$ is the shape function which is connected with $Y_m=\sin (m\mathit{\pi }/\alpha )\theta$ and $\alpha$ is the whole structural angle; $\theta$ is the angular coordinate and b is the width of the studied element. The stiffness matrix of the finite curvilinear strip element can be derived as

$$\begin{array}{c}\multicolumn{1}{c}{{\mathbf{K}}_e=\underset{V}{\int }{\mathbf{B}}^T\mathbf{DB}\mathrm{dV}=\underset{V}{\int }[{\mathbf{B}}_1{\mathbf{B}}_2\cdots {\mathbf{B}}_r{]}^T}\\ \multicolumn{1}{c}{\mathbf{D}[{\mathbf{B}}_1{\mathbf{B}}_2\cdots {\mathbf{B}}_r]\mathrm{dV}}\\ \multicolumn{1}{c}{=\underset{V}{\int }\left[\begin{array}{cccc}{\mathbf{B}}_1^T\mathbf{D}{\mathbf{B}}_1\hfill & {\mathbf{B}}_1^T\mathbf{D}{\mathbf{B}}_2\hfill & \cdots \hfill & {\mathbf{B}}_1^T\mathbf{D}{\mathbf{B}}_r\hfill \\ \multicolumn{1}{c}{{\mathbf{B}}_2^T\mathbf{D}{\mathbf{B}}_1}& {\mathbf{B}}_2^T\mathbf{D}{\mathbf{B}}_2\hfill & \cdots \hfill & {\mathbf{B}}_2^T\mathbf{D}{\mathbf{B}}_r\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{{\mathbf{B}}_r^T\mathbf{D}{\mathbf{B}}_1}& {\mathbf{B}}_r^T\mathbf{D}{\mathbf{B}}_2\hfill & \cdots \hfill & {\mathbf{B}}_r^T\mathbf{D}{\mathbf{B}}_r\hfill \end{array}\right]\mathrm{dV}}\\ \multicolumn{1}{c}{=\left[\begin{array}{cccc}{{\mathbf{K}}_e}_{11}\hfill & {{\mathbf{K}}_e}_{12}\hfill & \cdots \hfill & {{\mathbf{K}}_e}_{1r}\hfill \\ \multicolumn{1}{c}{{{\mathbf{K}}_e}_{21}}& {{\mathbf{K}}_e}_{22}\hfill & \cdots \hfill & {{\mathbf{K}}_e}_{2r}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{{{\mathbf{K}}_e}_{r1}}& {{\mathbf{K}}_e}_{r2}\hfill & \cdots \hfill & {{\mathbf{K}}_e}_{\mathrm{rr}}\hfill \end{array}\right]}\end{array}$$

(15)

where $\mathbf{D}$ is the elastic matrix and the strain matrix $\mathbf{B}$ can be determined by substituting equation (14) into equation (13). Then from equation (15), it can be obtained

$${{\mathbf{K}}_e}_{\mathrm{mn}}=\underset{V}{\int }{\mathbf{B}}_m^T\mathbf{D}{\mathbf{B}}_n\mathrm{dV}$$

(16)

Figure 2.

The finite curvilinear strip element.

For the orthogonality of $Y_m=\sin (m\mathit{\pi }/\alpha )\theta$ , equation (15) can be transformed into a principal-diagonal-block matrix as

$${\mathbf{K}}_e=\mathrm{diag}[\begin{array}{cccc}{{\mathbf{K}}_e}_{11}\hfill & {{\mathbf{K}}_e}_{22}\hfill & \cdots \hfill & {{\mathbf{K}}_e}_{\mathrm{rr}}\hfill \end{array}]$$

(17)

The mth controlling equation of the pinned curvilinear box girder is

$${\mathbf{K}}_{\mathrm{mm}}{\mathbf{U}}_m={\mathbf{R}}_m$$

(18)

where ${\mathbf{K}}_{\mathrm{mm}}=\sum _e{\mathbf{T}}^T{{\mathbf{K}}_e}_{\mathrm{mm}}\mathbf{T}$ ; $\mathbf{T}$ is the rotation matrix from the integral coordination to the local coordination; ${\mathbf{U}}_m$ and ${\mathbf{R}}_m$ are, respectively, the mth peak values of the deformation vector and load vector in the integral coordination. After the maximum value of the deformation vector ${\mathbf{U}}_m$ is summarized with the variable m varying from 1 to r, the deformation vector $\mathbf{U}$ of the pinned curvilinear box girder is attained as

$$\mathbf{U}=\sum _{m=1}^r{\mathbf{P}}_m{\mathbf{U}}_m$$

(19)

where ${\mathbf{P}}_m$ is the mth matrix which is linked to $\sin (m\mathit{\pi }/\alpha )\theta$ and $\cos (m\mathit{\pi }/\alpha )\theta$ .

Agglomeration theory for the hybrid indeterminate curvilinear box girder

As for the hybrid indeterminate curvilinear box girder, it is impossible to resolve the hybrid indeterminate issue only depending on the above finite curvilinear element method. Therefore, the fairly complicated arithmetic is solved with agglomeration theory. Releasing the interior restrictions of diaphragms and the exterior restrictions of middle supporters in Figure 3, the hybrid indeterminate box girder is converted into a determinate analytical system noted as a pinned curvilinear box girder. The equation of compatibility for structural deformation of the joints at the redundant restrictions can be expressed as

$$\mathbf{p}\mathbf{r}+\mathbf{\delta }=0$$

(20)

where $\mathbf{p}$ is the flexibility matrix for the hybrid indeterminate curvilinear box girder, $\mathbf{r}$ is the interior and exterior redundant force vector, and $\mathbf{\delta }$ is the deformation vector of the determinate system. In each section of the diaphragm, the interior redundant forces just constitute the self-equilibrium mechanical status. Selecting the ith diaphragm section and, for instance, taking the No. 1 node as the studied point shown in Figure 3, from equilibrium equation, it is obtained

$$\begin{array}{c}\sum \mathbf{H}=0:H_1=-H_2-H_3-\cdots -H_n\hfill \\ \sum \mathbf{V}=0:V_1=-V_2-V_3-\cdots -V_n\hfill \\ \sum \mathbf{M}=0:M_1=H_2{z}_2-V_2{x}_2-M_2+H_3{z}_3-V_3{x}_3\hfill \\ \multicolumn{1}{c}{-M_3+\cdots +H_n{z}_n-V_n{x}_n-M_n}\end{array}$$

(21)

where $H_k,V_k,M_k(k=1,\dots ,n)$ are the interior redundant forces of the kth linking node of the hybrid indeterminate box girder and $x_k,z_k(k=1,\dots ,n)$ are the coordinates of the kth linking node. Equation (21) can be written as

$${\mathbf{r}}_i={\mathbf{\mu }}_i{\mathbf{R}}_i$$

(22)

where ${\mathbf{\mu }}_i$ is the agglomerated matrix of the interior redundant forces of the ith diaphragm section and ${\mathbf{R}}_i$ is the vector of the agglomerated interior redundant forces, in which the component is independent of each other. The exterior redundant forces of the middle supporters are considered and then the total agglomerated equation can be derived as

$$\mathbf{r}=\mathbf{\mu }\mathbf{R}$$

(23)

where $\mathbf{\mu }$ is the agglomerated redundant force matrix and $\mathbf{R}$ is the agglomerated redundant force vector. Then the equation of compatibility for the hybrid indeterminate curvilinear box girder can be expressed as

$$\mathbf{FR}+\mathbf{\Delta }=0$$

(24)

where $\mathbf{\Delta }$ and $\mathbf{F}$ are, respectively, the deformation vector and the flexibility matrix after agglomeration is fulfilled, where $\mathbf{\Delta }={\mathbf{\mu }}^T\mathbf{\delta }$ and $\mathbf{F}={\mathbf{\mu }}^T\mathbf{p}\mathbf{\mu }$ . Substituting the agglomerated redundant force vector $\mathbf{R}$ obtained from equation (24) into equation (23), the vector $\mathbf{r}$ of the redundant forces of the hybrid indeterminate box is gained. Then let the redundant force vector $\mathbf{r}$ added on the basic system and the displacement field of the hybrid indeterminate curvilinear box can be achieved by equations (18) and (19).

Figure 3.

Interior redundant forces of the diaphragm.

Variable scale gradient theory for accumulative Bayesian detectionof displacement constants

Variable scale gradient theory

The main optimization methods may be assorted into two kinds: direct optimization method (simplex method, complex method, etc.) and gradient optimization method (variable scale method, Newton gradient method, etc.). ¹² Generally, the direct search methods are always inferior to the gradient search methods for the lower computational efficiency because of simply depending on the correction of the objective function.^16–18 The variable scale gradient method incessantly changes the spatial matrix scale to produce new search directions during the iterative processes and optimizes the derived objective function efficiently.

Combined with the finite curvilinear strip element method and variable scale gradient theory, accumulative Bayesian detection steps of displacement constants of the curvilinear box girder are achieved as follows:

• 1. Set the initial values ${\mathbf{E}}^0$ of displacement constants $\mathbf{E}$ of the curvilinear box girder and the convergence criterion ${\epsilon }_1$ and ${\epsilon }_2$ .

• 2. From equation (4), compute the accumulative Bayesian objective function $J({\mathbf{E}}^0)$ , and from equation (5), determine the gradient $\nabla J({\mathbf{E}}^0)$ of the accumulative Bayesian objective function at ${\mathbf{E}}^0$ . Set the initial variable scale matrix ${\mathbf{H}}^0$ equal to the unit matrix $\mathbf{I}$ . Let the iterative variable $k=0$ .

• 3. Determine the search direction vector by ${\mathbf{d}}^k=-{\mathbf{H}}^k\nabla J({\mathbf{E}}^k)$ .

• 4. From equation (4) and using the next 1D Fibonacci search method, confirm the optimal step length $\lambda$ to achieve

$$J({\mathbf{E}}^{k+1})=\underset{\mathrm{\lambda }}{\min }J({\mathbf{E}}^k+\mathbf{\lambda }{\mathbf{d}}^k)$$

(25)

Then compute the accumulative Bayesian objective function $J({\mathbf{E}}^{k+1})$ and the gradient $\nabla J({\mathbf{E}}^{k+1})$ .

• 5. For the following convergence judgment equations, if one of them is satisfied, it is that

$$\left|\frac{J({\mathbf{E}}^{k+1})-J({\mathbf{E}}^k)}{J({\mathbf{E}}^k)}\right|<{\epsilon }_1\mathrm{or}\Vert \frac{{\mathbf{E}}^{k+1}-{\mathbf{E}}^k}{{\mathbf{E}}^{k+1}}{\Vert }_2<{\epsilon }_2$$

(26)

The variable scale gradient iteration is convergent and the detection results of displacement constants $\mathbf{E}$ are $\hat{\mathbf{E}}={\mathbf{E}}^{k+1}$ . The iteration is terminated. Otherwise, continue the next step.

• 6. If $J({\mathbf{E}}^{k+1})>J({\mathbf{E}}^k)$ , then let ${\mathbf{E}}^0={\mathbf{E}}^{k+1}$ and return to step (2) to reiterate. Otherwise, continue the next step.

• 7. Compute the variable vector difference ${\mathbf{S}}^k$

$${\mathbf{S}}^k={\mathbf{E}}^{k+1}-{\mathbf{E}}^k$$

(27)

• 8. Compute the gradient vector difference ${\mathbf{Y}}^k$

$${\mathbf{Y}}^k=\nabla J({\mathbf{E}}^{k+1})-\nabla J({\mathbf{E}}^k)$$

(28)

• 9. Determine the next iterative variable scale matrix ${\mathbf{H}}^{k+1}$

$${\mathbf{H}}^{k+1}={\mathbf{H}}^k+\mathrm{\Delta }{\mathbf{H}}^k$$

(29)

$$\mathrm{\Delta }{\mathbf{H}}^k=\frac{{\mathbf{S}}^k{({\mathbf{S}}^k)}^T}{{({\mathbf{S}}^k)}^T{\mathbf{Y}}^k}-\frac{{\mathbf{H}}^k{\mathbf{Y}}^k{({\mathbf{Y}}^k)}^T{\mathbf{H}}^k}{{({\mathbf{Y}}^k)}^T{\mathbf{H}}^k{\mathbf{Y}}^k}$$

(30)

where $\mathrm{\Delta }{\mathbf{H}}^k$ is the incremental matrix of the variable scale matrix. Then let $k=k+1$ and return to step (3) to continue reiteration.

The Fibonacci series method

The 1D search of step length $\lambda$ is necessary in step (4) of accumulative Bayesian detection steps of displacement constants of the hybrid indeterminate curvilinear box girder and it is a fairly complicate issue in constant detection. The linear search methods primarily involve the golden section search method, polynomial interpolation search method, and Fibonacci series method. ¹⁷ Among these methods, the Fibonacci series method has comparatively better computational efficiency, ¹⁸ the main formulae of which are as follows

$${\lambda }_1={\lambda }_0+\frac{F_{j-2}}{F_j}({\lambda }_3-{\lambda }_0)$$

(31)

$${\lambda }_2={\lambda }_0+\frac{F_{j-1}}{F_j}({\lambda }_3-{\lambda }_0)$$

(32)

$$F_j=F_{j-1}+F_{j-2}(j⩾2)$$

(33)

where $[{\lambda }_0,{\lambda }_3]$ is the interval, where the optimal step length t is located, ${\lambda }_1$ and ${\lambda }_2$ are the interpolating step sizes, and $F_j$ is the essential Fibonacci series.

The detection analysis of the examples

Accumulative Bayesian detection of displacement constants $\mathbf{E}=[\begin{array}{ccc}{E}_1\hfill & E_2\hfill & E_3\hfill \end{array}{]}^T$ of the hybrid indeterminate curvilinear concrete box with interior and exterior redundant forces is studied in this article ( $E_1$ , $E_2$ , and $E_3$ are, respectively, the Young’s moduli of the top plate, web plate, and bottom plate, respectively). Interior redundant forces of the diaphragm and exterior redundant forces of the supporters are both in the middle span section. The detection analysis model of the hybrid indeterminate curvilinear box girder is selected from Zhao and Chen, ¹⁰ and the curvilinear strip element and the number of the typical line are shown in Figure 4. In Figure 5, the radius is $R=250\mathrm{cm}$ , the integral angle is $\alpha =0.64\mathrm{rad}$ , and the length of the middle axis is $L=160\mathrm{cm}$ . In Figure 6, joint points between the diaphragm and the box in the middle span section are labeled. Poisson’s ratio $\mu$ is 0.16. The true values of displacement constants $\mathbf{E}$ and the widths of the top plate, web plate, and bottom plate recorded as $d_1$ , $d_2$ , and $d_3$ are evaluated in Table 1. The uniform loads $p_1=125.0\mathrm{N}/\mathrm{cm}$ and $p_2=100.0\mathrm{N}/\mathrm{cm}$ are vertically loaded to the typical line of No. 3 and No. 12 of the curvilinear box girder, respectively. In order to carry out the accumulative Bayesian detection of displacement constants $\mathbf{E}$ of the hybrid indeterminate curvilinear box girder, the procedure named ABIVSM is developed. First, the mechanical analysis procedure of the hybrid indeterminate curvilinear box girder based on finite curvilinear strip element theory and agglomeration theory is employed, which has already been proved. ¹⁰ Choose No. 1–5 points in the 1/4 section plane shown in Figures 4 and 5 as the displacement measured points and measure the displacement of each referred point five times. The expectations and standard variances of the measured deformations are listed in Table 2.

Figure 4.

Number of curvilinear strip element and the typical line/cm.

Figure 5.

Restrictions of the hybrid indeterminate curvilinear box.

Figure 6.

Joint points between the box and the diaphragm.

Table 1.

True values of displacement constants and the widths of the curvilinear box girder.

Constant name	$E_{1t}$	$E_{2t}$	$E_{3t}$	$d_1$	$d_2$	$d_3$
Value	250 × 104 N/cm2	200 × 104 N/cm2	300 × 104 N/cm2	0.40 cm	0.60 cm	0.40 cm

Table 2.

Expectations and standard variances of the measured displacements/cm.

Selected points	Displacement expectations					Displacement standard variances
	$w_1$	$w_2$	$w_3$	$w_4$	$w_5$	${\sigma }_{w_1}$	${\sigma }_{w_2}$	${\sigma }_{w_3}$	${\sigma }_{w_4}$	${\sigma }_{w_5}$
1	0.174	0.185	0.172	0.181	0.178	0.128	0.136	0.131	0.133	0.137
2	0.190	0.188	0.183	0.184	0.187	0.141	0.143	0.147	0.148	0.154
3	0.191	0.190	0.196	0.195	0.194	0.167	0.166	0.162	0.165	0.161
4	0.189	0.198	0.190	0.197	0.193	0.159	0.168	0.163	0.164	0.158
5	0.201	0.197	0.200	0.195	0.199	0.155	0.163	0.156	0.158	0.162

Case 1

Accumulative Bayesian detection of displacement constants of the hybrid indeterminate curvilinear box girder when the priori information is precise, which means that the priori information of the curvilinear box structure is ${\mathbf{E}}_0=[250.0,200.0,300.0{]}^T$ . The two groups of the initial values of displacement constants are, respectively, set ${\mathbf{E}}_{1\mathrm{ini}}=[500.0,600.0,200.0{]}^T$ and ${\mathbf{E}}_{2\mathrm{ini}}=[100.0,100.0,200.0{]}^T$ . Suppose that the convergence criteria ${\epsilon }_1=0.001$ and ${\epsilon }_2=0.001$ and the constant variation coefficient is 0.1. Together with the data in Table 2, the iterative results of displacement constants are obtained as shown in Table 3 and Figure 7.

Table 3.

Results of accumulative Bayesian detection of displacement constants of the curvilinear box girder in Case 1 (10⁴ N/cm²).

Displacement parameters	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$
Initial value	500.0	600.0	200.0	100.0	100.0	200.0
Final value	250.37	200.33	300.25	250.24	200.45	299.81
Iterative times	9	9	9	14	14	14
Relative error (%)	0.15	0.16	0.08	0.10	0.23	0.06
Convergence criterion	${\epsilon }_1$	${\epsilon }_1$	${\epsilon }_1$	${\epsilon }_2$	${\epsilon }_2$	${\epsilon }_2$

Figure 7.

Accumulative Bayesian detection results of displacement constants in Case 1 (10⁴ N/cm²): (a) iterative results with ${\mathbf{E}}_{1\mathrm{ini}}$ and (b) iterative results with ${\mathbf{E}}_{2\mathrm{ini}}$ .

From Table 3 and Figure 7, it is proved that, when the priori information is precise, the iterative process of accumulative Bayesian detection of displacement constants of the hybrid indeterminate curvilinear box girder is steadily convergent to the true constant values. Whether the initial constant values are far from the true or not, the convergence is independent of the initial constant values. The results also indicate that, compared with the Powell direct optimal results,^18,19 the variable scale theory is more efficient because the computational times of the accumulative Bayesian objective function mainly resulting from the finite curvilinear strip element model are fewer. The result of the constant variation coefficient is about 0.078, which is ameliorated in comparison with the given coefficient. In this case, the iterative process can be convergent in conformity to the two convergence criteria when different initial constant values are selected and set.

Case 2

In order to achieve some other regularities of accumulative Bayesian detection of displacement constants of the hybrid indeterminate curvilinear box girder when the priori information is precise, the initial values of displacement constants ${\mathbf{E}}_{3\mathrm{ini}}=[400.0,500.0,200.0{]}^T$ and ${\mathbf{E}}_{4\mathrm{ini}}=[100.0,100.0,100.0{]}^T$ are, respectively, chosen. The other data are the same as in Case 1. The iterative results of accumulative Bayesian detection of displacement constants are listed in Table 4 and Figure 8.

Table 4.

Results of accumulative Bayesian detection of displacement constants of the curvilinear box girder in Case 2 (10⁴ N/cm²).

Systematic constants	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$
Initial value	400.0	500.0	200.0	100.0	100.0	100.0
Final value	250.37	200.45	300.45	249.34	200.37	300.32
Iterative times	10	10	10	12	12	12
Relative error (%)	0.15	0.23	0.15	0.27	0.19	0.11
Convergence criterion	${\epsilon }_2$	${\epsilon }_2$	${\epsilon }_2$	${\epsilon }_1$	${\epsilon }_1$	${\epsilon }_1$

Figure 8.

Accumulative Bayesian detection results of displacement constants in Case 2 (10⁴ N/cm²): (a) iterative results with ${\mathbf{E}}_{3\mathrm{ini}}$ and (b) iterative results with ${\mathbf{E}}_{4\mathrm{ini}}$ .

It is indicated from the results in Table 4 and Figure 8 that the iterative processes are still steadily convergent to the true values. Some other conclusions can be drawn as follows. First, supposing that the initial constant values become closer to the true values, the iterative times cannot always get fewer when the convergence criterion is satisfied. For example, compared with ${\mathbf{E}}_{1\mathrm{ini}}$ , ${\mathbf{E}}_{3\mathrm{ini}}$ is evidently closer to the true value ${\mathbf{E}}_{\mathrm{true}}$ . However, the iterative number of ${\mathbf{E}}_{3\mathrm{ini}}$ is 10, which is larger than that of ${\mathbf{E}}_{1\mathrm{ini}}$ (9 times). Second, the convergence accuracies are all very high and the relative errors are less than 1%, which are not distinctly improved when the initial constant values are nearer to the true values. Third, when one of the initial constant values is changed, the whole iterative times and relative errors may have probably altered. For example, compared with ${\mathbf{E}}_{2\mathrm{ini}}$ , the item $E_3$ in ${\mathbf{E}}_{4\mathrm{ini}}$ is intentionally altered and the rest are unaltered. Actually, the iterative times and relative errors of the unaltered items of $E_1$ and $E_2$ are subsequently influenced. The reason that leads to the discussed result is that, during the processes of the detection of the poly constants, the relationships between the poly constants are interrelated and interdependent.

Case 3

The third case is when the accumulative Bayesian detection of displacement constants of the hybrid indeterminate curvilinear box girder when the priori information is imprecise. This case is in accordance with practical engineering because the supposed priori information only dependent on engineering experience is hardly possible to coincide with the true value. Let priori information ${\mathbf{E}}_0=[200.0,250.0,250.0{]}^T$ . In order to make comparison conveniently, the initial values of the displacement constants and the other data are the same as in Case 1. The iterative results of accumulative Bayesian detection of displacement constants are listed in Table 5.

Table 5.

Results of accumulative Bayesian detection of displacement constants of the curvilinear box girder in Case 3 (10⁴ N/cm²).

Systematic constants	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$
Initial value	500.0	600.0	200.0	100.0	100.0	200.0
Final value	154.26	118.85	148.87	207.87	155.08	236.35
Iterative times	100	100	100	24	24	24
Relative error (%)				16.85	22.46	21.22
Convergence criterion	Divergent	Divergent	Divergent	${\epsilon }_2$	${\epsilon }_2$	${\epsilon }_2$

It can be found from Table 5 that if the iterative process of displacement constant is convergent, it only conforms to the second convergence criterion. The iterative results indicate that the whole relative errors are larger than 10% and the iterative processes are disconvergent to the true values. Without the judgment criterion of the true values, how to judge whether the priori information is applied properly is significant. Otherwise, it will make the displacement constants become convergent to false values, which will inevitably guide the practical engineering wrongly. Through considerable research, the regularity can be grasped and it is obtained that if the priori information satisfies the precise condition, the iterative process can be convergent by the two convergence criteria, which may produce different iterative times. If the priori information is imprecise, the iterative processes may be convergent only satisfying the second convergence criterion or divergent.

Conclusion

Accumulative Bayesian detection of displacement constants of the hybrid indeterminate curvilinear box girder is completed and the main conclusions are drawn as follows:

1. The accumulative Bayesian detection processes of displacement constants of the hybrid indeterminate curvilinear box girder are steadily convergent to the true values when the priori information is precise, which proves the accuracy of the derived detection analytical model and the reliability of the compiled procedure.

2. In accumulative Bayesian detection, the variable scale gradient method incessantly engenders new search directions during the iterative processes and optimizes the derived objective function efficiently, while the Powell optimal method is comparatively inferior for the lower computational efficiency because of its simple dependence on the revision of the objective function.

3. Compared with the routine Bayesian objective function, it is an improvement that the accumulative Bayesian objective function of displacement constants of the hybrid indeterminate curvilinear box girder can tackle the measured systematic responses of different times and different spots simultaneously, which can also consider the stochasticity of the displacement constants and systematic responses accurately.

4. With finite curvilinear strip element theory and agglomeration theory, a numerical mechanical method is put into practice without more elements in partition for the spatial mechanical analysis of the hybrid indeterminate curvilinear box girder.