METHODS FOR CALCULATING RECTANGULAR SECTION BEAMS MADE OF WOOD AND CONCRETE

The research aimed to study methods for calculating wood-concrete beams of rectangular cross-section when testing building structures according to the ultimate limit state. The article focuses on the comparison of theoretical methods for calculating structures and considers several methods of fastening the samples of a typical wood-concrete beam. There were obtained experimental data of the ultimate limit state for each sample and carried out a comparative analysis of the most advantageous scheme of fastening the sample parts. The scientific novelty is in the development of an algorithm for calculating composite wood-concrete beams of rectangular cross-sections. As a result, numerical comparison of the values for calculating a typical wood-concrete beam using two of the methods under consideration was given, experimental studies were carried out, as well as a comparative analysis of the obtained theoretical and experimental results.


Figure 1: Diagram of cross-sectional forces in bending
Let us consider the efforts in a bent wood-concrete beam with a rectangular section of width b, the height of a wooden element ht, and the height of reinforced concrete hc. In this case, the total height of the beam is h=ht+hc. Let us denote the height of the compressed zone as x, and assume that the neutral axis passes through the reinforced concrete element. The first condition for the balance of external and internal forces for the element under consideration: the sum of all forces is zero: Using (1), we express the height of the compressed zone: Equation (2) shows that the compressed zone height is directly proportional to the wood tensile strength and the height of the wood element, according to (2) and Fig.1. Figure 2: Dependence of the compressed zone height on the height of the wooden element: 1 -B20-grade concrete, first-grade pine wood; 2 -B20-grade concrete, second-grade pine wood; 3 -concrete B30-grade, first-grade pine wood.
Let us write down the condition for the strength of the normal section of the considered combined wood-concrete bending beam: To calculate the considered bending combined beam, you can apply the method using the modification factor n [1]: where: Ec is the elasticity modulus of concrete in bending, Et -is the elasticity modulus of wood in tension.
Applying the modification factor n, one can express the values of the idealized cross-section in the following formulas [1]:

= +
where: Ai is the area of an idealized section,Ac is the cross-sectional area of the concrete part, At is cross-sectional area of a wooden part,Ii is the inertia moment of the idealized section,Ic is the inertia moment of the concrete part,It is the inertia moment of the wooden part.The inertia moment of the cross-section of a timber-concrete beam is defined as: The distance between the gravity center of the wood piece and the idealized cross-section can be calculated as: where: e is the distance between the gravity center of the wood and concrete part, a_c -is the distance between the gravity center of the concrete part and the idealized cross-section.The normal stresses distribution in the cross-section of a wood-concrete composite structure is determined by the formulas: The tensile strengths at the upper (Mu,c1) and lower (Mu,c2) edges of the concrete part can be obtained from the following strength conditions: where: Rb is the concrete compressive strength, Rbt is the concrete tensile strength.The ultimate strength (Mu,t) in a wooden part should be determined from the strength condition (12): where: ft,0,d -wood tensile strength, fmd -wood bending strength. From the condition for the limit moment of resistance, it can be expressed the equation: Let us observe a 1-meter long wood-concrete beam loaded with concentrated forces in 1/3 of the span, with a section of 100x75mm made of elements: concrete 50x75x1000mm, wood 50x75x1000mm. Second-grade pine is taken as the material of the wooden element, and the concrete element was made from B15-grade concrete.

EXPERIMENTAL SECTION
In the laboratory of Southwestern State University, tests of wood-concrete beams of combined cross-section under the action of short duration loads were carried out. types of connections connecting the upper (concrete) and lower (wooden) structural elements (Fig. 4). There was used a testing plant in the experimental research, the diagram and general view of which are shown in Fig.  5. The load on the samples was transferred through the metal distribution element 3, the mass of which is taken into account in the total load. The load is transferred to the sample in thirds of the span, according to the design scheme (Fig. 5). To exclude the crumpling of the sample on the supports and in the places where the load is applied, metal gaskets 4 and 4.1 were used. The loading of the tested beam was carried out with a stepwise increasing load with a step of 50 kgf (0.49 kN). The value of the deformations was determined after each load application step. Results. During the experiment, the samples were tested. The first of them is shown on the Fig.6. It was found that the ultimate load for samples 1, 2, and 3 was 3.93; 5.26 and 4.66 kN, respectively. The experimental research results are presented graphically in Fig.  6. The plotted dependence graph (Fig. 7) clearly shows that the first sample was deformed linearly up to a load of 3.44 kN, and the third sample was deformed linearly up to a load of 3.68 kN. It is noteworthy that the second sample was deformed linearly up to fracture, which occurred at 5.28 kN.

DISCUSSION
Linear deformation of the second sample may be explained by the type of connection used in it, which turned out to be the most effective in the operation of this wood-concrete beam, also allowing the structure to work longer according to the conceived scheme and withstand more load than other samples.
Based on a comparison of the theoretically obtained values and experimental data, we can see that the proposed calculation method allows, with the accuracy necessary in the context of the experiment, to determine the critical values of the force "F" and momentum "M" for the considered type of structure. It should be noted that during experimental studies, the observed destruction of the testing samples occurs at the more high-load levels compared to the theoretical data. This may occur because these studies were conducted under a short duration load, in contrast to the studies of European scientists. Also, it is worth taking into account that during the experiment there were created wood-concrete samples of various fastening methods, and using the obtained results can be helpful in further researches on this issue.