After discretizing the structure into a finite-element model with n degrees of freedom, the static equilibrium equation can be expressed as:

(1)

where, K(p) represents the global stiffness matrix of the structure; p , U and F represent the parameter vector to-be-identified, the static displacement vector and the load vector of the structure, respectively. Correspondingly, the structure’s displacement response vector U and strain response vector ε under the load F can be expressed as:

(2)

where, B represents the strain-displacement relationship matrix.

As with the dynamic response test, general loading and testing can only be conducted with a small degree of freedom in current structures. Furthermore, loading and testing usually do not follow the same sequence of degrees of freedom. It is assumed that the structural response test will be carried out under nlc linearly-independent working conditions. Under the i^th working condition, assuming that the number of degrees of freedom tested in the displacement response test is m, that the number of degrees of freedom untested is l, and that the structural load vector is F_i, then the corresponding displacement and strain test values become U_mⁱ and ε_mⁱ, respectively. Under the i^th working condition, the theoretical values of static response of the structure are U_i and ε_i, respectively. For correspondence between analysis results and measured results, U_i can be divided into a tested component U_Ti (m dimensions) and an untested component U_Ni (l dimensions); F_i can be divided into F_Ti and F_Ni. On this basis, Eq. (1) can be converted into:

(3)

The following equation can be obtained from Eq. (3):

(4)

After substituting Eq. (4) into Eq. (3), U_Ti can be shown as:

(5)

To simplify Eq. (5), it can be expressed as:

(6)

Similarly, the strain ε_Ti corresponding to the measured part has a similar form:

(7)

where, H_uⁱ and H_εⁱ represent the displacement matrix and the strain observation matrix, respectively.

The displacement and strain residual vectors corresponding to the i^th working condition are therefore:

(8)

(9)

Due to unavoidable errors with displacement and strain test data in structural and load test processes, and taking into consideration that errors may vary with the degrees of freedom of testing and loading, the displacement-weighted matrix (W_u), the strain-weighted matrix (W_ε) and the load-weighted matrix (W_F) have been introduced with regard to each working condition. These matrices are in accordance with the precisions of various displacement measuring points, strain measuring points and loading conditions. W_u, W_ε and W_F are all diagonal matrices and the diagonal elements are the weighted values of the observed results of static displacement vectors, strain vectors or load vectors under various working conditions. For example, in the i^th working condition, diagonal elements of W_ui are the weighted values of the static displacement vector in E_uⁱ(p).

(10)

In the i^th working condition, W_εi and W_Fi have the same form as W_ui. For repeated trials, the diagonal elements of W_ui and W_εi can be selected as the reciprocal of variance of observed results, and the diagonal elements of W_Fi can be selected as the reciprocal of variance of loads. However, for a single trial, the diagonal elements can be chosen according to the same rule of thumb. After weighted correction, the displacement and strain residual vector corresponding to the i^th working condition can be expressed as:

(11)

(12)

By using the head-tail connection of the displacement and strain residual vector under each working condition, the overall residual vector of the static response of the structure can be obtained.

The magnitude-weighted matrix W_ss can also be introduced to correct under two considerations: (1)the magnitude of the displacement data and the strain data in the residual vector need to be maintained on the same order; and (2)consideration of the relatively significant differences in the magnitudes of different test data will prevent low-magnitude test data being submerged by high-magnitude test data. According to Cui, W_ss can therefore be represented as [11]:

(13)

Assuming that the number of dimensions of all the static response test data under all working conditions is SM, and taking the minimum variance solution as the solution to each parameter to-be-identified, then the static response-based structural parameter identification problem can be summarized as the optimization problem:

(14)

where, e_i (p) represents the i^th element of the SM-dimensional vector E_st (p); and p^L, p^U are the upper and lower limits of p.

The relationship between the change of cable force and the change of displacement will be derived in this section. If the variations of displacement and cable force can be simultaneously measured, the parameters to be solved can then be identified by using the optimization method.

The strain energy of the structure can be expressed as [12, 13]:

(15)

If the influence of shear stress is neglected, the strain energy of the structure can be further expressed by introducing the finite-element concept:

(16)

where, m represents the number of beam elements; l_i, E_i, I_i and A_i represent the length, elastic modulus, sectional moment of inertia and sectional area of the i^th element, respectively; M_i^l and M_i^r represent the bending moments at the left and right ends of the i^th element, respectively; N_i^l and N_i^r represent the axial forces at the left and right ends of the i^th element, respectively.

Here, the bending moment and the axial force can be expressed as a function of external forces:

(17)

where, F_DLP represents dead load, live load and pre-stressing force; and T represents the cable force. Matrices B and C are determined by element characteristics and structural forms. Without imposing other constraint conditions, ∂U/∂T_i = 0 indicates that the displacement of a notch of the stay cable relative to the unknown cable force T_i is 0, and on this basis the following equation can be established:

(18)

where, Φ represents the influence matrix, and the value in the i^th row and the j^th column represents the variation of the i^th target value (including displacement, stress, counter-force, etc.) after the cable force of the j^th stay cable goes through element change; and δ represents the actual variation of the target value. This equation is in fact an equation of force method where Φ represents the flexibility matrix; u_DLP represents the displacement vector caused by other loads; and δ=−u_DLP.

Jiujiang Bridge is a single-pylon, double-cable plane PC cable-stayed bridge based on the fixed pier-pylon-beam system. This bridge, being operational since June 1988, has a span combination of 2×160m, as shown in Fig. (1) and Fig. (2) illustrates the general cross-sectional profile. The geometric shape of the girder and cable force tests were conducted in July, 2000 and August, 2008;both tests were undertaken under similar conditions. In addition, by the time of the first test, the bridge had been in service for more than a decade and concrete creep had ceased. Therefore, it was presumed that creep exerted no influence on girder deformation from 2000 to 2008. As described in the previous section, without considering the influence of stay cable relaxation and temperature on cable force and the geometric shape of the girder, the variation of external loads may give rise to a corresponding displacement, and the change in the shape of the cable-stayed bridge girder is only caused by the change of cable force.

Fig. (1). Bridge type layout.

Fig. (2). General cross-sectional profile of the girder.

The spatial finite-element model of the cable-stayed bridge is shown in (Fig. 3); the pylon, stay cable and bridge deck system are simplified and they have been applied with corresponding loads.

Fig. (3). Spatial finite-element model.

(1) Structural members: The girder, pier and pylon all adopted BEAM4 elements and the cable adopted the LINK10 element.

(2) Materials and sections: The concrete elastic modulus of the girder adopted the result measured by the on-site rebound method, and its sectional area and sectional moment of inertia referred to the design drawings.

(3) Loads: Given the materials and parameters of the structure, its actual state of internal force only depends on four factors: dead load of the structure, counter-force of the support, permanent pre-stressing force inside the girder and cable force of the stay cable [14]. (i) The dead load of the structure was determined according to the design drawings. (ii) The counter-force of the support was calculated by simulating the state evolution process of the bridge (including the construction stage of the bridge and the operation stage after bridge completion) based on existing design data and taking into account the influence of various factors on the structure. (iii) The permanent pre-stressing force was calculated using the equivalent load method and calculation of pre-stressing force loss according to the General Code for Design of Highway Bridges and Culverts (JTG D60-2004) [15]. (iv) The cable force refers to the result of the full-bridge cable force measurement undertaken in 2008 [16].

Based on the sensitivity of different parameters, selecting parameters more sensitive to displacement change for correction can improve model correction efficiency [17]. As known from sensitivity analysis, the mass of the girder exerts no influence on deformation; stiffness is the key parameter influencing girder deformation. Given that it is very difficult to accurately obtain the current concrete elastic modulus value of the girder, section area and moment of inertia of the girder were taken as constant values (referring to the values given by the design drawings), and concrete elastic modulus as a corrected parameter was adopted. As the bridge has different types of girder sections at the site of mid-span, side span and pylon, the girder was divided into six zones in the longitudinal direction (as shown in Fig. 4), each of which was assigned with a corrected parameter value. Table 1 shows the variables and value ranges of the parameters to-be-identified.

Fig. (4). Schematic diagram of identification sections used to measure girder stiffness.

After the determination of parameters to-be-identified, the parameter identification of the girder became an issue of structural parameter optimization. Based on the girder’s geometric shape and cable force data of the bridge obtained from the two tests (2000 and 2008), the change of cable force was adopted as the real variable parameter optimization, thus making the displacement change value of the model approach the measured value. In the model, girder displacement caused by the change of measured cable force was U_ai (calculated displacement). Taking U_ai as a design variable, the absolute value of the relative error between calculated displacement and measured displacement was defined as the objective function:

(19)

where, U_ti represents the measured displacement difference (of the girder’s geometric shape) between 2008 and 2000; and U_ai represents the calculated displacement. The number of measuring points is reflected by the number 36.

ANSYS optimization module was employed to create the corresponding optimization analysis file for optimized iterative analysis by a total of 50 iterative steps. The optimal solution was obtained after the convergence of the program ended at the 12^th step. The final value of the objective function was min(objection_function). After solving the optimal solution by the first-order optimization method, the zero-order optimization method was then employed for verification. The two methods obtained the same result and presented a high level of consistency, suggesting that the optimization result is a global solution rather than solely a local solution. Fig. (5) shows the convergence of the objective function and Table 1 shows the values of various design variables under the optimal result. A comparison between the calculated displacement and the measured displacement values of various measuring points after parameter optimization is shown in (Fig. 6).

Fig. (5). Convergence of objective function iteration.

Fig. (6). Comparison between measured and calculated displacement increments after parameter identification.

(Note: The displacement measuring point is located at the connection between the stay cable and the girder)

Results for the theoretical displacement and the measured displacement Fig. (6) show a relatively significant error; the average difference was 36.72%. Results for the corrected displacement and measured displacement showed a difference of 5.32%, indicating a good fit. As shown in Table 1, the coefficients of elastic modulus of various zones after identification ranged between 0.782 and 0.854, with a mean value of 0.812. This suggests that the mean decline in the stiffness of the girder was about 18.8%. To test the degree of influence of measured displacement error on the parameter identification results, for the purpose of parameter optimization, the measured displacement value U_ti was increased and decreased by 5%, respectively. When U_ti increased by 5%, the mean value of the coefficients of elastic modulus (BEAM_E) decreased by 3.16%; when U_ti decreased by 5%, BEAM_E values increased by 3.94%. These results suggest that a small input error would not result in a large output error, and that the error of the measured displacement U_ti caused no significant disturbance to the identification results.

A comparison between the results of internal force identification and the values measured by the stress release method in 2007 are shown in Table 2 [18, 19]. The mean value of the identification results was 2 MPa less than that of the measured results; the results provide consistent rules of stress distribution. The difference in result between the two sets of results may be due to different temperature conditions. However, under the dead load state, the girder stress was always in a compressed state.

The crack resistance and compression resistance of the girder were calculated in the normal service stage. The calculation results of the normal section crack resistance of the girder are shown in Fig. (7). Under the combination for short-term action effects, the top flange had a relatively high compressive stress reserve. However, the compressive stress reserve of the bottom flange was already very low; a tensile stress of 0.87MPa was recorded at the bottom flange position near Cable Z15#. According to the requirements of the Generalized codes for design of highway bridges and culverts (JTG D60-2004) [15], tensile stress is not allowed.

Fig. (7). Combination for short-term action effects: maximum stress values of the top-flange and bottom-flange normal sections of the girder.

The calculation results for the normal section compression resistance of the girder are shown in Fig. (8). Under the characteristic combination, the maximum compressive stress (-15.77MPa) occurred near Cable Z4#, already approaching the specified allowable value of -16.2 MPa [15]. The girder was therefore already in an extremely dangerous working state and adjustment of the internal force was required.

Taking U as the bending strain energy of the structure, the following objective function can be used:

(20)

where,

and represent the matrices constituted by the bending moment and the axial force on the left and right ends of the element under the action of dead load (including dead weight, pre-stressing force and current cable force); and represent the influence matrices of the bending moment and the axial force on both ends of the element under the action of individual element cable forces; and {T} represents the adjusted cable force vector. {B} is a diagonal matrix having the following diagonal elements:

As Jiujiang Bridge is a single-pylon symmetric structure, to prevent cable force adjustment from causing an additional bending moment to the main pylon, it is assumed that the stay cables at symmetric positions on both sides of the pylon (having the same cable number, such as Z10# and J10#) have the same adjusted cable force. It is also assumed that the four horizontal cables (A, B, C and D) corresponding to the same cable number also have the same adjusted cable force. In total there were 18 adjusted cable force variables.

After determining the objective function and constraint conditions, the problem of cable force optimization was converted into a quadratic programming problem in its standard form:

In fact, the model for the cable force optimization of cable-stayed bridges is a special constrained extreme value model, expressed in the form of a sum of squares or weighted sum of squares. [H] in the model is therefore a positive definite matrix. On this premise, the optimization problem of this model is converted into a convex quadratic programming problem [21], to which there is a unique global solution. MATLAB toolbox was used to solve the quadratic programming problem and provide a solution to the cable force optimization problem by using the “quadprog” function in the optimization toolbox.

(1) Results for the optimization of cable force are shown in Fig. (9). After cable force optimization, Cable 8# had the maximum adjusted cable force (-118.8kN) and Cable 3# had the minimum adjusted cable force (6.2kN). It can also be seen that various groups had relatively uniform cable forces.

Fig. (9). Results of optimization of cable force.

(The distribution of A,B,C and D are shown in Figs. (1 and 2).

(2) Results for cable displacement after optimization are shown in Fig. (10). Results show that the geometric shape of the girder was restored to the level recorded in 2000. The maximum displacement increment (27.83 mm) was observed at cable position 13#.

Fig. (10). Displacement increment comparison diagram after optimization of cable force.

(3) Stress and internal force results after optimization for the girder are shown in Fig. (11). The top and bottom flanges of the girder are both in the compression situation under the state of dead load, the combination for short-term action effects and the characteristic combination. Neither the crack resistance value nor the compression resistance value of the girder in the checking calculation exceeded the specified limits [15]; the maximum compressive stress of the top and bottom flanges were -10.9Mpa and -14.6Mpa, respectively, indicating that stress had been effectively controlled.

Fig. (11). Girderstress diagram after optimization of cable force.