Investigation of Actual In-Plane Geometric Imperfections of Steel Tied-Arch Bridges

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Unique data on real geometric imperfections of existing tied-arch bridges for advanced nonlinear buckling analysis.

Abstract

The complex global analysis of the steel tied-arch bridges may require specific data in some cases, e.g., residual stress or real geometrical imperfections. The presented research is focused on the investigation of the actual in-plane shape of the steel tied-arch bridges. To obtain the actual shapes of the arches of existing bridges, 3D terrestrial laser scanning was applied. Data were then investigated as a group in order to find the relationship within the tied-arch bridges’ vertical imperfections. Firstly, the detailed surveyed points in the selected position were mathematically determined to estimate the measured data. The next step included the creation of curves presenting the real shape of the bridges’ arches course using numerical modelling. After the pre-programming, the calculations and related numerical modelling were performed in the Matlab environment. These modelled arch-bridges courses were approximated by using regression analysis. The approximating curves from the regression analysis represent the theoretical course of the arches. The difference between the theoretical shape and the processed measured values may be presented as the examined in-plane imperfections. Subsequently, the paper presents comparisons from the evaluation of the individual imperfections on the investigated bridges and a comparison of the measurement with the standardised shape of the imperfections.

1. Introduction

Steel tied-arch bridges are one of the most suitable types of bridge superstructures due to their shape, which is close to a beam deflection curvature from the external load. Together with the favourable architectural aspect, this was the reason for their application as a part of the railway and road transportation network in history and in the present. In general, most of these bridges were built many decades ago, and they are still in use despite adverse external influences and often neglected maintenance. In some specific cases, it is necessary (or will be necessary) to monitor the real status and functionality of these bridge superstructures.

In general, the most important information about the structures and structural elements is obtained from experimental verification of the structures and their parts [1,2,3] from the diagnostic surveys [4,5] and the continuous monitoring of the building structures and the transport structures [6,7,8].

All experimentally and diagnostically determined data can be subsequently applied in further theoretical analyses. This data collection and application system is also preferably implemented in the analysis of the bridge structures, which are essential elements of the transport infrastructure.

The global analysis of the arch bridges may become more complicated in some cases [9,10,11], especially in the case of existing bridges [12,13,14]. A number of the input parameters influence the real behaviour of the tied-arch bridges and their resistance determination. The steel arches are loaded by a combination of compressive force and bending moments. These types of bridges are prone to stability loss both in- and out-of the arch plane [15,16,17]. The buckling length in the arch plane is affected by the position of the hangers, the way they connect, the stiffness of the arch-to-girder joint, and the overall arrangement. The out-of-plane buckling length from the plane of the arch is influenced mainly by the use of the upper longitudinal bracing and also by the construction of the arch-to-girder joint [10]. In order to verify the arch stability, in some specific cases, it is appropriate to perform a more detailed geometrical and materially nonlinear analysis, taking the second-order theory and the imperfections into account.

In the majority of the currently published outputs, the real geometric imperfections measurement is, in particular, focused on the smaller structural elements scanned using laboratory 3D scanners [18,19] or using contact measurement principles [20]. The submitted research aims to collect data on the tied-arch bridges’ imperfections in situ. Consequently, the real imperfections shape and their amplitudes may be taken into account within the computational models instead of the possible theoretic imperfect shape [21]. The shape of the geometrical imperfections entering into these analyses is usually considered in terms of existing technical standards (e.g., the shape of the theoretical elastic buckling modes). The application of the real measured imperfections into the analyses seems to be an alternative approach. For this reason, the actual shapes of the tied-arch bridges were measured and compared with their proposed theoretical shape. The results of the measurements are influenced by the actual arches’ deformations from their self-weight as well as by the quality of their construction and assembly. Due to the terms and conditions of the task, it was necessary to choose an effective and precise measurement method that meets the demanding requirements on the density of measured data, the precision of the final pre-processed data, and the accuracy of the model, which is compared with the theoretical model after processing.

All in all, there are several methods for capturing and measuring the location and the shape of the examined objects in a geodetic way. The shape of the tied-arch bridges may be expressed according to the spatial polar coordinates from the measurement using UMS (Universal measuring station) or TS (total station) and consequent measurement processing. Another way of object shape expression includes 3D terrestrial laser scanning technology. According to the mentioned preset conditions and requirements, this method was chosen to fulfil the research aim—the analysis of the tied-arch bridges’ imperfection concerning the hanger systems within the tied-arch bridges located in Slovakia-Poland-Czechia cross-border territory.

In general, the application of 3D terrestrial laser scanning is a part of many kinds of research and publications. Its benefits and applications have been implemented in various engineering or natural fields so far [22,23]. As previously mentioned, this method’s choice for the vertical imperfections investigation was based on several preset research conditions and requirements. These requirements and conditions encompass non-contact, effective, and precise measurement, as well as the ability to establish an appropriate data density to ensure the inclusion of important or strategic bridge-point and capture the overall bridge course and behaviour.

The physical nature of the 3D laser scanning measuring is not a subject of this paper, as many other innumerable articles, research, and publications are devoted in detail to the fundamentals, principles, and practical application of 3D terrestrial laser scanning [24,25,26,27,28].

An interesting alternative approach is to quantify the real geometric imperfections of arch bridges based on utilising on-site strain measurements. The actual imperfections can be determined by comparing the measured strain values with the strain values obtained from a linear combination of theoretical imperfections (calculated using sophisticated finite-element models). This procedure was performed to investigate out-of-plane imperfections of tied-arch bridges [29].

2. Structural Arrangement of the Bridges

There are many different types of arch bridges depending on their structural system, the position of the bridge deck, the shape of the arch, or the ratio of the stiffness of their structural members. Due to the simpler conditions of their foundation and the transfer of the support reactions, tied-arch bridges are often preferred over true arches. This type of steel bridge with a bottom bridge deck has found wide application on roads and railways, especially when the bridging obstacles are of medium and larger spans. The bridge superstructure is mostly constructed with one or two steel arches, the main girders, the hangers, the bridge deck, and the upper longitudinal bracing (Figure 1).

In most cases, the theoretical shape of the axis of the arch copies the shape of a circle or parabola (Figure 2a); exceptionally, it can be a catenary. In older tied-arch bridges, the shape of the arch is often polygonally angled (Figure 2b).

The main girders are stressed by the bending moments and large tensile forces, and at the ends, they are mostly rigidly connected to the arches. The hangers that connect the main girders to the arch have mostly pinned joints, and their arrangement can be various (Figure 3). If there is an upper longitudinal bracing on the bridge, it is usually designed as a frame bracing or a truss bracing system.

3. Investigated Tied-Arch Bridges

Nine bridge superstructures were chosen as a representative sample for the analysis of the actual arch shape in their plane and their comparison with the theoretical ideal shape. Selected bridges have different hanger arrangements, and they are part of a road, railway, or pedestrian walkway. Two selected bridges have a network-shaped arrangement of hangers, two bridges have more massive vertical hangers, the next four bridges have a slender vertical hangers arrangement, and one bridge has a modified Nielsen-type system of the hangers’ arrangement.

3.1. Břeclav Railway Bridge

The railway bridge called “Oskar” (Figure 4a) lies on the main line between Břeclav and Vienna. It consists of a pair of separate steel bridge superstructures with a total theoretical span of 97.5 m and skewness of up to 45° (there is a separate superstructure for each track). The cross-sections of the arch and the main girder are closed, the bridge deck is orthotropic, and the network arrangement of the hangers is used.

3.2. Brodno Footbridge

The steel footbridge (Figure 4b) over the Kysuca River is situated in the district of Žilina city, called Brodno. The span of the footbridge is 75 m, and the axial distance between the individual arches is 3.4 m. The hangers are made of high-strength steel bars and arranged in a network shape. The main girder and the arch have box cross-sections. The bridge deck consists of I-shaped steel cross beams on which walkable composite gratings are placed.

3.3. Mohelnice Road Bridge

The road bridge (Figure 5a) over the watercourse Morava is situated in the non-urban area of the town of Mohelnice on the road Nr. 444. The total span of the bridge is 46 m, the hangers are arranged vertically, and their distance is only 3 m. The main girder and arch have closed cross-sections, and the bridge deck is made of steel cross beams and reinforced concrete slab. The bridge has no upper longitudinal bracing.

3.4. Olomouc Road Bridge I

The road bridge (Figure 5b) over the Morava river is located on “U dětského domova” street. The theoretical span of the bridge is 64.9 m, the hangers are arranged vertically, and their distance is 3.0 m. The cross-section of the arch is a closed asymmetrical steel box, and the main girder is a concrete-encased steel beam. There is no upper longitudinal bracing on this bridge either.

3.5. Nedvědice Road Bridge

The bridge (Figure 6) on the border of the Vysočina Region over the Svratka River has a theoretical span of 30 m. The steel arches are without the upper longitudinal bracing, and the hangers have a modified Nielsen-type arrangement. The bridge deck consists of steel-concrete composite beams.

3.6. Olomouc Road Bridge II

The road bridge (Figure 7a) over the Morava River is located on “Heydukova” Street in Olomouc. The theoretical span of the bridge is 63 m, and the main girders are connected to the arch by nine massive hangers of circular cross-sections. The cross-section of the arch is a closed steel box, and the cross-section of the main girders is also created as a steel box. The system of composite steel to concrete beams forms the bridge deck. The arches are axially distant at 13.77 m, and they are braced by the combination of the massive portal frame and the truss upper longitudinal bracing.

3.7. Karviná Railway Bridge

This bridge (Figure 7b) consists of two single-track independent steel superstructures placed side by side. The superstructure acts as a three-span continuous girder, while the middle span (60 m) is stiffened by the arch. The arches have a frame system of upper longitudinal bracing, and the hangers are vertical.

3.8. Milówka Railway Bridge

The one-track railway bridge (Figure 8a) on line Nr. 139 “Katowice—Zwardoń” over the river Sola has a span of 75 m. The arches have a closed box cross-section, and the axial distance of the arches is 5.0 m. The main girder is connected to the arch by nine vertical hangers of the circular cross-section. The steel bridge deck is orthotropic with trapezoidal ribs, and the upper longitudinal frame bracing is used on the bridge.

3.9. Karviná Road Bridge

This bridge (Figure 8b) carries a four-lane road (I/59) and a pair of sidewalks over the Olša River. The superstructure acts as a three-span continuous girder, while the middle span (67.37 m) is stiffened by the arch. The axial distance of the arches is 18.16 m, and they have a rectangular closed cross-section. The bridge deck is made of a system of steel-to-concrete beams. The bridge has no upper longitudinal bracing, and the hangers are vertical and quite slender.

4. Selected Method of Bridges Shape Measurement

As previously mentioned, before the investigation, it was considered a matter of choosing the appropriate method for the reliable and complex determination of the arch shape. The non-invasive measurement, sufficient precision, and the amount of examined tied-arch bridges were the main reasons for choosing the method of 3D laser scanning. The course of the measurement (station position, number of stations, etc.) depended on each scanned bridge type and other pre-defined conditions. However, the process of the measurement, in general, was the same in every case. On every bridge, a single arch side has been measured, depending on the position accessibility. The mentioned pre-defined conditions include the bridge accessibility and the position, the possibility of setting a proper station for the laser scanner, etc. The measurement was performed with regard to the aim of capturing the course of the actual shape of the arch as accurately as possible. During the measurement of the railway bridges, there were more extensive intervals between trains, allowing measurements to be taken during these gaps without any influence from the live load on the bridge. In the case of road bridges, there may have been some traffic on the bridge (but outside rush hours and without heavy trucks occurring on the bridge during the measurements). However, the bridges were measured multiple times from the various positions, and for the post-processing, only the positions that were minimally affected by the traffic were selected. Additionally, the final point cloud represents the average position of the investigated bridge elements, which means that the short-term effect from the live load (such as small vehicles, a gust of wind, etc.) would have only had a negligible impact on the measurement.

The arch shape was always measured from at least two stations with a 3D laser scanner Leica ScanStation [30]. The stations’ positions were chosen in order to capture the tied-arch bridge in the most suitable way to eliminate atmospheric errors and reduce the influences caused by the traffic (Figure 9).

According to these facts, the stations were not forced centring, as the measuring environment was not satisfactory for this method of measurement. Another reason why the stations were not forced centred was to avoid the centring error, which might have become up to 2–3 mm using laser centring from the laser scanner. Whereas the paper deals with the in-plane vertical imperfections, the measurement itself was performed within the local spatial coordinate system.

This kind of measurement ensured the required precision, and that was the reason for choosing such a measuring process for the tied-arch bridges. The overall precision is ±3 mm, but the relative precision between the points presenting the shape of the arch is ±2 mm. The registration between the scans was made by using visual registration or by registration on the number of targets, depending on each case [31,32,33]. There were two cases of the measurement—in the first one, the target registrations were used by employing targets placed on the bridge itself. In the second, the targets were not used; the registration was processed by using “visual registration”. It was meant to match the point clouds using geometrical objects—i.e., sharp parts of the bridge structure. After the measurement and its’ registration, the final point clouds were clipped, and outer, frontal lower points in regular sections were chosen. The bridge arch was represented by the point cloud with spatial coordinates x, y, and z for every single point. The point clouds (pre-processed scans) of investigated bridges are displayed in Figure 10.

The sections were placed on every single meter from the arch beginning (see an example in Figure 11). The 3D coordinates of these points were selected and represented the input data for the following processing.

5. Post-Processing

5.1. Background

The next process was divided into a few steps. The first step was the translation of the mean plane by using the least square method (see graphical illustration in Figure 12).

The result of the estimation is presented by the two-dimensional coordinates when one dimension is the distance from the bridge beginning, i.e., stationing in the plane direction; the second dimension is the unchanged elevation of the point. The next step included the application of the nonlinear regression analysis [34]. Due to the kind of curve used during the bridge creation, the formulas used represent the circle or parabola course, and the question was used to estimate the parameter’s value. The result of the calculation was the comparison of the ideal theoretical shape and the shape provided by measured data [35]. According to the second estimation (parabolic or circular arch shape estimation), numerical modelling of the calculated values and the consequent nonlinear regression analysis application and approximation lead to the vertical imperfections examination. These results represent the input data for the first part of the research. The following estimation—approximation of the vertical imperfections course and the values related to the tied-arch bridge length using the numerical modelling of the curve—led to the vertical imperfections symmetry examination.

5.2. Nonlinear Regression Analysis

It is necessary to work with the best approximation and the bridge arch shape-describing function relationship to find the estimated properties fitting curve. The function relationship has to be estimated, and by performing this estimation for the points from the point cloud in the chosen step, it is possible to find out the fitting curve estimating properties and to examine the difference and distance from the measured points in relation to the fitting curve. The estimated values have to be set by using the proper estimating method—the Gauss-Mark model in our case.

The Gauss-Mark model may be defined as a model for the statistical approach to the linear model leading to the Gauss-Markov Theorem [36,37]. According to [38], this model is a basis for the estimations and the linear model with conditions for the unknown quantities or, according to [39], the estimation of an intermediate measurement with conditions.

$$\mathit{E}\left(\mathit{l}\right)=\mathit{A}\mathit{x},$$

(1)

$${\mathit{B}}^{\mathit{T}}\mathit{x}+\mathit{u}=0,$$

(2)

$$\mathit{C}={\sigma }_0^2{\mathit{P}}^{-1},$$

(3)

$$\mathit{x}=\mathit{d}\mathit{x}+{\mathit{x}}_0,$$

(4)

$$\mathit{l}={\mathit{l}}^{\mathbf{\prime }}+\mathit{v},$$

(5)

where E(l) is a vector of n rows and l columns mean value of random measurements expressed by vector l realised by a vector of estimated values; A is a matric of n rows and k columns set coefficients; x is a vector of k rows and l columns unknown parameters; B^T is a matrix of r rows and k columns known parameters (r ≤ k); and u is a vector of closures of the size r rows and l columns. Matrix P with dimension n rows and n columns includes the weight coefficients. Vector x₀ includes the approximate values of parameters x with the same dimension, and the vector dx expresses the corrections of these parameters. If the input data are of the same variance C, the stochastic model may be written:

$$\mathit{C}={\sigma }_0^2\mathit{E}.$$

(6)

The estimation and calculation using the Gauss-Mark model may be effectively soluted by the block matrix:

$$\left(\begin{array}{c}\mathit{k}\\ -\mathit{d}\mathit{x}\end{array}\right)={\left(\begin{array}{cc}{\mathit{B}}^{\mathit{T}}\mathit{Q}\mathit{B}& \mathit{A}\\ {\mathit{A}}^{\mathit{T}}& 0\end{array}\right)}^{-1}\left(\begin{array}{c}\mathit{u}\\ 0\end{array}\right),$$

(7)

in which k is a vector of correlates, and from which:

$$\mathit{v}=-\mathit{Q}\mathit{B}\mathit{k},$$

(8)

$$\mathit{Q}={\mathit{P}}^{-1}.$$

(9)

The undeviated estimation of a singular variance:

$${\widehat{\sigma }}^2=\frac{{\mathit{v}}^{\mathit{T}}\mathit{P}\mathit{v}}{n-k+r}.$$

(10)

The unknown parameters estimation accuracy is set by a covariance matrix of estimation:

$$\mathit{C}\left(\widehat{x}\right)={\widehat{\sigma }}^2\left[{\left({\mathit{A}}^{\mathit{T}}\mathit{P}\mathit{A}\right)}^{-1}-{\left({\mathit{A}}^{\mathit{T}}\mathit{P}\mathit{A}\right)}^{-1}\mathit{B}{\mathit{Q}}_{\mathit{x}\mathit{x}}{\mathit{B}}^{\mathit{T}}{\left({\mathit{A}}^{\mathit{T}}\mathit{P}\mathit{A}\right)}^{-1}\right],$$

(11)

in which:

$${\mathit{Q}}_{\mathit{x}\mathit{x}}=-{\left[{\mathit{B}}^{\mathit{T}}{\left({\mathit{A}}^{\mathit{T}}\mathit{P}\mathit{A}\right)}^{-1}\mathit{B}\right]}^{-1}.$$

(12)

In general, the tied-arch bridges may be divided into the sub-categories in which the examined bridge part (arch) is represented by the functional relationship for:

(a)

circle;

(b)

parabolic function.

5.2.1. An Estimation of the Fitting Curve Properties on the Circular Arch-Shaped Examined Part of a Bridge using the Gauss-Mark Model

The functional relationship of a circular arch is, in general, defined as:

$$a^2+b^2-r^2=0$$

(13)

According to the measuring errors and estimation using the Gauss-Mark model, it is necessary to presuppose that the Formula (13) is not perfectly fulfilled. The formula presents a condition that must be fulfilled after the estimation, and the differences from the condition present the closures, i.e., vector u.

In Formula (13), the “a” parameter presents the change in the position—stationing, i.e., the change in a horizontal way, and the “b” parameter presents the change in the height, i.e., the change in a vertical way.

We can transform Formula (13) into:

$$X^2+Z^2-R^2=0$$

(14)

The measured data represent the approximate values that need to be estimated, and our goal is to estimate and approximate the overall course of an examined part of circular arch bridges based on Formula (14).

Formula (14) represents the basis for the estimated values solved by the Gauss-Mark model. The coordinates of the circular arch centre do not acquire “the zero values” for the origin, so Formula (14) needs to be defined for the i-number of measured points as:

$${\left(X_i-X_S\right)}^2+{\left(Z_i-Z_S\right)}^2-R^2=0$$

(15)

Formula (15) is re-written in the form of a function:

$$f_0\left(X_S,Z_S,R,X_i,Z_i\right)={\left(X_i-X_S\right)}^2+{\left(Z_i-Z_S\right)}^2-R^2$$

(16)

Whereas the squared values of the parameters occur in Formula (16), the formula is not linear. To fulfil the entire matrix in the Gauss-Mark model, the formula, in other words, condition or function (16), must become linear. To make it linear, we used Taylor’s Theorem.

The general Taylor series for the two-dimensional random variable [25]:

$$f\left(x^{\prime }+v_x,y^{\prime }+v_y,\right)=f\left(x^{\prime },y^{\prime }\right)+\frac{\partial f}{\partial x^{\prime }}{v}_x+\frac{\partial f}{\partial y^{\prime }}{v}_y$$

(17)

Nonlinear function (16) is derivated by the measured and the unknown parameters. The Taylor series for the function (16):

$$\begin{gathered} f_0\left(X_S+{dX}_S,Z_S+{dZ}_S,R+dR,X_i+{dX}_i,Z_i+{dZ}_i\right)= \\ {f}_0+\frac{\partial f_0}{\partial X_S}{dX}_S+\frac{\partial f_0}{\partial Z_S}{dZ}_S+\frac{\partial f_0}{\partial R}dR+\frac{\partial f_0}{\partial X_i}{dX}_i+\frac{\partial f_0}{\partial Z_i}{dZ}_i\hspace{1em}for i=1,\dots ,n \end{gathered}$$

(18)

According to function (16), X_i and Z_i are measured data, and X_S, Z_S, and R are the unknown parameters, i.e., n = 3.

The partial derivatives of the function with respect to the unknown parameters (i.e., the parameters multiplying the corrections ${dX}_S,{dZ}_S,dR$) represent the elements in the A-matrix:

$${\mathit{A}}_{(n,3)}=\left(\begin{array}{ccc}\frac{\partial f_0}{\partial X_S}& \frac{\partial f_0}{\partial Z_S}& \frac{\partial f_0}{\partial R}\\ ⋮& ⋮& ⋮\\ \frac{\partial f_0}{\partial X_S}& \frac{\partial f_0}{\partial Z_S}& \frac{\partial f_0}{\partial R}\end{array}\right).$$

(19)

The partial derivatives of the function with respect to the measured parameters (i.e., the parameters multiplying the corrections (dX_i, dZ_i) represent the elements in the B-matrix:

$${\mathit{B}}_{(2n,n)}=\left(\begin{array}{c}\begin{array}{cc}\frac{\partial f_0}{\partial X_1}& 0\end{array}\begin{array}{cc}0& 0\end{array}\\ \begin{array}{cc}\frac{\partial f_0}{\partial Z_1}& 0\end{array}\begin{array}{cc}0& 0\end{array}\\ \begin{array}{cc}0& \frac{\partial f_0}{\partial X_2}\end{array}\begin{array}{cc}0& 0\end{array}\\ \begin{array}{cc}0& \frac{\partial f_0}{\partial Z_2}\end{array}\begin{array}{cc}0& 0\end{array}\\ \begin{array}{cc}0& 0\end{array}\begin{array}{cc}\ddots & 0\end{array}\\ \begin{array}{cc}0& 0\end{array}\begin{array}{cc}0& \frac{\partial f_0}{\partial X_n}\end{array}\\ \begin{array}{cc}0& 0\end{array}\begin{array}{cc}0& \frac{\partial f_0}{\partial Z_n}\end{array}\end{array}\right).$$

(20)

The closures vector u consists of non-met conditions from Formula (15) as the measured data—including measuring errors and the influence of the environment during the measuring process—cause the left part of an equation to not equate to zero:

$${\mathit{u}}_{\left(n,1\right)}=\left(\begin{array}{c}{f}_0^1\\ \begin{array}{c}{f}_0^2\\ ⋮\end{array}\\ n\end{array}\right).$$

(21)

To find the corrections for the unknown parameters $X_S,Z_S,R$, it is necessary to calculate their rough values $X_{S0},Z_{S0},R_0$ that are about to be estimated after the Gauss-Mark model application. It is possible to calculate them using the system of three equations with three unknown parameters, or the other solution comes from the matrix multiplication using three optional points from the point cloud:

$$\left(\begin{array}{c}{X}_{S0}\\ Z_{S0}\end{array}\right)={\left(\begin{array}{cc}{X}_1-X_2& Z_1-Z_2\\ X_1-X_3& Z_1-Z_3\end{array}\right)}^{-1}\left(\begin{array}{c}{K}_1\\ K_2\end{array}\right)$$

(22)

$$K_1=\frac{1}{2}\left(X_1^2-X_2^2+Z_1^2-Z_2^2\right)$$

(23)

$$K_2=\frac{1}{2}\left(X_1^2-X_3^2+Z_1^2-Z_3^2\right).$$

(24)

The third unknown parameter R is subsequently set from the calculated centre coordinates $X_{S0},Z_{S0}$ using Formula (15). An analogous approach, in general, was used for second-degree parabola-shaped arcs.

5.2.2. An Estimation of the Fitting Curve Properties on the Parabola Arch-Shaped Examined Part of a Bridge Using the Gauss-Mark Model

The second group of tied-arch bridges has a parabolic shape that is defined by:

$${y=ax}^2+bx+c$$

(25)

The formula representing the parabola is presupposed not to be fulfilled for estimation using the Gauss-Mark model. In the case of parabolic tied-arch bridges, the coordinate “x” represents the stationing, and coordinate “y” is the related height change, i.e., the “z” parameter:

$$Z_i={aX}_i^2+{bX}_i+c$$

(26)

The assumption for the Gauss-Mark estimation model:

$$Z_i-\left({aX}_i^2+{bX}_i+c\right)=0$$

(27)

The estimation goal is to find the most likely functional formula for the examined tied-arch bridges course representation. Formula (27) is re-written in the form of a function:

$$f_0\left(a,b,c,X_i,Z_i\right)=Z_i-a{X}_i^2-b{X}_i-c$$

(28)

Similar to the previous section, before the Gauss-Mark model application, the input formula must be linearised using the Taylor series by deriving from both measured and unknown parameters. The Taylor series for function (28) takes the form:

$$\begin{gathered} f_0\left(a+da,b+db,c+dc,X_i+{dX}_i,Z_i+{dZ}_i\right)= \\ {=f}_0+\frac{\partial f_0}{\partial a}da+\frac{\partial f_0}{\partial b}db+\frac{\partial f_0}{\partial c}dc+\frac{\partial f_0}{\partial X_i}{dX}_i+\frac{\partial f_0}{\partial Z_i}{dZ}_i\hspace{1em}for i=1,\dots ,n \end{gathered}$$

(29)

where X_i and Z_i are measured data and a, b, and c are the unknown parameters, i.e., n = 3.

The matrix A (as in Section 5.2.1) consists of partial derivatives of the function f₀ by the unknown parameters—a, b, and c in this case:

$${\mathit{A}}_{(n,3)}=\left(\begin{array}{ccc}\frac{\partial f_0}{\partial a}& \frac{\partial f_0}{\partial b}& \frac{\partial f_0}{\partial c}\\ ⋮& ⋮& ⋮\\ \frac{\partial f_0}{\partial a}& \frac{\partial f_0}{\partial b}& \frac{\partial f_0}{\partial c}\end{array}\right)$$

(30)

Similar to the previous section, the partial derivatives by the measured parameters $X_i,Z_i$ represent the elements in the B-matrix, and the vector of closures consists of non-met conditions from Formula (27). Before the estimation of parameters a, b, and c representing the adjusted tied-arch bridge course, the approximate values a₀, b₀, and c₀ were calculated using the Gauss-elimination method. The parameters a₀, b₀, and c₀ were the input data for the estimation and adjustment with the Gauss-Mark model.

6. Results and Comparison of the Measurements

The main idea of the measurement and the post-processing was to evaluate the theoretical shape of the steel arches with the actual, imperfect shape and to compare the obtained data among the bridges. The investigated actual vertical geometrical imperfections (the real shape of the arch) depend on several factors. The shape of existing arch bridges in the vertical plane is influenced not only by the structural design, stiffness, the position of hangers, the arch-to-girder joint, or the assembly procedure, but it is also significantly influenced by the dead load of the bridge itself and the temperature.

The measured data were processed by the procedure presented in the previous sections. In accordance with the information from the design project of bridges and with the fitting curve parameters calculation, arch bridges may be divided into two groups: parabolic arch bridges (Olomouc Road Bridge I, Olomouc Road Bridge II, Nedvědice Road Bridge, Mohelnice Road Bridge, Břeclav Railway Bridge, Karviná Railway Bridge, Karviná Road Bridge, and Milówka Railway Bridge), and a circular arch bridge (Brodno Footbridge).

The investigated length of the arch represents almost its whole theoretical length but without the short end sections that are part of the arch-to-girder joint. In this relatively small area (at the end of the arch), the geometry of the arch often changes due to the structural modifications resulting from the stress concentration but also for aesthetic reasons. Due to the relevance of the study, the elimination of possible errors as well as a certain ambiguity of the input data obtained in the area of the arch-to-girder joint, the whole analysis was performed only on a defined part of the arc–examined length l_arch and with the corresponding rise of arch f (Figure 13).

The evaluated data from the measurement of the steel arches are graphically presented in Figure 14. The imperfections of the arches’ shape are for clarity when presented on an appropriate larger scale (from 10:1 to 40:1), and the horizontal length of the arch is given in relative, percentage form. As mentioned above, this study does not consider the total length of the arches but only (so-called) the examined length of the arch.

An evaluation and an appropriate interpretation of the measured data and the mutual comparison of the imperfections on the individual bridges is a relatively complex task. For the most illustrative comparison of the investigated arches imperfections, the parameters below were used:

The area delimited by the real (imperfect) and ideal shape of the arch (area M+ above the theoretical curve of the arch; area M− under the theoretical curve of the arch)—see Figure 15:

• The maximum measured amplitude (+w; −w)—see Figure 15;
• The number of transition points (the crossing of the theoretical curve and the imperfect real shape of the arch (see Figure 15);
• The average length between the transition points.

Another investigative parameter is a symmetry coefficient. This coefficient was calculated as an area ratio between the inverse values related to the total examined arch length. The inverse values are represented by the area bounded between two curves—one curve runs through the basic imperfections’ course, and the second curve is a symmetrically inverted curve axially symmetric according to the bridge symmetry axis (Figure 16).

Moreover, the real geometrical imperfections were compared with the theoretical shape of the imperfection according to the standard EN 1993-2 [17]. The theoretical imperfection shape for in-plane buckling of the arches is considered sinusoidal. The maximum amplitude is considered 1/500 of the horizontal length of the arch (i.e., 2‰ of l_horizontal). For the evaluation of the theoretical standardised imperfect shape of the arch, similar parameters are used for the real shape of the arch (real in-plane geometrical imperfections):

• The area that is delimited by the standardised imperfect shape of the arch and ideal shape of the arch (Area S+ above the ideal theoretical curve of the arch; Area S- under the ideal theoretical curve of the arch)—see Figure 17;
• The maximum standardised amplitude (+e₀; −e₀)—see Figure 17;
• The number of transition points (the crossing of the ideal theoretical curve and standardised imperfect shape of the arch)—see Figure 17;
• The average length between the transition points.

The comparison of the measured imperfections and standardised sinusoidal imperfections for individual bridges is illustratively presented in Figure 18.

More detailed outputs from the analysis of individual bridge arches’ mutual comparison and the arches’ actual and standardised imperfections comparison in their plane are summarised in

Table 1. Comparison of measured geometrical imperfections and equivalent imperfections according to EN 1993-2.

PARAMETER	Length of Arch larch	Max. Measured Amplitude w−	Max. Measured Amplitude w+	Absolute Max. Measured Amplitude∣w∣	Area M−	Area M+	Total Area M	Ratio Area M/larch	Rise of the Arch f	Horizontal Length of the Arch lhorizontal	Ratio w/lhorizontal
BRIDGE	[m]	[m]	[m]	[m]	[m2]	[m2]	[m2]	[m]	[m]	[m]	[-]
Milówka (RWB)	70.49	−0.058	0.055	0.058	0.311	0.790	1.101	0.016	9.57	67.0	1/1159
Karviná (RDB)	58.93	−0.066	0.046	0.066	0.700	1.003	1.704	0.029	7.22	56.5	1/863
Karviná (RWB)	55.30	−0.061	0.062	0.062	0.794	1.094	1.889	0.034	8.22	52.0	1/843
Břeclav (RWB)	87.98	−0.078	0.072	0.078	1.161	1.438	2.598	0.030	10.60	84.5	1/1079
Mohelnice (RDB)	46.03	−0.014	0.029	0.029	0.153	0.138	0.291	0.006	6.96	43.0	1/1507
Olomouc I (RDB)	60.98	−0.081	0.042	0.081	1.713	0.436	2.148	0.035	6.67	59.0	1/729
Nedvědice (RDB)	30.89	−0.012	0.018	0.018	0.065	0.098	0.163	0.005	4.62	29.0	1/1625
Brodno (FTB)	67.05	−0.054	0.055	0.055	0.442	0.548	0.990	0.015	6.91	65.0	1/1183
Olomouc II (RDB)	66.31	−0.049	0.055	0.055	0.448	0.589	1.037	0.016	8.06	63.5	1/1161
PARAMETER	Ratio f/lhorizontal	Index of symmetry	No. of transition points (measurement)	No. of transition points (standard EN 1993-2)	Average length between transition points (measurement)	Average length between transition points (EN 1993-2)	Max. amplitude acc. to EN 1993-2∣e0∣	Area S+; S−	Total area S	Ratio max e0/max w	Ratio area S/area M
BRIDGE	[-]	[m]	[-]	[-]	[m]	[m]	[m]	[m2]	[m2]	[-]	[-]
Milówka (RWB)	0.14	0.03585	37	1	1.86	35.25	0.134	3.007	6.013	2.32	5.462
Karviná (RDB)	0.13	0.00451	4	1	11.79	29.47	0.113	2.120	4.239	1.73	2.489
Karviná (RWB)	0.16	0.00814	4	1	11.06	27.65	0.104	1.831	3.661	1.69	1.938
Břeclav (RWB)	0.13	0.01569	9	1	8.80	43.99	0.169	4.733	9.466	2.16	3.643
Mohelnice (RDB)	0.16	0.00998	7	1	5.75	23.02	0.086	1.260	2.520	3.01	8.660
Olomouc I (RDB)	0.11	0.06072	3	1	15.25	30.49	0.118	2.290	4.581	1.46	2.133
Nedvědice (RDB)	0.16	0.00803	2	1	10.30	15.45	0.058	0.570	1.141	3.25	7.002
Brodno (FTB)	0.11	0.03408	29	1	2.24	33.53	0.130	2.775	5.549	2.37	5.604
Olomouc II (RDB)	0.13	0.02825	20	1	3.16	33.16	0.127	2.681	5.361	2.32	5.170
(RDB)—road bridge; (RWB)—railway bridge; (FTB)—footbridge

.

The ratio f/l_horizontal (rise-to-span ratio) is at the level of 0.11–0.16 for all examined bridges. The ratio between the area delimited by the real (imperfect) and the ideal shape of the arch in relation to the examined length of the arch (ratio M/l_arch) is at the level of 0.005–0.035. The maximum measured imperfection (amplitude) reaches the values between 1/843–1/1625 of the horizontal length of the arch. The comparison of the proportion of the areas under and above the theoretical shape (M− and M+; see Figure 15) of the arch is presented in Figure 19.

The comparison of the real geometrical imperfections with the equivalent geometrical imperfections according to EN 1993-2 [17] brings interesting results (Figure 20). The maximum predicted equivalent imperfections (amplitude w_max) are, on average, 2.26 times larger than the measured values (amplitude e₀). Within the comparison of the areas (total area S and total area M according to Figure 17 and Figure 15), the areas provided by the equivalent geometrical imperfection shape are, on average, 4.68 times larger than the areas based on the measured values.

The geometric imperfections course seems to be rather random; a certain similarity was recorded only between the bridges Karviná Road Bridge, Karviná Railway Bridge, Břeclav Railway Bridge, and Olomouc Road Bridge I. Based on the measurements of a selected group of the tied-arch bridge measurements, unambiguous differences in the shape of the imperfections depending on the arrangement of the hangers were not directly confirmed.

7. Discussion and Conclusions

The paper shows the modern view on the interdisciplinary evaluation of the existing arch bridges. The main research findings and their unique contributions can be summarised in the following points:

• The research on the geometrical imperfections of steel structures is, in the vast majority of cases, investigated on the structural members with limited dimensions in-labo, thus the presented research is unique in terms of its focus on examining large steel structural parts (bridge arches) in situ;
• Terrestrial laser scanning appears to be a suitable method with satisfactory accuracy for the investigation of geometrical imperfections on large steel structural objects;
• The research yielded a small but interesting and unique statistical set of information about the geometrical imperfections of steel arch bridges in their plane;
• There are many parameters that could be applied to the evaluation of geometrical imperfection, e.g., area comparison of S-areas (see above), the number and distance of transition points, coefficient of symmetry, etc.;
• The maximum amplitude of imperfection measured on the examined bridges reaches the values between 1/843–1/1625 of the horizontal length of the arch;
• The unambiguous differences in the shape of the imperfections depending on the arrangement of the hangers were not directly confirmed;
• The real shape of the arch imperfections could be used in further more complex nonlinear analyses with imperfections (GNIA of GMNIA) of the bridge superstructure included;
• There are significant size differences between measured geometrical imperfections and equivalent geometrical imperfections of steel arch bridges, according to EN 1993-2.

The last conclusion point should be further discussed. The equivalent geometrical imperfection, according to EN 1993-2, includes not only basic arch geometric imperfections but also structural imperfections, which usually cover material imperfections (e.g., residual stress) and imperfections caused by other effects (e.g., small inaccuracies that arise in the welded connections of assembly parts). So, for the correct comparison of the in situ measurement and values defined by standard EN 1993-2, there should be the data supplemented by the experimental investigation of residual stresses on existing steel bridges. Moreover, the presented research outputs were focused only on the in-plane imperfections, but for the global analyses, there should also be taken into account out-of-plane buckling. Ongoing research is, therefore, currently focused on the following issues:

• A further collection of the data about geometric imperfections of steel arch bridges in order to create larger statistical data;
• The evaluation of the out-of-plane geometric imperfections of arches and investigation of the influence of upper longitudinal bracing;
• The measurement of residual stress on box-shaped cross-sections, typically applied for arches of steel bridges via the hole-drilling method.