BETON APPLICATION OF NUMERICAL METHODS IN ANALYSIS OF TIMBER CONCRETE COMPOSITE SYSTEM

Kompozitni konstrukcijski sistemi podrazumevaju racionalno spajanje elemenata od odgovarajućih materijala, tako da se optimalno iskoriste njihova svojstva. Spregnute konstrukcije imaju najširu primenu u inženjerskim konstrukcijama velikog raspona [12], ali mogu se uspešno primeniti i u stambenim i poslovnim objektima. Adekvatnim spajanjem konstrukcijskih elemenata istih ili različitih fizičko-mehaničkih karakteristika u integralni poprečni presek, postiže se osnovni cilj postupka, tj. povećava se nosivost sistema u odnosu na pojedinačne elemente. U zavisnosti od primenjenih materijala, spregnute konstrukcije koje se često primenjuju u građevinarstvu uglavnom su tipa drvo–drvo, beton–beton, čelik–beton i drvo-beton. Budući da su spregnute konstrukcije – napravljene od različitih materijala i različitim načinima spajanja – dostigle veoma visok stepen primene u građevinarstvu u poslednjih nekoliko decenija, neophodna je njihova preciznija analiza, kao i preciznije projektovanje. Poznato je da upotrebljeni tip sredstava za sprezanje najviše utiče na globalno ponašanje spregnutih konstrukcija. Stoga, od ključnog značaja je to kako da se uvede problem ponašanja veze između spregnutih materijala u analizi i proračunu.


INTRODUCTION
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.
U EN1995 [5] usvojen je pojednostavljeni manuelni postupak proračuna ("γ-metod"), koji se u praksi široko primenjuje. Ovaj metod prvobitno je primenio Mohler (1956), razmatrajući problem klizanja u spoju između spregnutih elemenata (drvo-drvo), s mehaničkim spojnim sredstvima, ali uz odgovarajuće modifikacije, ovaj postupak se može primeniti i na druge tipove spregnutih konstrukcija, kao što su konstrukcije tipa drvo-beton. "Gama" metod razvijen je za statički sistem proste grede izložene sinusoidnom opterećenju q=q0·sin(π·x/L). U ovom slučaju postoji jednostavno rešenje u zatvorenom obliku, koje se može primeniti i na druge vrste opterećenja, a zbog malog odstupanja od tačnog analitičkog rešenja diferencijalne jednačine. Ova metoda zasniva se na efektivnoj krutosti spregnutog sistema i teoriji elastičnog sprezanja, imajući u vidu konzervativni efekat raspodele sila unutar nosača, i skoro u potpunosti pokriva sve parametre koji utiču na 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 t hese 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, w ith 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 elastic ponašanje SDB konstrukcija.
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 function 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 r esults, 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].

GOVERNING EQUATION OF TCC SYSTEM
The theory of elastic coupling [17], [11] is used for the calculation of TCC structures, where mechanical fasteners are used.
The basic assumptions of the theory of elasticity that are introduced are as follows: • timber and concrete are isotropic, elastic materials and Hook's law applies, • Bernoulli's hypothesis is valid, i.e. plane sections initially perpendicular to the midsurface will remain plane and perpendicular on deformed axis, • coupling means are set at certain distances and can be considered as equivalent continuous connection with the constant elasticity along the beam, • cross sections of concrete and timber are constant along the span, • concrete and timber have equal deflections at each point of the connection, • axial force acts at the centre of gravity (centroid) of the concrete section.
In TCC structure, one element slips (v)over the other along TC interface in the case of bending. Sliding of elements is prevented by the coupling means with appearance of interlayer slip (shear in contact interface) force Ts with compression force N1 and the bending moment M1 in the upper and the tensile force N2 and the bending moment M2 in the lower element of the structure, Figure 1 (where notation A and I represent geometrical properties of cross-sections of upper and lower element, while E represent the modulus of elasticity of applied materials). The intensities of forces depend on the stiffness and deformability of the coupling means and its slip modulus K [2]. Figure 1. Interlayer slip of TCC beam [2] Kada se razmatra spregnuta greda drvo-beton, statičkog sistema proste grede, opterećena ravnomerno raspodeljenim opterećenjem q(x), bez spoljašnje aksijalne sile, problem elastičnog sprezanja može se predstaviti diferencijalnom jednačinom drugog reda u aksijalne sile u betonu:

Slika 1. Klizanje u kontaktnoj ravni SDB nosača [2]
The problem of partial composite action could be represented with differential equation of the second order in the function of the axial force in concrete while observing the composite timber-concrete simply supported beam system with uniformly distributed load q(x) without an external axial force: (1) gde je: where are: s -rastojanje spojnih sredstava za sprezanje; r -rastojanje između težišta betonskog i drvenog dela preseka.
Solving the differential equations (1 or 4) is a complex task, especially for different load cases and/or boundary conditions. In the literature, analytical solutions for different load cases and support conditions could be found. According to [17], analytical solutions for axial force N, interlayer slip force Ts and vertical displacement w, for a simply supported beam system and continuous load, are given by the equations (6-8).
A demonstration of the qualitative change of the axial force N i.e. slip force Ts in TCC system is shown in Figure 2.

APROXIMATE METHODS
The problems of the theory of elasticity are described by means of the differential formulation (differential equations and corresponding boundary conditions) or in the variational formulation in the form of the functional. Although the solutions to these problems in the mathematical sense exist as unambiguous, finding analytical solutions is a delicate and often unsolvable task. Therefore, the approximate methods are often used to find solutions to these problems. Of particular interest are those methods in which the assumption of a solution in the form of approximate or trial function is used as the baseline, wherein one of the most commonly applied weight residual methods is Galerkin method, and commonly applied variational method is Ritz method.
The unknown function u(x) of problem's differential equation has to be approximated by the approximate solution ū(x), expression (9), that could be represented as a superposition of products of known basis functions Φm and unknown coefficients cm.

Metoda reziduuma
Neka je posmatrani fizički problem, u domenu Ω, koji može da bude 1D do 3D, definisan diferencijalnom jednačinom: conditions. As regards other conditions, the choice of functions Φm is generally arbitrary, but the quality of the solution largely depends on the choice of functions Φm. It is desirable that the functions Φm also satisfies natural boundary conditions, and thattheir shape qualitatively corresponds to the exact analytical solution. Therefore, qualitative knowledge of the nature of the solution is very useful in order to avoid the wrong choice of functions that in their form represent a rough deviation from the analytical solution.
Galerkin method has wider application than Ritz's because it can solve even those problems in which the functional does not exist. In the mechanics of deformable bodies, these two methods are equivalent, as they give results of the same accuracy. By choosing the same trial functions in Ritz and Galerkin method, the same coefficients of cm (i.e. same solutions) will be obtained.
Metode reziduuma sastoje se u nalaženju funkcija ū(x) za koje će integralna jednačina (12) biti zadovo-The scalar product of the two vectors is equal to zero if these vectors are mutually orthogonal. Accordingly, the integral equation (12) is a condition of the orthogonality of the residual vector to the selected vector of weight functions.
In this way, the approximate solution ū(x) approximates the exact solution u(x). All solutions ū(x) that satisfy (10) must satisfy (12) regardless of weight functions' choice. The dimension of the weight functions vector corresponds to the number of unknown coefficients cm of the considered problem.
As one of the basic variants of the residual method, which adopts weight functions as basis functions Φm for which the required solution is approximated, is Galerkin's method [16].
Based on the differential equations of the elastic coupling (1 or 4) and the condition (12), it is possible to define the following relations for determining the problem of the TCC girder trough the axial force in the concrete or trough displacements for the case of a TCC beam loaded with continuous load q, according to fallowing expressions: Integralna formulacija koja u sebi implicitno sadrži diferencijalnu jednačinu problema, naziva se slaba formulacija (13 ili 14) koja izražava uslove i relacije koje moraju biti zadovoljene u prosečnom ili integralnom smislu.
Kada da se posmatra jednodimenzionalni linijski problem s domenom definisanosti x∈[x1,x2], funkcional (potencijalna energija) izražava se putem integrala I(u) u celom domenu: An integral formulation that implicitly contains a differential equation of the problem is called a weak formulation (13 or 14) that expresses the conditions and relations that must be satisfied in the average, or in an integral sense.
Since the differential equation of the problem is of even order (2r = 4), it is possible to reduce the required order of derivation in the trial functions by partial integration of the expression (12) from r = 4 to r = 2. By partial integration of expressions (13 or 14) is achieving that selected trial functions must satisfy only the essential conditions, that have to be satisfied by the selection of trial functions themselves, while the force conditions are already included into the formulation of the partial integration problem.
By solving the integrals, a system of n equations by unknown coefficients cmis obtained andan approximate solution for the required function u(x) could be derived by determination of coefficients cm.

Variational method
As the Ritz method is based on a variational formulation, it is necessary to satisfy the requirement of extremum of a functional that describes the problem under consideration. To solve problems in the mechanics of deformable bodies, the functional is equal to the total potential energy, and the stationary value corresponds to its minimum value. In the theory of structures this method is the most famous variation procedure. The reason for that is that there is a functional in the form of potential energy [13]. In case of one-dimensional beam problem with the defined domain x∈[x1,x2], functional (potential energy) is expressed as an integral I(u)over the entire domain: where is: Π (...) -represents the functional of functionsu(x), du(x)/dx, d 2 u(x)/dx,... Extremum of a functional is represented by requirements that the first variation of the functional be zero: (16) ili zapisano u razvijenom obliku: or shown in the developed form: Kako su koeficijenti c1,c2,...,cn međusobno nezavisni parametri, onda se δΠ=0 svodi na sledeći uslov: Since c1,c2,...,cn are mutually independent parameters, then δΠ=0 is represented by following condition: (18) što predstavlja sistem algebarskih jednačina po nepoznatim koeficijentima cm.
Based on the differential formulation of the partially composite problem, it is possible to define a functional according to variation principles [8]. As the mechanical fasteners are commonly used for coupling in TCC, a certain displacements (i.e. an interlayer slip) occur on the TC interface due to the external load. Besides the strain energy due to internal forces (M1, N1, M2 i N2 where N=N1=-N2), it is also necessary to take into account the strain energy due to interlayer slip. Functional, or total potential energy of the composite system, in the case of simply supported beam with uniform distributed load q(x), can be shown in the following form [19]: (19) (20) where is: Wi -potencijalna energija deformacije; We -potencijal sila. Kako moment savijanja M(x) možemo izraziti preko pomeranja w(x), koristeći uslov jednakih rotacija spregnutih elemenata (drveta i betona), uvodeći odnos g(x), izraz (20) prikazujemo u sledećem obliku: Wi-strain energy due to internal forces, We -potential energy due to external forces. As the bending moment M(x) can be expressed by deflection w(x), using the condition of equal rotations of the composite members (timber and concrete), introducing the relation g(x), the expression (20) is represented in the following form: (23) gde je: where: Uvedeni odnos g(x) izveden je iz uslova kompatibilnosti pomeranja na spoju dva elementa, koji može da se zapiše u sledećem obliku: The introduced relation g(x) was derived from the compatibility of displacements at the interface of the two subelements, that could be written in the following form: Poznajući rad unutrašnjih sila Wi, određen je funkcional za elastično spregnuti SDB nosač. Kako se u izrazu g(x) javlja normalna sila N(x), za rešavanje problema, pored pretpostavljanja probne funkcije za pomeranje w(x), potrebno je pretpostaviti i probnu funkciju za normalnu silu N(x). Primenom varijacionih principa na funkcional, Ritz-ovom metodom, možemo rešiti problem elastičnog sprezanja, odnosno odrediti pomeranje nosača i unutrašnje sile u spregnutom nosaču.
Knowing the strain energy due to internal forces Wi, functional for a partial TCC system is determined. As in the expression g(x) the normal force N(x) appears, for solving the problem, beside assumed trial function of displacement w(x), it is also necessary to assume the trial function for N(x). By applying variation principles to a functional, with Ritz method, the problem of partially composite system can be solved, which means to determine displacement and internal forces in the composite members.

APPROXIMATION OF THE SOLUTION -TRIAL FUNCTIONS
Suitable, and therefore, most often, trial functions are polynomials or trigonometric functions. Trial functions should satisfy the following conditions: • to be continuous and differentiable till the necessary order, • in addition to essential, they also have to satisfy natural boundary conditions, • to correspond qualitatively by the form to the analytical solution, • to be complete, e.g. in the case of polynomials of a certain degree, all members of the lower degrees should also be included.
Based on the well-known solutions from the literature, equations (6) and (8), as well as on the qualitative flow of the solution (Fig. 2), three functions (hyperbolic, sinusoidal and polynomial functions) were selected for trial functions in this paper. The adopted trial functions describe the law of the change of the axial force N, slip forces Ts (N') and displacement w along the composite girder and qualitatively correspond to the solutions (Fig. 2). By means of expressions (27, 28 and 29), selected trials for Nand/orw are given, while on Fig. 3 the shape of the function and its first derivative along the beam are shown.
In Galerkin method, one of three suggested trial functions has to be adopted (for N or w) according to chosen integral formulation (Eq 13 or 14). In Ritz method it is necessary to adopt two trial functions(for N and w).
Analiza rezultata dobijenih primenom različitih probnih funkcija za aproksimativno rešenje pomeranja w, pokazuje da se minimalno odstupanje javlja ukoliko se usvoji funkcija oblika polinoma, a maksimalno ako se usvoji hiperbolična funkcija. Modeli w-HIP i w-SIN imaju značajna odstupanja kod napona (σ1,b and σ2,t) i to čak do 41%, a za sile klizanja i za smičuće sile (Ts and Fs) čak i do 28%. Model w-HIP pokazuje znatno manje The floor structure consists of a glulam beams that are coupled with concrete slab by vertically arranged dowel type fasteners. In this paper, the slip modulus K is determined by the Gelfi model [6]. The floor structure is loaded by the self-weight of the structural elements g, by additional permanent load dg, as well as by the imposed load p. It is considered that the timber glulam beams will be supported in the stage of pouring and hardening of the concrete slab, and the composite section will receive imposed and total permanent load. It is possible to analyze the part of the composite floor structure separately (glulam beam with the effective width of the concrete slab), because in analyzed TCC floor system all concrete slabs are one-way and glulam beams are simply supported with uniformly distributed load. Numerical analysis according to Galerkin and Ritz method of TCC structure was performed and several subprograms/codes are written in MATLAB 2014 [15]. The simplified "γ-procedure" was also performed in order to obtain the referent values suggested by Eurocode. From Figs. 5 and 6, it can be noticed that numerical results of group A models have smaller differences in relation to the analytical solution than the models of group B. Analysis of results obtained by different trial functions for the approximate solution of the normal force N shows that a minimal deviation occurs when a hyperbolic function is adopted, and the maximum one if it is the sinusoidal function. The models N-POL and N-SIN have significant deviations in stresses (σ1,b and σ2,t) even up to 21%, and minor deviations in the slip and shear forces (Ts and Fs) up to 6.5%. All values are smaller than those obtained by analytical solutions. Comparing the Group A models with results of the " γ " method, none of the models has greater deviations in absolute sense, but it can be noticed that the N-POL and N-SIN models give smaller values. Analysis of results obtained by different trial functions for the approximate displacement w solution shows that a minimal deviation occurs if the polynomial function is adopted, and the maximum one if it is a hyperbolic function. The w-HIP and w-SIN models have significant deviations in stresses (σ1,b and σ2,t) even up to 41%, and for slip and shear forces (Ts and Fs) even up to 28%. Model w-HIP shows significantly smaller values vrednosti u poređenju sa analitičkim rešenjem. Poredeći modele grupe B s rezultatima "γ-metode", može se uočiti je da najbolje podudaranje sa "γ-metodom" ima model w-POL. comparing to analytical. By comparing the models of group B with results of the "γ-method", it can be seen that the best match with the "γ-method" has the w-POL model.  Figure 6. Percentage deviations of models of group B in relation to the analytical solution

Numerical analysis by Ritz method
For the purpose of numerical analysis, the group of models is defined on the basis of simultaneous selection of two trial functions for axial force N(x) anddeflection w(x): Models of Group C (w-POL,N-POL; w-POL,N-SIN; w-SIN,N-SIN). All effects analyzed by Galerkin method are calculated according Ritz as well and their percentage deviations compared to analytical solution are presented on Fig 7. Results of simplified " γprocedure" were also compared with analytical solution.  The results obtained by variant model (w-POL,N-SIN) are on the safe side because they give a slightly higher values (up to 3%) for internal forces and displacements, while deviations for normal stresses and slip/ shear forces, arise up to 19% and 15% respectively, comparing to analytical solution. The reason for such increase lays in the fact that normal stresses and slip/ shear forces are derived values from baseline unknowns w(x) and N(x), so the cumulative errors are higher. It is obvious that the selection of trial functions for w(x) and N(x) has the significant impact on final result, as well as on derived statical values. It can be also noted that model (w-POL,N-SIN) has the best match with the approximate "γ-method" proposed in EN 1995. Although variant model (w-POL,N-POL) shows smaller deviations from variant model (w-SIN,N-SIN), comparing these two models with analytical solutions it can be seen that obtained values overestimate or underestimate analytical ones, but both models are not on the safe side.

Verification of Galerkin method
In order to verify the application of Galerkin's method in TCC system, the comparison of numerical obtained values and experimental data was done. As it was shown that the N-HIP modeldescribes the problem of TCC partial composite action on the best way, this model was applied for comparative analysis with experimental test results of TCC (EP1 and EP2) beams [18], where mechanical fasteners were used. The diagrams in Figure 8 show the stresses in cross-section of constitutive elements of TCC beams in the middle of the span, in relation to force intensity throughout experimental loading phases (F = 6, 12, 18, 24, 30 and 36 kN). The shaded surfaces represent the envelopes of the results obtained from experiments for beams EP1 and EP2, while the full and dashed lines represent the results of the numerical analysis by N-HIP model and by "γ-method" respectively. Considering the stresses (σ1,b and σ2,t) on the contact of two materials, deviations from the experimental results can be noticed, significantly for stressσ2,t at loads F = 18 and 24 kN. A good match with experimental results on the top and bottom sides of the cross-section as well as in bottom of the concrete slab is obvious.

CONCLUSION
Based on the presented analysis using Galerkin's method and Ritz method, it can be concluded that the selection of trial functions in problem formulation has the major influence on the final effect values of obtained and presented results. An important influence is also the choice of baseline unknowns (w and N) for Galerkin's method. Also, the qualitative knowledge of solution nature can significantly contribute tothe reduction of errors in obtained results related to analytical solution. In presented analysis by Galerkin's method the proposed N-HIP model qualifies as the most appropriate in order to solve the problem of partial coupling. When using a variational formulation, functional could be defined trough one or more unknowns (forces/displacements), while the unknowns that are chosen as basic will be determined with more accuracy than the derived ones.
As the weighted residual method or the variation formulation in the FEM is quite usual form, presented the Galerkin and Ritz methods could be successfully applied when defining a composite FE for partially timberconcrete composite systems, thereby enabling an efficient engineering tool in practice. With introduction of Eurocode for timber-concrete composite structures [4], it is expected that more precise definition of basic input parameter -slip modulus for different types of fasteners K, will contribute to the more realistic response of composite beams in numerical analysis.

ACKNOWLEDGMENTS
The work has been done within the scientific research project TR 36043 "Development and application of a comprehensive approach to the design of new and safety assessment of existing structures for seismic risk reduction in Serbia", which is funded by the Ministry of Science of Serbia.