The displacement-computing algorithm presented in this paper is based on the results of measurements of a bridge in Ostróda (Poland). A wildlife migration route and a local road pass under the bridge. There are four two-lane carriageways, separated by median strips, on the bridge. Hence, the width of the bridge is large, amounting to
The shells of corrugated-steel buried structures are made of metal plates. Using circular segments, one can make shells having complex cross-sectional shapes. The typical geometric feature of such shells is a circular plate (with radius of curvature
Shells made of corrugated plates are characterised by high stiffness [2], but only when buried in soil, i.e. when the soil–steel structures are put into service. As the thickness of the soil surcharge increases, the effect of live loads (a system of concentrated forces) decreases, as shown by the experimental results reported in this paper. Owing to road pavement or track superstructure rigidity also, the effects generated by vehicles are considerably reduced [2]. The range of the action of loads situated on the embankment, i.e. outside the shell, usually amounts to the shell’s span length
Investigations of soil–steel structures under a load moving on the roadway usually involve dynamic testing, e.g. Bęben [3] and Mellat et al. [4]. Moreover, static testing plays an important role in determining the mechanics of structures such as shells sunk in the soil medium, as shown previously [2, 5, 6, 7, 8, 9, 10]. The influence functions used in bridge models, e.g. the study by Machelski [11], can be useful in such analyses. Influence functions can also be determined on the basis of measurements performed on soil–steel structures in service, subjected to a load in the form of a system of concentrated forces produced by vehicle wheels [2]. Then, the interaction of all the members, including the pavement (which, in typical bridges, is considered to be a secondary element not belonging to the structure), is taken into account.
The results of acceptance tests based on the concept of a moving load in the quasi-static sense are used in this paper. Two trucks driving parallel to each other along the road axis constituted the load. The trucks would drive stepwise, i.e. would stop at distances
Sequence of locations
xp/L | –4/4 | –3/4 | –2/4 | –1/4 | 0 | 1/4 | 2/4 |
---|---|---|---|---|---|---|---|
Primary travel | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
Secondary travel | – | 11 | 10 | 9 | 8 | 7 | 6 |
A fragment of the soil–steel structure calculation model is shown in Fig. 2. An important component of such a structure is the pavement with its foundation. The material of this component and its physical properties are different from those of the backfill. Studies of corrugated-steel buried structures have shown that the backfill greatly participates in carrying operational loads. The layered structure of the backfill is crucially important for the load-carrying capacity of the soil–steel structure, determining the way in which the whole soil–steel structure deforms, as the shell deformation functions presented in this paper demonstrate. In this structural system, the soil–shell interface plays an important role.
Corrugated plates in soil–steel structure models are rendered as orthotropic shells or as a mesh-like beam structure [11]. The beam system facilitates computations in many ways, e.g. it becomes possible to formulate influence functions for the internal forces in the shell, which helps to determine the disadvantageous locations of loads on the bridge roadway. Such a representation of the corrugated plate was adopted in the following considerations.
In order to determine the displacement of any shell point in the radial direction
Calculation result
Integration is conducted over the whole length of the circumferential section, i.e. along an arc with length
Obviously, the algorithm for calculating displacement
If unit strains in two points of the corrugation, as in the beam cross-section shown in Fig. 3, are to be calculated, two equations should be formulated as follows:
one for the crest, i.e.
and one for the valley, i.e.
From the difference between these values, one gets
Equation (4) takes into account that
Therefore, . (1) can be now presented in the following form:
Equations (1) and (6) are identical, but for different load functions. The effects of external loads in Eq. (1) are expressed by internal forces, while in Eq. (6), they are expressed by unit strains. In both the solutions, the planar cross-section principle as applied to beam systems is used.
The advantage of the solution expressed by Eq. (6) is that the problem size is reduced to the shell’s circumferential section isolated from the structure. The values of εg and εD contain complete information about the bridge’s geometric structure (3D) and the shell support conditions, whereby the actual characteristics of the backfill material and the pavement material are specified. The distribution of the unit strains (function) along the length of the circumferential section takes into account the actual principles of the interaction between the structural members, e.g. the slip of the soil relative to the shell in the interfacial zone. Functions εg(s) and εD(s) reflect the changes in load location. Thus, the measured unit strains, identical as the directly measured displacement
Unit strains in the soil–steel structure were measured using strain gauge sensors stuck on the corrugated plate’s surface accessible from beneath. In Fig. 3, the results of the measurements are denoted as εg and εD. The measuring points were regularly spaced in the circumferential direction of the shell band, as shown in Fig. 4. The measured unit strains εg and εD are used to determine the other characteristic deformations brought to a common line, as shown in Fig. 3. Hence, the deformation of the crest on the (inaccessible) soil backfill side is calculated as follows:
Of primary importance is the value of the unit strain on the central axis, calculated from the following formula:
Figure 3 shows the geometric dependences (based on the planar cross-section principle) for unit strains in the cross-section of the circumferential section. It follows from them that the radius of curvature of beam ρ and its curvature κ are interrelated by the strength dependence given in the following formula:
Fig. 4. Two points, namely 6 and 12, symmetrically situated relative to the shell crown 9, were selected. The
The shape of the graphs indicates that their form depends on the direction of vehicle travel: the primary travel and the secondary travel. The primary travel would start from the location of the vehicles beyond the structure (
The interdependences between εg(
Figure 7 shows functions
The shape of the graphs plotted for the three points indicates that they differ for both the primary and the secondary travel, when the points 6 and 12 are asymmetrically located also. The arrangement of the vehicle axles has little bearing on the lack of symmetry – the vehicles are not positioned symmetrically relative to the shell crown during the primary and secondary travel (Fig. 1).
By comparing the effects of bending and compression, one can determine the eccentricity (
The graphs in Figs. 5 and 6 show slight deviations of the resultant form the of the cross-section core, considered to relate to
Hence, the conclusion is reached that the compression should be qualified as compression with a small eccentricity.
When the geometric relations in Eq. (9) for curvature change are introduced into Eq. (6), one gets the following equation:
Functions κ(s) and ε(s) are obtained as results of in situ measurements, and so these are in 3D geometry. Naturally, the quantities contain all the (earlier-mentioned) effects contributing to shell deformation. From Eq. (1), only the tracking functions
Because of the complex shapes of the functions in Eq. (13), instead of the integral form of the algorithm, it is convenient to use a matrix algorithm, as in the following equation:
In this approach, when the arc line is divided into several segments, each having length
Assuming that functions ε(s) and κ(s) are continuous, when continuous tracking functions
When the function is discrete at the point of application of the concentrated force, as, e.g. in the case of
When the displacement function
The changes in the shell crown deflection (point 9), derived as results yielded by different measuring techniques and denoted as T – tensometric, I – inductive and G – laser geodesy, are compared in Fig. 8. The deflection measured by the inductive sensors is regarded as the primary one, also because of its accuracy of 0.01 mm. Graph I was obtained during the continuous two-way travel of the trucks (the additional measurement diagram). Considering the sensitivity of the strain gauges, the calculation results based on the tensometric measurements rank equally with the inductive sensor measurements (compare Fig. 5 and Fig. 6). The results yielded by the geodetic laser techniques are regarded as secondary since their accuracy is lower by one order, amounting to 0.1 mm. However, due to the laser measurement, it was possible to determine displacements in the whole area of the circumferential section and not only at selected points as in the case of the T and I measurements.
The graphs for the primary travel presented in Fig. 8 are similar. The similarity between graphs G and I is natural in this case. In the case of inductive sensor measurements, a displacement is the result of the deformation of the whole circumferential section as a component of the analysed structure. In the case of tensometric measurements, deformations from the entire length of the shell’s circumferential section are added up (integrated). Thus, when the number of points is sufficiently large and the displacement functions are quite regular, as in Figs.5 and 6, the results are bound to agree. It should be noted that the tensometric and inductive measurement results agree for the entire travel of the trucks and not only for one location specified by
The result of displacement calculation, as in Eq. (14), consists of two factors stemming from bending and the axial force. Figure 9 shows the diagrams for the variation of the two factors obtained when calculating the deflection of point 6 of the shell. The proportions of the components vary greatly and depend also on the type of displacement (
Figures 10 and 11 show selected examples of calculations using Eq. (17) and the results of tensometric measurements in the soil–steel structure’s points given in Fig. 4. Using the general algorithm, one can calculate displacements at any selected shell points, in the form of components belonging to two groups:
In the proposed algorithm, it is possible to select a point (not necessarily consistent with the configuration shown in Fig. 4) and a displacement direction. As a rule, the value of resultant
The positive characteristic of the behaviour of the soil–steel structure shell is that its deformation tends towards the initial state, i.e. towards the zero value of displacement after the full load cycle [7]. This happens in many situations (most often in the case of a shell crown deflection). Residual displacements often occur at points situated on the shell’s side surface [5, 7]. Such a situation is shown in Fig. 10, 11 and 12.
Figure 12 shows the shell deflection function
The effects of loads that change their location, i.e. moving loads, but with static characteristics, were analysed. The results of tensometric measurements performed for a dense grid of points located on the corrugated plate’s circumferential section were used to calculate displacements characterising the deformation of a shell buried in soil. Consequently, the solution comprises all the soil–steel structure’s members, namely the corrugated plate, the backfill and the pavement with its foundation, and it takes into account all the natural (real) principles governing the interactions between the members. This means that the structure’s geometry and physical properties, as well as the load, are faithfully (accurately) reproduced in 3D. A convenient feature of the algorithm is that the functions of the component (directional) displacements
The proposed algorithm for converting unit strains into displacements yields accurate results. The advantage of this measuring method (electric resistance extensometry) is that there is no need for the solid scaffold used for displacement measurements. This can be useful when the access to the underside of the shell is difficult because of, for instance, boggy ground with a watercourse [8]. Another convenience is that any displacement direction can be selected. In order to directly measure displacements, one must properly fix (which poses difficulty) the sensor to the bent plate and build a stable measuring stand. The main advantage of the algorithm is the limitless number of points and their locations on the analysed circumferential section of the shell, as well as the arbitrary directions of displacements, whereby the deformations of the shell can be faithfully reproduced.
The results of the measurements show that the changes in unit strains differ between the primary travel and the secondary travel. This was observed previously when measuring the displacements of the shell in soil–steel structures [7]. The characteristic feature of the experimental results (unit strains) and the calculated geometric effects (e.g. ρ) and displacements is a hysteresis loop [5, 13]. The problem of the behaviour of the backfill–shell interface in such a hysteresis loop remains unsolved. Considering the two components of the solution in Eq. (17) and the diagrams shown in Fig. 9, one can notice that the deformation of the structure is due to the equivalent effects of bending and compression.
The algorithm requires a dense grid of measuring points. The form of the solution to be used when the number of measuring points is small has been presented in an earlier report [6]. In this case, a collocation algorithm for generating a 2D model (as in Fig. 1) is useful. Then, it is necessary to map the circumferential section of the whole structure, i.e. not only of the shell but also of the other members, namely the backfill and the pavement with the load, as shown in Fig. 2. Thus, in the solution described by Machelski and Janusz [6], the structure (3D geometry) is reduced to a planar (2D) model, as in Fig. 1.