Keywords

1 Introduction

One of the most common types of bridges constructions are T-girder bridge deck structures with welded rigid so called “frame” reinforcement cages [1,2,3,4,5]. These are road bridges with and without of transverse beams and main beams, simply supported, regular structure, small and medium span, which were designed for loads of class H-18, NK-80. The specificity of such girder structures is that with a rather thin shell of T-beams geometry, i.e. with low own weight, the cross sections are supersaturated with reinforcement steel (percentage of reinforcement up to 5%), which is not a conventional feature of classical reinforced concrete elements. Along the girder welded reinforcement cages are made with breaks in the rods [6,7,8,9,10]. It means that the amount of reinforcement and, accordingly, the stiffness in the cross sections where the rods was broken is variable, i.e. the spatial system of the girder structure along the girder is irregular. These girder structures have been in operation for about 30–40 years. They were designed by the method of allowable stresses, which indicates the presence of certain strength reserves that need to be used rationally, for example in determining how to reconstruct, widening and increase of structure load bearing capacity. To do this, it is important to have a real values of the girder structures cross-sections stress-strained state of the elements.

This task has already been solved earlier in KADI, LPI, BildorNDI and in the other institutions [11,12,13]. There were studies the bridge girder structures on Plexiglas models and reinforced concrete as well. As a result of tests, experimental data of sections vertical displacements were obtained, which determined the experimental particles of external force/bending moments, while mistakenly believing that the stiffness of the sections remains constant and equal to the design value Ecired according to Eq. (1):

$$ f_{ij} = \frac{{Fw_{ij} }}{{\sum\limits_{i,j = 1}^{n,m} {w_{ij} } }}, $$
(1)

where F is the external test load; fij - the share of external test load on the i-th beam in the j-th section; wij- vertical component of displacement of the i-th beam in the j-th section; n is the number of beams in cross section; m is the number of calculated cross-sections along the span.

The proportion of external force (experimental bending moments) was also determined by another known method from the measured support reactions of beams with regular and irregular stiffness in the cross section of girder bridge deck structures, assuming that the share of external load transmitted to the i-th beam of the span system is equal to the sum of the support reactions of this beam according to Eq. (2):

$$ f_{i} = R_{Ai} + R_{Bi} ;\,i = 1,2,3 \ldots n, $$
(2)

where RAi and RBi – support reactions of i-th beam.

It should be noted that the Eq. (1), (2) are valid for elastic spatial beam systems with regular in cross section stiffness, assuming that the fraction of force Fi acts only in the transverse cross section of bridge deck where external force F in the form of concentrated force is applied.

As can be seen, the previous investigations did not determine the experimental bending moments and stiffness, taking into account the nonlinear behavior of reinforced concrete sections deformation with cracks. This issue was first solved in the laboratory R&D-88 of National University “Lviv Polytechnic” [14, 15] by developing a new method for determining the experimental bending moments, taking into account cracking and nonlinearity of RC sections deformation [16,17,18]. The essence of the method was that the bending moments in the cross sections of the reinforced concrete beam under a certain scheme of application of external load and certain conditions of supporting are a function of the average relative deformations of concrete and reinforcement steel of these sections in Eqs. (3), (4):

$$ M_{i} = F(\varepsilon_{cmi} ), $$
(3)
$$ M_{i} = F(\varepsilon_{smi} ), $$
(4)

where εcmi, εsmi experimental values of average relative deformations of concrete and steel; i = 1, 2, 3,…, n - the number of cross sections of the beam that was investigated; M – beam’s bending moment in the i-th section.

Therefore, after experimental testing the bridge span structure according to a similar loading scheme and obtaining experimental values of average relative deformations of concrete and steel in beams cross sections in the girder structure found the appropriate experimental bending moments over the entire load range – in the elastic and elastic-plastic stages performance.

Further development the method of experimental moments determining in girder span structures sections taking into account the nonlinearity of RC elements based on deformation diagrams of bridge reference beams should be continued in the direction of using the differential equation of bent axis with curvature determining by measured vertical displacements of sections and experimental average strains of concrete and steel as well. These issues were addressed in this paper.

2 Experimental Research Methodology and Main Results

Tests of the full scale T-beam length 13,5 m with rigid/frame reinforcement were made (Fig. 1). This beam was separated from adjacent beams of the bridge deck superstructure by cutting of welded metal plates of T-beam shelves and diaphragms. Then a testing metal structure system of individual design and execution, which consisted of a truss length of 15,0 m and a height of 2,5 m was installed. By means of ties and traverses system the external testing load was transferred to the existing bridge supports (Fig. 1a, b). Measurements of deformations were performed with mechanical microindicators based on 250 mm distance, which allowed to obtain the values of the average relative deformations of the RC sections. To measure the deformations of the reinforcement steel, special holders were welded at the level of the lower reinforcing rod center of gravity (Fig. 1, c). Vertical displacements were measured by deflectors 6PAO in the same sections. Crack formation was observed to determine the width of crack opening with a microscope. If possible, all measuring instruments were duplicated. As a result, according to the experimental data in the investigated sections, reference diagrams of concrete and steel deformations were received, as well as graphical dependences the change in the initial stiffness of the cross sections of the beams. Figure 2 presents these diagrams for the most characteristic section 2-2 in the middle part of the beam.

Fig. 1.
figure 1

The bridge beam with rigid/frame reinforcement and testing steel truss system general view: a - facade; b - cross section; c - a general view of mechanical watch type micro indicators installation.

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Fig. 2.
figure 2

The main experimental results of reference beam: a - reference graphs of concrete and steel relative deformations; b - reference beam stiffness diagram.

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The bridge deck superstructure was tested using a similar method. The experimental data matrix of the relative deformations εc, εs and vertical displacements wz in the cross sections of the beams was obtained. Consider as an example the most characteristic section 2–2 in the middle part of the span under loading with a concentration force F = 800 kN. In this section, the following values of relative deformations were obtained: εc = 0,0005721, εs = 0,001205. Next, according to the deformation diagrams of reference beam (Fig. 2,a) determine the experimental bending moments in the elastic-plastic stage of operation, acting in the first beam from the force F = 800 kN: Ms2-2 = 1175 kNm (for εs), Mc2-2 = 1201 kNm (for εc,). The average experimental bending moment was Mexp2-2 = 1186 kNm. Similarly it was determined for all sections thought the deck superstructure [1].

Consider the differential Eq. (5) of the beam bended axis:

$$ y^{\prime \prime } = \frac{M}{B}, $$
(5)

where y″ - cross section curvature under bending; B - stiffness of the section in the area with cracks.

The real experimental cross sections curvature can be determined from experimental data of average relative deformations Eq. (6):

$$ y^{\prime \prime } = \frac{1}{r} = \frac{{\varepsilon_{cm} + \varepsilon_{sm} }}{h}, $$
(6)

where εcm and εsm - experimental values of average relative deformations of concrete and steel; h - the distance from the concrete upper fiber to the center of gravity of the reinforcing welded cage lower rod.

Having found the value of the experimental curvature and the corresponding stiffness of the section in the area with cracks, from Eq. (5) it is easy to determine the value of Mexp in this section.

The experimental curvature can also be determined by the measured vertical displacements by drawing their longitudinal graphs in the beams of the deck girder structure [2]. According to the experimental graph of vertical displacements (Fig. 3,a), it is possible to write the equation of curvature, which is based on regression analysis (Fig. 3,b). For the first beam of the deck girder superstructure when loaded with a force F = 800 kN in section 2–2, the Eq. (7) will have the form:

Fig. 3.
figure 3

Vertical displacement of first beam of deck superstructure: a – experimental; b – by regression analysis.

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$$ y = 0.010750 + 0.01568x - 0.00002x^{2} + 5.0973e - 9x^{3} - 1.5934e - 13x^{4} , $$
(7)

where y - vertical displacement, cm; x - cross-sectional distance, cm.

By differentiating the Eq. (7) twice by x, we obtain the following Eq. (8) of curvature:

$$ \frac{{d^{2} y}}{{dx^{2} }} = y^{\prime \prime } = - 0.00004 + 30.582e - 9x - 19.121e - 13x^{2} . $$
(8)

Substituting the section coordinate x = 540 cm, we obtain y″ = 2,355·10−5 cm−1.

The experimental curvature value Eq. (9) in the same cross section by the Eq. (6):

$$ y^{\prime \prime } = \frac{1}{r} = \frac{{\varepsilon_{cm} + \varepsilon_{sm} }}{h} = 2.235 \cdot 10^{ - 5} {\text{cm}}^{ - 1} . $$
(9)

As could see, these values obtained by two independent methods are almost the same, which indicates the possibility of their use independently of each other.

3 Algorithm of Determining the Experimental Bending Moments by Experimental Curvature

From tests data of a reference bridge beam similar on a design, conditions of supporting and loading to beams of a girder deck superstructure, to draw the diagram (M–Bexp).

By the tests of the girder deck superstructure get experimental data of the average relative deformations and displacements; determine the experimental curvature of the bended axis of the sections by Eqs. (6), (7). By iterative method from the diagram (M–Bexp) determine the appropriate experimental cross-sectional stiffness in the area with cracks, while:

  • in the first step to accept B = Ec Ired and determine accordingly MI;

  • in the second step for MI determine the stiffness BII and calculate appropriate moment MII;

  • in the third and subsequent steps, repeat the steps as in step II; calculation is ending if: (Mi − M i-1)/Mi < 0,05.

4 An Example the Experimental Bending Moments Determining

As an example, we define the experimental bending moments in section 2-2 according to the diagram (M-Bexp) (Fig. 2, b) and the above algorithm:

  • when determined the y″ by the formula (6)

    • I-st step MI = 2,235960 = 2145 kNm;

    • II-nd step MII = 2,235505 = 1128,6 kNm;

    • III-rd step MIII = 2,235510 = 1139,8 kNm.

As can be seen, already in the third step of iterations the stiffness practically does not change (because the diagram has a gentle character, which is a characteristic feature of the reinforced beams with frame cage reinforcement) and the necessary convergence of results is reached.

  • when determined the y″ by the Eq. (7)

    • I-st step MI = 2,355960 = 2310 kNm;

    • II-nd step MII = 2,355505 = 1215 kNm;

    • III-rd step MIII = 2,355510 = 1201 kNm

Comparing the obtained values of the experimental moments in the cross section 2–2 of the first beam of the bridge deck girder structure, determined by the methodology and the above mentioned method found that the discrepancy is: for Eq. (6) – 3,5%, for Eq. (7) - + 1,6%, which indicates a practical coincidence of results according to the proposed algorithm and method.

5 Conclusions

The experimental bending moments in the nonlinear area of section deformation determined by the proposed algorithm showed practically accurate and close to the real object results. In addition, this technique has its advantages: there is no need to determine the experimental average relative deformations εcm, εcm when testing girder deck structures and, as a consequence, reducing the complexity of work in preparation for testing, as well as reducing the time of testing and in-house work important in engineering practice.