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Composite construction systems consider the rational structural composition of the right materials at the right places in order to optimally exploit their properties. Composite structures have the widest application in large-span engineering constructions [12], but they can be applied successfully in residential and commercial buildings. By adequate coupling of the constructive elements of the same or different physicalmechanical characteristics into an integral cross-section, the basic goal of the procedure is achieved, i.e. the capacity of the system is increased in relation to the individual elements. Depending on the applied materials, composite structures that are often in use in the construction industry generally are timber-timber, concrete-concrete, steel-concrete and timber-concrete.

Since the composite structures, made by different materials and methods of joining, have reached very high level of application in the construction industry in last several decades, there is a demand for their more precise analysis and design. It is known that the type of used fasteners mostly influence the overall behaviour of the coupled structures. Therefore, it is of crucial importance how to introduce the problem of the connection behaviour between coupled materials into the analysis
and design.

Coupling of the constitutive elements can be achieved in different ways, where one of the most common procedure is the use of number of individual shear connectors (mechanical fasteners, anchors,...). Shear connectors should ensure bond of two different materials, transferring the shear forces between two elements, enabling the composite action of the structure. The interest of researchers and constructors as well as numerous studies and research works refer to these types of fasteners since the application of dowel type connectors is the most common in timber-concrete composite structures (TCC), and additionally the behaviour of the overall construction depends on their behaviour. The use of mechanical fasteners for coupling two different materials such as timber and concrete shows that the behaviour of the TCC system is very complex, since the fasteners allow certain interlayer slip that leads to partially interaction (elastic composite action). Therefore, the analysis and design of TCC structures requires consideration of the interlayer slip between the sub-elements.

Considering one-dimensional problem, the first theories for partial composite action for beams subjected to static loads were developed by Newmark (1943,1951), Granholm (1949), Pleshov (1952) and Goodman (1967). The application of partial composite action theory was performed by Girhammar and Gopu (1991) in analysis of columns with interlayer slip subjected to one particular axial loading case which was extended and generalized in their further work. Based on previous research and analysis, they presented an exact static analysis of partial composite structures with interlayer slip [7] and afterwards in papers [8] , [9] , [10] they proposed an exact and simplified methods for analysis of the partial composite structures applied to the beams and columns. In Serbia, in the field of timberconcrete composites, the theoretical basis for analysis of partially composite system was given by B.Stevanović (1994) [17] and later on by Lj.Kozaric [11] and R.Cvetkovic [3], which was followed by experimental data.

The theory of partial (elastic) composite action is based on the corresponding assumptions of the theory of elasticity and takes into account the interlayer slip in the connection at their calculation. The exact calculation of the partial composite action implies solving differential equations where closed form solutions can be formulated only for some particular (simple) cases of boundary and loads conditions.

In EN1995 [5] the simplified manual design procedure ("γ-method") widespread in practice is adopted. This method was originally applied by Mohler (1956), considering the problem of interlayer slip between composite members (timber-timber) coupled with mechanical fasteners, but, with appropriate modifications, this procedure can be applied to the other types of composite constructions such as timberconcrete system. "Gamma" method was developed in the case of simply supported beam subjected to sinusoidal load q=q0·sin(π·x/L). In this case, there is a simple closed-form solution, that could be applied to the other types of loads as well, due to a slight deviation from the exact analytical solution of the differential equation. This method is based on the effective stiffness of the composite system and on the theory of elasticcoupling, taking into account the conservative effect of the distribution of forces within the girders, and so far most fully covers all the parameters that affect the behaviour of TCC.

Also, for the calculation of composite systems, it is possible to apply approximate methods based on the differential [14] or the variation formulation [13].

The differential formulation is based on the derivation of differential equations that describe the problem in a particular domain, where the solution depends on the boundary conditions. In solving the problem, it is necessary to find unknown  unction that satisfies differential equation as well as the boundary conditions. By solving the derived differential equations, an analytical solution of the problem arises, where the closed-form solution can be obtained only for a limited number of simpler design models. If the design model is complex, then the approximate methods are most commonly used and suitable for obtaining an acceptable solution. Residue methods are in such cases a convenient way to formulate a numerical solution.

In the variational formulation of the problem, it is necessary to find unknown function or several functions that satisfy the requirement of functional stationarity, where the unknown function must also satisfy the corresponding additional conditions that are not implicitly contained in the functional. In order to apply the variational formulation, it is necessary that functional exists for considered problem.

Numerous methods and procedures for determination of approximate solutions have been developed based on the differential and variational formulation of the problem. The Galerkin method is the most frequently applied one from the residue methods, while the Ritz method is most often used for variational formulation.

Finite element method (FEM) is one of the most used numerical methods in structural analysis where the final element formulations is based on the solution of differential equations by residual methods or using the variation formulation.FEM based on the Galerkin method (or other weighted residual methods) can be applied to a much broader set of differential equations because it is not necessary to have a proper variational form as it is the case when using Rayleigh-Ritz based FEM [1]. Based on the previous exposition, it can be concluded that the application of simplified and/or approximate numerical methods for the analysis and design of TCC structures is welcome and recommended. Therefore, the approximate methods based on differential or variational formulation [16] are widely used, because they can be implemented in structural analysis software in order to provide a specific tool for engineers for designing partial composite structures.

This paper presents the Galerkin method in the analysis of the TCC system [14]. The selection of trial functions that describe the problem of elastic composite action as well as their influence on the final results was analyzed. For comparison of the obtained results, analysis were performed according to analytical solution [17] and the "gamma" method [5]. On the basis of the proposed numerical models, a model that best describes the problem of elastic coupling was chosen for further comparative analysis with the experimental data [18]. In addition, the use of the Ritz method was also presented in the analysis of the TCC system. The obtained results according to the Ritz method with different trial functions were analyzed and compared to the analytical and the "gamma" method solutions. All analysis were performed using MATLAB software [15].

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Dragan Manojlović - MSc, dig, Departman za građevinarstvo i geodeziju, Fakultet tehničkih nauka, Univerzitet u Novom Sadu
Tatjana Kočetov-Mišulić, Dr - dig, Departman za građevinarstvo i geodeziju, Fakultet tehničkih nauka, Univerzitet u Novom Sadu
Aleksandra Radujković, Dr - dig, Departman za građevinarstvo i geodeziju, Fakultet tehničkih nauka, Univerzitet u Novom Sadu

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