Updated Bayes identification of displacement constants of curve indeterminate box girder with variable scale theory

Abstract

For curve indeterminate box girder, updated Bayes identification model of displacement constants was derived and studied with the variable-scale optimization theory. First, the updated Bayes objective function of displacement constants of the structure was founded. The gradient matrix of the objective function to displacement constants and the calculative covariance matrix were both deduced. Then, with finite curve strip element method, mechanical analysis of curve indeterminate box girder was completed. With automatic search scheme of quadratic parabola interpolation for optimal step length, the variable scale theory was utilized to optimize the updated Bayes objective function. Then, the identification steps were expounded, and the identification procedure was developed. Through typical examples, it is achieved that the updated Bayes identification model of displacement constants has numerical stability and perfect convergence. The stochastic performances of systematic parameters and systematic responses are simultaneously deliberated in updated Bayes objective function, which can synchronously take the actual measured information at different times into account. The variable-scale optimization method continually changes the spatial matrix scale to generate renewed search directions during the iterations, which certainly accelerates the identification of the displacement constants.

Introduction

The curve indeterminate box girder is commonly used in civil engineering, and especially, the curve indeterminate box girder often appears in bridge engineering. The perfect mechanical performances such as larger torsion stiffness and lighter weight naturally exist in this kind of thin-walled structures.^1–3 The research results of mechanical analysis of the curve indeterminate box girder have ceaselessly emerged because of the importance in civil engineering.^4,5 In Ye et al., ⁶ the layered shell element based on the supposition of the ignorance of the transversal shearing stress is deduced. Then, in Zhang, ⁷ the degraded solid element is applied for the curve indeterminate box girder, and with failure criteria of concrete material, the mechanical behaviors after cracking are studied in detail. Compared with the layered shell element, the finite strip element has fewer nodes and elements in structural analysis, and the mechanical responses of the curve indeterminate box girder are completed in Zhang et al. ⁸ In practical engineering, before the analysis of the curve indeterminate box girder, the displacement constants including elastic modulus and Poisson’s ratio must be mastered; otherwise, the prearrangement cannot be put into practice,^9,10 and this problem is quite worth researching. During the available achievements, the Kalman filtering theory which has the advantages of auto-revision and auto-optimization has been applied in the identification of the displacement constants, which shows that if the initial constants are given improper, then the Kalman correction matrix might be divergent. ¹¹ The Powell direct optimization method has been successfully used in the constant identification of the curve indeterminate box girder. ¹² In Powell iterative processes, the improvement of the displacement constants only depends on the revision of the objective function without any search direction, which consequentially leads to lower computational efficiency. ¹³ In Zhang et al., ¹⁴ one-dimensional optimization method called Fibonacci series search method is employed to improve the computational efficiency, but it cannot conquer the inherent disadvantage of the Powell direct optimization method. The gradient optimization methods such as the variable-scale optimization method can smooth over the defect just disserted to a certain extent.^15–19 The motivation of this article is how to derive the mechanical identification model of displacement constants of curve indeterminate box girder with updated Bayes theory.

Thus, the updated Bayes objective function expression of displacement constants of the curve indeterminate box girder is derived and the finite curve-strip-shaped element for the specific structure is attained. The major contributions of this article also lie in that the variable-scale optimization method is combined to establish the identification model of displacement constants of the curve indeterminate box girder. And, through analysis of some classic examples, several important conclusions of identification of displacement constants of the box structure are achieved. And, the updated Bayes objective function can tackle the measured systematic responses of different times and different spots simultaneously, which can also consider the randomness of the displacement constants and systematic responses accurately. The updated Bayes identification model and the variable-scale optimization method can also be used in other kinds of structures.

Updated Bayes theory for displacement constants of curve indeterminate box girder

In the identification model of displacement constants of the curve indeterminate box girder, the displacement constants are generally considered as stochastic variables which are usually recorded as the stochastic vector to complete the identification of poly constants. In identification of research achievements such as identification of models performance of fracture toughness of polymeric particle nanocomposites, identification of material interfaces in piezoelectric structures, and identification of typical flaws in piezoelectric structures using extended finite element method (FEM),^20–22 Bayes theory is widely applied because one of the superiorities lies in its taking the stochastic property of systematic parameters and systematic responses into account efficiently; however, a much more efficient Bayes objective function in this article is derived below. From Bayes theorem, ¹³ it can be attained as

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

(1)

where $\mathit{E}=[E_1{E}_2\cdots E_m{]}^T$ , m is the dimension of the vector $\mathit{E}$ . $f(\mathit{E})$ is the priori distributing density function, $f({\mathit{W}}^{*}|\mathit{E})$ is the conditional distributing density function of the systematic response, $f({\mathit{W}}^{*})$ is the systematic response distributing density function, and $f(\mathit{E}|{\mathit{W}}^{*})$ is the posterior distributing density function. It is assumed that displacement constants $\mathit{E}$ are conformed to normal distributing, and then, the formula of the priori distributing function $f(\mathit{E})$ is

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

(2)

where ${\mathit{E}}_0$ is the expectation vector of displacement constants $\mathit{E}$ , ${\mathit{C}}_{\mathit{E}}$ is the covariance matrix, and $\left|{\mathit{C}}_{\mathit{E}}\right|$ is the determinant of the covariance matrix ${\mathit{C}}_{\mathit{E}}$ .

In structural engineering, the systematic responses at the testing points must be measured several times and the collected displacement data ${\mathit{W}}_i^{*}$ at the ith time are all samples of ${\mathit{W}}^{*}$ . If the common Bayes objective function is derived to detect constants, then there are many repetitive assignments. Therefore, the updated Bayes objective function of stochastic displacement constants of the curve indeterminate box girder is derived. The joint density function of ${\mathit{W}}_i^{*}$ is $\underset{i=1}{\overset{n}{\Pi }}f({\mathit{W}}_i^{*}|\mathit{E})$ , where n is the number of measured times of systematic responses. From maximum likelihood theory, it can be achieved as

$$\begin{array}{c}\multicolumn{1}{c}{f({\mathit{W}}^{*}|\mathit{E})=(2\pi {)}^{-\frac{\mathrm{mn}}{2}}\underset{i=1}{\overset{n}{\Pi }}{\left|{\mathit{C}}_{{\mathit{W}}_i^{*}}\right|}^{-\frac{1}{2}}\times \exp }\\ \multicolumn{1}{c}{[-\frac{1}{2}\sum _{i=1}^n{({\mathit{W}}_i^{*}-{\mathit{W}}_i)}^T{\mathit{C}}_{{\mathit{W}}_i^{*}}^{-1}({\mathit{W}}_i^{*}-{\mathit{W}}_i)]}\end{array}$$

(3)

where ${\mathit{W}}_i$ is the systematic response vector from the calculative result. ²³ Setting equations (2) and (3) into equation (1), the updated Bayes objective function J may be deduced into

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

(4)

Evidently, ${\mathit{W}}_i$ in equation (4) is the implicit function of displacement constants $\mathit{E}$ . For the sake of attaining the identification results of displacement constants with variable-scale optimal theory, from equation (4), the gradient matrix of updated objective function J to displacement constants $\mathit{E}$ are derived as

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

(5)

${\mathit{W}}_i(\mathit{E})$ is expanded according to Taylor’s formula at the expectation point and keeping just the first two items,^24,25 it can be derived as

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

(6)

where the sensitivity matrix of systematic response to displacement constants ${\mathit{G}}_i(\overline{\mathit{E}})={(\partial {\mathit{W}}_i/\partial \mathit{E})|}_{\mathit{E}=\overline{\mathit{E}}}$ . Sub-stituting equation (6) directly into equation (5), it can be achieved as

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial J}{\partial \mathit{E}}=\sum _{i=1}^{n}2{\mathit{G}}_i^T{\mathit{C}}_{{\mathit{W}}_i^{*}}^{-1}({\overline{\mathit{W}}}_i+{\mathit{G}}_i\mathit{E}-{\mathit{G}}_i\overline{\mathit{E}}-{\mathit{W}}_i^{*})}\\ \multicolumn{1}{c}{+2{\mathit{C}}_{\mathit{E}}^{-1}(\mathit{E}-{\mathit{E}}_0)}\end{array}$$

(7)

where ${\overline{\mathit{W}}}_i={\mathit{W}}_i(\overline{\mathit{E}})$ . When the objective function takes the minimum, equation (7) is obviously equal to zero. Then, it can be derived as

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

(8)

Presuming $\mathit{H}=\sum _{i=1}^n{\mathit{G}}_i^T{\mathit{C}}_{{\mathit{W}}_i^{\ast }}^{-1}{\mathit{G}}_i+{\mathit{C}}_{\mathit{E}}^{-1},\mathit{M}={\mathit{H}}^{-1}$ , $[{\mathit{G}}_1^T{\mathit{C}}_{{\mathit{W}}_1^{*}}^{-1},$ ${\mathit{G}}_2^T{\mathit{C}}_{{\mathit{W}}_2^{*}}^{-1},\dots ,{\mathit{G}}_n^T{\mathit{C}}_{{\mathit{W}}_n^{*}}^{-1}]$ , and from equation (8), the identification values $\hat{\mathit{E}}$ of displacement constants $\mathit{E}$ of curve indeterminate box girder can be rewritten as

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

(9)

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

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

(10)

where ${\mathit{C}}_{{\mathit{W}}^{*}}=\mathrm{diag}({\mathit{C}}_{{\mathit{W}}_1^{*}},{\mathit{C}}_{{\mathit{W}}_2^{*}},\dots ,{\mathit{C}}_{{\mathit{W}}_n^{*}})$ , and ${\mathit{C}}_{{\mathit{W}}_i^{*}}$ is the covariance matrix of collected systematic responses at the ith time. With the nonsingular property of ${\mathit{C}}_{\mathit{E}}$ and ${\mathit{C}}_{{\mathit{W}}^{*}}$ , the above equation can be derived as

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

(11)

Considering $\mathit{G}=[{\mathit{G}}_1,{\mathit{G}}_2,\dots ,{\mathit{G}}_n{]}^T$ , the series summation form for equation (11) can be achieved as

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

(12)

Finite curve-strip-shaped element method for curve indeterminate box girder

Governing equation of finite curve-strip-shaped element for simply supported box girder

In equation (4), the updated objective function of displacement constants of curve indeterminate box girder, ${\mathit{W}}_i$ is the computational vector of systematic responses. For this kind of box structure, finite curve-strip-shaped element method utilizes different kinds of conic angles $\beta$ shown in Figure 1 to describe different connections between strains and displacements. The equations are expressed as

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

(13)

Figure 1.

Finite curve-strip-shaped element of box girder: (a) geometrical description of curve box, (b) element of top plate or bottom plate, and (c) element of web plate.

Aimed to simply supported box girder, displacement field function of finite curve-strip-shaped element through shape function interposition is achieved as

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

(14)

where $\mathit{f}=[u,v,w{]}^T$ is the displacement field function in the whole coordinates. ${\mathit{f}}_m=[u_m,v_m,w_m{]}^T$ is the mth displacement field function and ${\mathit{\delta }}_m$ is the peak displacement field vector of the nodal line in the element form. ${\mathit{N}}_m$ is the shape function matrix determined by the harmonic sinusoids $Y_m=\sin (m\pi /\alpha )\theta$ . In Figure 1, $\alpha$ shown is the curvature angle and $\theta$ is the angular coordinate and b is the width of the curve-strip-shaped element. The stiffness matrix of finite curve-strip-shaped element is derived as

$${\mathit{K}}_e=\underset{V}{\int }{\mathit{B}}^T\mathit{DB}\mathrm{dV}=\underset{V}{\int }{[{\mathit{B}}_1{\mathit{B}}_2\cdots {\mathit{B}}_r]}^T\mathit{D}[{\mathit{B}}_1{\mathit{B}}_2\cdots {\mathit{B}}_r]\mathrm{dV}$$

(15)

where $\mathit{D}$ is the elastic matrix and $\mathit{B}$ is the strain matrix determined by substituting equation (14) into equation (13). In equation (15), it can be achieved as

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

(16)

Due to the orthogonal property of the harmonic sinusoids $Y_m=\sin (m\pi /\alpha )\theta$ , equation (15) can be derived into the form of principle diagonal block matrix as

$${\mathit{K}}_e=\mathrm{diag}[{{\mathit{K}}_e}_{11}{{\mathit{K}}_e}_{22}\cdots {{\mathit{K}}_e}_{\mathrm{rr}}]$$

(17)

The mth item of the governing equation of pinned curve box is

$${\mathit{K}}_{\mathrm{mm}}{\mathit{W}}_m={\mathit{R}}_m$$

(18)

where ${\mathit{W}}_m$ is the mth item of the peak value of displacement vector and ${\mathit{R}}_m$ is the one of the peak value of the load vector in the whole coordinates. ${\mathit{K}}_{\mathrm{mm}}=\sum _e{\mathit{T}}^T{{\mathit{K}}_e}_{\mathrm{mm}}\mathit{T}$ and $\mathit{T}$ is transitional matrix from the whole coordinates to the local coordinates. After summation of the peak value ${\mathit{W}}_m$ of displacement vector when m varies from 1 to r, the displacement vector $\mathit{W}$ for the pinned curve box structure is

$$\mathit{W}=\sum _{m=1}^r{\mathit{P}}_m(\sin \frac{m\pi }{\alpha }\theta ,\cos \frac{m\pi }{\alpha }\theta ){\mathit{W}}_m$$

(19)

where ${\mathit{P}}_m$ is defined as the mth item of interpolation matrix which is linked with harmonic trigonometric functions $\sin (m\pi /\alpha )\theta$ and $\cos (m\pi /\alpha )\theta$ .

Force agglomeration theory for curve indeterminate box girder

As for curve indeterminate box girder, it is impossible to resolve the curve indeterminate problem only depending on the derived finite curve element method. Therefore, the requisite joint algorithm is deduced with force agglomeration theory. Removing the external restrictions of middle supporters and the internal restrictions of diaphragms shown in Figure 2, the indeterminate box girder structure is converted into statically determinate system noted as simply supported box girder. Compatibility equations of structural deformation of the points at the external and internal restrictions can be presented as

$$\mathit{hr}+\mathit{q}=0$$

(20)

where $\mathit{h}$ is the flexibility matrix of indeterminate box girder, $\mathit{r}$ is the vector of the external and internal redundant forces, and $\mathit{q}$ is the displacement vector of the basic system.

Figure 2.

Inner redundant forces of diaphragm.

Then, in the section where each diaphragm lies, the internal redundant forces just constitute the self-equilibrium mechanical system. Select the ith diaphragm section and, for instance, take the No. 1 point into consideration shown in Figure 2, then from equilibrium relationship, it is achieved as

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

(21)

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

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

(22)

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

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

(23)

where $\mathit{\mu }$ is the agglomerating redundant force matrix and $\mathit{R}$ is the agglomerating redundant force vector. From principle of virtual work and equation (20), compatibility condition expression of indeterminate box girder can be derived as

$$\mathit{HR}+\mathit{Q}=0$$

(24)

where $\mathit{Q}$ and $\mathit{H}$ are, respectively, the displacement vector and the flexibility matrix after force agglomeration is completed, where

$$\mathit{Q}={\mathit{\mu }}^T\mathit{q}$$

(25)

$$\mathit{H}={\mathit{\mu }}^T\mathit{h}\mathit{\mu }$$

(26)

Substituting the agglomerating redundant force vector $\mathit{R}$ obtained from equation (24) into equation (23), the vector $\mathit{r}$ of the redundant forces of the curve indeterminate box is gained. Then, add the redundant force vector $\mathit{r}$ on the basically determinate system and the displacement field of the curve indeterminate box girder can be obtained through equations (18) and (19).

The identification model of displacement constants with updated Bayes theory

The current optimal methods may be mainly divided into two categories called direct search strategy (simplex method, complex method, etc.) and gradient search strategy (variable scale method, Newton gradient method, etc.). ¹² Generally, the direct search methods are comparatively inferior to the gradient search methods for the lower computational efficiency because of its only simply depending on the revision of the objective function.^13,14 Variable-scale optimal method ceaselessly alters the spatial matrix scale to produce new search directions during the iterative processes and optimizes the objective function efficiently. During the variable scale steps of displacement constants of curve indeterminate box girder, determination of the optimal step size is involved, which belongs to a fairly complicated question in constant identification analysis. In the achieved references, the one-dimensional search algorism is probably directed to golden section algorism and Fibonacci series search algorism because they can guarantee the accuracy of the calculation and the simplicity of the program. But the iterative times are always larger, which inevitably leads to a lower efficiency because the two methods belong to linear interpolation search.^26–28 The quadratic parabola interpolation method is adopted, which can make certain the interval of the optimal step size and implement the assignment automatically.

Combined with finite strip element method, the flowchart for the identification model of displacement constants of curve indeterminate box girder with updated Bayes theory is shown in Figure 3, and the corresponding steps in the mechanical identification model are achieved as follows:

• 1. Choose the initial values ${\mathit{E}}^0$ of displacement constants $\mathit{E}$ of curve indeterminate box girder and set the convergence criterion ${\epsilon }_1$ and ${\epsilon }_2$ .

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

• 3. Determine the search direction vector by

  $${\mathit{d}}^k=-{\mathit{H}}^k\nabla J({\mathit{E}}^k)$$

  (27)

• 4. From equation (4), confirm the optimal step length $\lambda$ to make

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

  (28)

Figure 3.

The flowchart for the mechanical identification model of the displacement constants.

The one-dimensional quadratic parabola interpolation search method is involved in equation (28), which can be decomposed into the next three steps.

• 5. Decision of the domain in which the optimal step size $\lambda$ lies in. Presuppose the initial step size ${\lambda }_1$ and a step size increment ${\lambda }_0$ and prescribe ${\lambda }_2={\lambda }_1+{\lambda }_0$ .

• 6. If $J({\lambda }_1)\ge J({\lambda }_2)$ , then the step length increment expression ${\lambda }_l={\lambda }_{l-1}+2^{l-2}{\lambda }_0$ , where $l\ge 3$ is computed. The computation is not suspended until $J({\lambda }_l)\ge J({\lambda }_{l-1})$ . If $J({\lambda }_1)<J({\lambda }_2)$ , then the other step length increment expression ${\lambda }_l={\lambda }_{l-1}-2^{l-3}{\lambda }_0$ , where $l\ge 3$ is computed. Similarly, the computation is not suspended until $J({\lambda }_l)\ge J({\lambda }_{l-1})$ . When the iteration is completed, the domain in which the optimal step size $\lambda$ lies in is determined and noted as $[{\lambda }_a,{\lambda }_d]$ .

• 7. Interpolating of the optimal step size $\lambda$ . Combined with the extremum specialty of the updated Bayes objective function, the expression of the optimal step size $\lambda$ is attained through relative deduction

$$\lambda =\frac{1}{2}({\lambda }_a+{\lambda }_d-\frac{{\lambda }_b}{{\lambda }_c})$$

(29)

$${\lambda }_b=\frac{J({\lambda }_d)-J({\lambda }_a)}{{\lambda }_d-{\lambda }_a}$$

(30)

$${\lambda }_c=\frac{1}{{\lambda }_e-{\lambda }_d}[\frac{J({\lambda }_e)-J({\lambda }_a)}{{\lambda }_e-{\lambda }_a}-{\lambda }_b]$$

(31)

where ${\lambda }_a$ and ${\lambda }_d$ are, respectively, the two endpoint values of the interval the optimal step size $\lambda$ lies in. ${\lambda }_b$ and ${\lambda }_c$ are generally the auxiliary variables. ${\lambda }_e$ is the middle point of the interval $[{\lambda }_a,{\lambda }_d]$ .

• 8. When the optimal step length $\lambda$ is achieved, compute the updated Bayes objective function $J({\mathit{E}}^{k+1})$ and the gradient $\nabla J({\mathit{E}}^{k+1})$ .

• 9. For the following convergence judgment equations, if either of the next two criteria is satisfied, it is that

  $$\left|\frac{J({\mathit{E}}^{k+1})-J({\mathit{E}}^k)}{J({\mathit{E}}^k)}\right|<{\epsilon }_1$$

  (32)
  $$\Vert \frac{{\mathit{E}}^{k+1}-{\mathit{E}}^k}{{\mathit{E}}^{k+1}}{\Vert }_2<{\epsilon }_2$$

  (33)

In equation (32), the mathematical symbol |·| means absolute value. In equation (33), the mathematical symbol $\Vert ·{\Vert }_2$ means the second norm.

The iterative computation is convergent, which means that the identification results of displacement constants $\mathit{E}$ are $\hat{\mathit{E}}={\mathit{E}}^{k+1}$ . The iteration is terminative and the procedure jumps to the final step (13). Otherwise, proceed with the following step.

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

• 11. Compute the variable vector difference ${\mathit{S}}^k$ and the gradient vector difference ${\mathit{Y}}^k$ in conformity with the following equations

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

  (34)
  $${\mathit{Y}}^k=\nabla J({\mathit{E}}^{k+1})-\nabla J({\mathit{E}}^k)$$

  (35)

• 12. Determine the iterative variable scale matrix ${\mathit{H}}^{k+1}$

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

  (36)
  $$\mathrm{\Delta }{\mathit{H}}^k=\frac{{\mathit{S}}^k{({\mathit{S}}^k)}^T}{{({\mathit{S}}^k)}^T{\mathit{Y}}^k}-\frac{{\mathit{H}}^k{\mathit{Y}}^k{({\mathit{Y}}^k)}^T{\mathit{H}}^k}{{({\mathit{Y}}^k)}^T{\mathit{H}}^k{\mathit{Y}}^k}$$

  (37)

Then, let $k=k+1$ and return to step (3) to proceed with reiteration.

• 13. The covariance ${\mathit{C}}_{\hat{\mathit{E}}}$ of the displacement constants $\mathit{E}$ is achieved from equation (12).

Analysis of the examples

The variable scale of displacement constants $\mathit{E}=[E_1{E}_2{E}_3{]}^T$ ( $E_1$ , $E_2$ , and $E_3$ are, respectively, the elastic modulus of different parts of the box, noted at the top plate, web plate, and bottom plate) of curve indeterminate box girder in Figure 4 is studied in different cases. Internal redundant forces of diaphragm and external redundant forces of supporters are altogether in the midspan section in Figures 5 and 6. Through element partition, the serial numbers of the curve-strip-shaped elements and the nodal lines are shown in Figure 4. The length of the middle axis is $L=120\mathrm{cm}$ , the radius is $R=300\mathrm{cm}$ , and the angle is $\alpha =0.4\mathrm{rad}$ . The actual displacement constants $\mathit{E}$ and Poisson ratio $\mu$ together with the widths of the top plate, web plate, and bottom plate noted as $t_1$ , $t_2$ , and $t_3$ are assigned in Table 1. And, the coefficient of variation of displacement constants $\mathit{E}$ is set as 0.1, with the supposition that each constant is independent. The vertical uniformly distributed loads $p_1$ and $p_2$ , which are both equal to 10 N/cm, are, respectively, placed to the nodal lines of No. 2 and No. 4 of the continuous box girder and $m=30$ is given to the series item. In order to put the identification of displacement constants $\mathit{E}$ of curve indeterminate box girder into practice, the corresponding procedure development is completed, in which the subprogram of curve indeterminate box girder is called. The six points (Nos 1–4, 10–11; shown in Figure 3) in midspan cross section are selected as acquisitive points of displacement data which are measured for five times at each referred point. The expectations and standard deviations of the acquired systematic responses are shown in Table 2.

Figure 4.

Numbers of curve-strip-shaped elements and nodal lines (cm).

Figure 5.

Vertical view of continuous box girder.

Figure 6.

Connection points between box and diaphragm.

Table 1.

Actual values of displacement constants and other parameters of the curve indeterminate box girder.

Parameter	$E_{1\mathrm{true}}$	$E_{2\mathrm{true}}$	$E_{3\mathrm{true}}$	$\mu$	$t_1$	$t_2$	$t_3$
Unit	104 N/cm2	104 N/cm2	104 N/cm2	–	mm	mm	mm
Value	250	220	280	0.16	4	5	4

Table 2.

Expectations and standard deviations of the systematic responses (10⁻² cm).

Selected points	Displacement expectations					Displacement standard deviations
	$W_1$	$W_2$	$W_3$	$W_4$	$W_5$	${\sigma }_{W_1}$	${\sigma }_{W_2}$	${\sigma }_{W_3}$	${\sigma }_{W_4}$	${\sigma }_{W_5}$
1	10.65	10.69	10.61	10.68	10.62	1.53	1.55	1.51	1.57	1.54
2	8.91	8.98	8.93	8.96	8.94	1.28	1.22	1.23	1.31	1.26
3	7.94	7.92	7.97	7.98	7.91	1.27	1.21	1.32	1.24	1.31
4	8.91	8.87	8.84	8.96	8.93	1.94	1.89	1.85	1.91	1.86
10	7.93	7.94	7.96	7.91	7.95	1.49	1.45	1.42	1.41	1.44
11	7.96	7.91	7.88	7.85	7.94	1.33	1.28	1.31	1.26	1.25

Case 1

The identification of displacement constants of the curve indeterminate box girder when the priori information is accurate, which indicates that the priori information of the continuous box girder is ${\mathit{E}}_0=[250.0,220.0,280.0{]}^T$ . The two groups of the initial constant values of displacement constants are, respectively, set ${\mathit{E}}_{G1}=[100.0,100.0,100.0{]}^T$ and ${\mathit{E}}_{G2}=[400.0,400.0,400.0{]}^T$ . Suppose the convergence criteria ${\epsilon }_1=0.1$ , ${\epsilon }_2=0.1$ , and the constant variation coefficient is 0.1. Together with the data in Table 2, the iteration results of displacement constants are attained in Table 3 and Figure 7. The iteration achievements of logarithm of updated Bayes objective function is shown in Figure 8.

Table 3.

Identification results of displacement constants of continuous box girder in Case 1 (10⁴ N/cm²).

Displacement constants	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$
Initial value	100.0	100.0	100.0	400.0	400.0	400.0
Final value	250.11	220.42	279.73	250.12	219.87	280.23
Iterative times	28	28	28	18	18	18
Relative error (%)	0.04	0.19	0.10	0.05	0.06	0.09
convergence criteria	${\epsilon }_1$	${\epsilon }_1$	${\epsilon }_1$	${\epsilon }_2$	${\epsilon }_2$	${\epsilon }_2$

Figure 7.

Variable scale results of displacement constants in Case 1 (10⁴ N/cm²): (a) iterative results when the first group is set and (b) iterative results when the second group is set.

Figure 8.

Iterative results of logarithm of updated Bayes objective function in Case 1.

From the results in Table 3 and Figure 7, it is indicated that when the priori information is accurate, the optimization process of the mechanical identification model of displacement constants of the curve indeterminate box girder converges to the actual parameter values stably. Iterative results of logarithm of updated Bayes objective function in Figure 8 are steadily decreased. The results indicate that whether the initial constant values are close to the actual parameter values or not, the convergence of the mechanical identification model is unassociated with the initial constant values. The advantage of the proposed method lies in that compared with the Powell direct optimal results,^13,14 the mechanical identification model is more efficient because the computational times of the derived Bayes objective function mainly resulting from the finite strip element model are fewer. The final coefficient of variation of constant is about 0.08, which is ameliorated in comparison with the given value. In this case, the optimization process can converge according to the two convergence criteria if different initial constant values are assigned.

Case 2

In order to achieve some other regularities of the mechanical identification model of displacement constants of curve indeterminate box girder while the priori information is accurate, the initial values of the third group ${\mathit{E}}_{G3}=[400.0,100.0,400.0{]}^T$ and the fourth group ${\mathit{E}}_{G4}=[100.0,400.0,100.0{]}^T$ are, respectively, chosen. The rest data are the same as Case 1. The iterative achievements of the mechanical identification model of displacement constants are listed in Table 4 and Figure 9. The iterative results of logarithm of updated Bayes objective function are shown in Figure 10.

Table 4.

Identification results of displacement constants of continuous box girder in Case 2 (10⁴ N/cm²).

Displacement constants	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$
Initial value	400.0	100.0	400.0	100.0	400.0	100.0
Final value	249.87	220.01	280.04	250.87	219.03	279.71
Iterative times	24	24	24	36	36	36
Relative error (%)	0.05	0.01	0.01	0.35	0.44	0.10
Convergence criteria	${\epsilon }_2$	${\epsilon }_2$	${\epsilon }_2$	${\epsilon }_1$	${\epsilon }_1$	${\epsilon }_1$

Figure 9.

Variable scale results of displacement constants in Case 2 (10⁴ N/cm²): (a) iterative results when the third group is set and (b) iterative results when the fourth group is set.

Figure 10.

Iterative results of logarithm of updated Bayes objective function in Case 2.

From the results in Table 4 and Figure 9, it is shown that the optimization processes still converge to the actual values steadily. Iterative results of logarithm of updated Bayes objective function in Figure 10 are also steadily decreased. Some other conclusions can be drawn as follows. First, supposed that the initial constant values are closer to the actual values, the iteration times are not necessarily reduced when either of the convergence criteria is satisfied. For example, compared with the second group, the third group is evidently closer to the actual value ${\mathit{E}}_{\mathrm{true}}$ . However, the iteration times of the third group are 24 times, which are more than that of the second group (18 times). Second, the convergence accuracies are all very high and the relative errors are less than 2%, which are indistinctively ameliorated when the initial constant assignments keep nearer to the actual data. Third, one of the initial constant values is changed, the whole iterative times and relative errors may probably be altered. For example, compared with the first group, the item $E_2$ in the fourth group is intentionally altered and the rest are unaltered. Actually, the iterative times and relative errors of the unaltered items of $E_1$ and $E_3$ are subsequently influenced. The reason for the above result lies in that during the identification processes of multiple constants, the relationships between the poly constants are interrelated and interdependent.

Case 3

The identification analysis of displacement constants of curve indeterminate box girder when the priori information is inaccurate. This case is consistent with the practical engineering because the presupposed priori information only dependent on engineering experience is hardly possible to agree with the actual value. Let priori information ${\mathit{E}}_0=[120.0,120.0,120.0{]}^T$ . Conveniently to make comparisons, the initial values of the displacement constants are set $E_{G5}={[100.0,100.0,100.0]}^T$ and ${\mathit{E}}_{G6}={[400.0,400.0,400.0]}^T$ , and the rest data are kept the same as Case 1. The iterative achievements of the variable scale of displacement constants are attained in Table 5.

Table 5.

Identification results of displacement constants of continuous box girder in Case 3 (10⁴ N/cm²).

Displacement constants	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$
Initial value	100.0	100.0	100.0	400.0	400.0	400.0
Final value	131.24	125.86	156.71	286.38	257.32	326.84
Iterative times	150	150	150	26	26	26
Relative error (%)	–	–	–	14.55	16.96	16.73
Convergence criteria	Divergent	Divergent	Divergent	${\epsilon }_2$	${\epsilon }_2$	${\epsilon }_2$

It can be found from Table 5 that if the iteration processes of systematic constant converge, then it only conforms to the second convergence criterion. The iterative results indicate that all of the relative errors are more than 10%, which means that the iterative processes are divergent. The reason is that combined with equation (4), when the priori information is inaccurate, it is improbable to make both of the items in the right of equation (4) converge to zero synchronously. Thus, although the iterative processes may converge, they cannot satisfy the first convergence criterion. Lacking the evaluating criterion of the accuracy of the actual values, how to judge whether the priori information is assigned appropriate is of great significance. Otherwise, the inappropriate assignment will lead the displacement constants to converge to wrong values, which would influence the engineering design by error. Through much research work, the regularity may be summarized and achieved that if the priori information satisfies the accurate condition, then the iteration processes can converge by both of the convergence criteria with different iterative times. If the priori information is inaccurate, then the iteration processes may diverge or converge only by the second convergence criterion.

Conclusion

The mechanical identification model of displacement constants of curve indeterminate box girder is completed with updated Bayes theory, and the main conclusions are drawn:

1. The identification processes of displacement constants of curve indeterminate box girder are steadily convergent to the actual values when the priori information is precise, which proves the accuracy of the derived mechanical identification model and the reliability of the compiled procedure;

2. During the iterative process, the mechanical identification model with variable-scale optimization theory incessantly engenders new search directions. It optimizes the derived objective function more efficiently than the Powell method, which depends simply on the revision of the objective function.

3. Compared with the routine Bayes objective function, it is an improvement that the updated Bayes objective function in the mechanical identification model can tackle the measured systematic responses of different times and different spots simultaneously, which can also consider the randomness of the displacement constants and systematic responses accurately.