Lukas Blesak1, Sandra R.S. Monteiro2,
Alfredo M.P.G. Dias3, Frantisek Wald1
1Department of Steel and Timber Structures, Czech Technical University in Prague, Czech Republic
2Department of Civil Engineering, University of Coimbra, Portugal
3Department of Civil Engineering, University of Coimbra, Portugal
Numerical simulation based on non-linear (NL) behaviour material definition of concrete in both compression and tension as a part of a coupled timber-concrete load-bearing system is presented in this paper. The main goal of a NL material modelling performed and introduced herein is to define the influence of concrete non-linearity over transverse loading distribution horizontally across a slab loaded locally.
Numerical simulations based both on linear elastic (LE) and NL material behaviour were compared with the results obtained from several real-size experiments whilst the effects of several specific phenomena will be published later. NL material definition of concrete is also termed as a “damaged material model” in certain references.
This work is a continuation of the research performed at the Department of Civil Engineering, University of Coimbra, which’s result had been analysed and further published in paper  and will be presented in a doctoral thesis .
Coupled timber-concrete load-bearing systems dispose of several advantages, such as lower deformations, higher load-carrying capacity, lower noise transmission, decrease in vibrations  and many others, which are well known and need not to be described in details. Both deformations and loading distribution of the loading applied over a coupled timber-concrete system is majorly influenced by the coupling system and the stiffness properties of the used structural elements. Tangential and axial stiffness of coupling in prefabricated, semi-prefabricated and in-situ casted systems is well known and described in the relevant standards , other papers [4, 5, 6] and many other. However, phenomena of local loading distribution in transverse direction – perpendicular to beams’ axis, has not been analysed sufficient yet and the references in this field of research can hardly be found, e.g. .
This problem is naturally highly effected by concrete material properties and its NL behaviour. Concrete is neither homogeneous nor isotropic. The non-isotropic feature of concrete makes its compressive strength different from its tensile strength , and so NL material models are preferably used more than LE models. However, its justness needs to be considered carefully.
Cracks can dramatically reduce the long-term durability performance of concrete, e.g. due to permitting increased fluid ingress , therefore preventing cracks from its initiation and a further evolution, for example by using high strength concrete, e.g. fibre concrete, may increase durability of concrete slabs significantly.
In this paper, the phenomena of transverse loading distribution over a coupled timber-concrete load-bearing slab is analysed, numerical simulations being compared with the real-size experiment are introduced and consequent results are summarized and discussed further.
Several real-size experiment set-ups were considered to analyse a global behaviour of a thin concrete slab coupled with overall five glued laminated timber beams focused on transverse loading distribution when being loaded locally. To make the results and further discussion clearer, only one experiment set-up is described below.
Concrete slab with thickness 50 mm (t) sized 3.39 x 3.48 m (L x B, L – dimension in the direction of beams axes) was coupled with overall 5 glued laminated beams GL28h with cross-section 100 x 200 mm (b x h), placed 0.75 m from one another. Concrete slab was provided with a structural steel rebar mesh sized 100 x 100 mm (m x n), bar diameter equal to 5 mm (t), placed in the middle of the slab thickness. Material properties of timber beams and concrete mixture were experimentally tested and the proper values were applied in the further analysis. Between concrete slab and timber beams a layer of pine wood shuttering (PWS) with thickness 20 mm (tpws) was assembled. Coupling was ensured by studs with diameter 8 mm, placed in vertical position each 100 mm in the beam axis direction. The structure was placed on supports fixed in the vertical direction and free in both horizontal directions at the end of each beam supporting its bottom edge. Structure was loaded locally in several places, always right above a beam – in the middle or in a quarter of span, through a steel plate dimensioned 200 x 200 x 40 mm; deflections were recorded in the middle and in the quarters of the beams spans; vertical reactions were recorded in all supports. Local vertical loading was applied with its value equal to 40 – 50 % of the ultimate loading estimated based on a previous numerical modelling. The experiment set-up can be seen in Figure 1. Beams were marked No. 1 to 5 from one side to another.
Four loading scenarios were opted for numerical simulations. Beams No. 2 and 3, loaded in the middle and in a quarter of span, respectively.
Loading scenarios are titled and defined as follows:
- B2 1/2 – Beam No. 2 loaded in the middle of span, F = 49.9 kN (43 % of ultimate load)
- B2 1/4 – Beam No. 2 loaded in the quarter of span, F = 51.7 kN (40 % of ultimate load)
- B3 1/2 – Beam No. 3 loaded in the middle of span, F = 57.5 kN (50 % of ultimate load)
- B3 1/4 – Beam No. 3 loaded in the quarter of span, F = 51.7 kN (40 % of ultimate load)
Software ATENA Science based on FEM was used for both LE and NL simulations. This software was chosen for its advanced definition of concrete post-crack behaviour, taking fracture energy Gf and other parameters definition into account [15, 16], which enables cracks initiation and evolution being observed. The basic input data applied in the numerical simulation is plotted in Figure 2.
For concrete slab simulation, CCIsoShellBrick elements with 4 cross-section layers were used. Reinforcement layer with the appertaining cross-section area and material properties was placed in the middle of the concrete slab. The same elements with 4 cross-section layers were used for PWS simulation. 3D solid elements were considered for glued laminated timber simulation. Type of elements and their geometry were opted based on the necessary computing requirements (dimension ratios, number of iteration points in a cross-section etc.) together with the computing efficiency based mainly on the results precision and computing time.
Modulus of elasticity (MOE) of glued laminated timber was defined experimentally by testing each of the beams respectively. The appertaining values of MOE for each beam were considered in the numerical model. Material properties of concrete were applied based on the results of compression tests (maximal stress in compression sav in tested cube failure – average value from all the tested specimen) performed on the subjected concrete mixture and a further numerical modelling. Default material model “SOLID Concrete Cementitious2” was applied. The default material model was chosen based on the results of compression tests and also considering the most appropriate material properties of concrete in tension. Detailed material description and particular input values definition can be found in [15, 16].
Two material interlayers (connection layers) Ka and Kc (Figure 2) were applied in the model. Shear (tangential) stiffness of timber-concrete coupling was defined in a previous research based on the experimental results. The applied data are plotted in Figure 3.
Average value of tangential stiffness KTT.b was considered (marked as “Model” in Figure 3). As the timber-concrete coupling is modelled through two shear planes, “glued laminated timber + PWS” and “PWS + concrete”, partial values of KTT.b marked as KTT.a were defined as understood from formula (1).
F: is the force acting over the connection
?slip: slip movement
b: beam width
ms: distance between screws in beam axis direction
Normal stiffness (in the vertical direction – axis “z”) KNN.a = KNN.c ˜ 8.
Tangential stiffness KTT.c was assumed to be equal to 0. In the experimental set-up assembly a plastic film with thickness approx. 0.1 mm was placed between PWS and concrete slab in order to prevent any shear stiffness or friction between the connected elements, see Figure 4.
Material for timber elements was defined as LE in all numerical models, whilst a global MOE was applied for each beam respectively; material for concrete was varied from LE to NL. The particular input data are listed in Table 1.
Exponential crack opening law was considered for tension behaviour of subjected concrete. The function of crack opening was derived experimentally  and the fracture energy value is based on . Material properties of concrete, particularly in tension, are of a crucial importance. In order to simulate the structural behaviour as precious as possible, the strength class and default material model of concrete was defined following an inverse analysis taking the experimental data into account.
For the experiment presented herein, biaxial stress state in a thin slab is taken into account. Based on :
- biaxial stress state is affected by micro-cracking
- when concrete is compressed in biaxial stress state:
- compressive and tensile strength is higher than its uniaxial strength
- tensile ductility is greater than that under uniaxial compression
- elastic limit is shifted up
Resulting the facts listed above, material properties of a particular concrete mixture, based on the standard definition, may not represent the very realistic behaviour of an analysed experiment. Failure modes, and so further results, depend on various stress combinations and possibly other phenomena which are being analyzed in an ongoing research. The same problems were observed in the research dealing with steel fibre reinforced concrete material and its proper numerical definition .
Primarily the vertical reactions obtained from numerical simulations and experiment were compared. Results are depicted in Figures 5 to 8. When evaluating the results, two numerical models were considered:
- models marked as “elastic” – material parameters of all the applied materials considered to be LE – defined by a relevant MOE and Poisson’s coefficient
- models marked as “Cem2” – timber elements material considered to be LE; concrete elements material considered to be NL (damaged material model)
- results marked as “exp.” – measured experimentally
Reactions expressed in % in the following graphs represent a percentage contribution of the particular beam in the overall loading distribution to the vertical supporting system. Reaction on both beam ends were summed up and divided by the actions of mechanical loading – overall force acting over the structure. It should be noted, that LE material model was also performed neglecting the presence of PWS  and similar results were obtained as when PWS was considered. As the ratio of slab bending stiffness and PWS bending stiffness was evaluated to approx. 1/40 ? 2.5 %, contribution of PWS bending stiffness may be neglected in this case (concrete slab 50 mm thick, PWS 20 mm thick, Econcrete ˜ 3 x Epine.wood).
As can be seen in Figure 5, contribution of the particular beams to the loading force distribution towards the vertical supports varies from the experimental results in case of the simulation taking NL behaviour of concrete into account. On the other hand, the LE simulation shows results similar to the experimental ones.
The bigger slab curvature is achieved, the bigger differences between the experimental and NL models were observed. The same phenomena was observed in case of all the loading scenarios.
Even NL material definition is generally considered to be more realistic, models with LE material definition gave results closer to the experimental ones in this case. However, it is necessary to mention the fact, that in case of transverse loading distribution the material properties of concrete in tension need to be considered properly. NL material models showed to be rather sensitive to the input data precision, a proper mesh size application, suitable way of loading and other, see also .
In order to investigate further the differences listed above, the progressive cracks pattern was observed for a chosen loading scenario – B3 – 1/2. The loading action over the slab was applied in 50 steps; the crack patterns in step No. 20, 50 are depicted in Figure 9. Only the middle part of the slab is shown – the area where beams No. 2, 3 and 4 are located.
As shell elements were applied in the modelling, the cracks are depicted throughout the slab thickness. The real position regarding the slab edge (top, bottom) can be seen in Figure 10, where width of cracks in the direction parallel to the beams axes is depicted.
The initial crack position along the loaded beam in the place of loading can be seen from the introduced crack opening patterns. The moment cracks occur – the ultimate tension strength in concrete is reached, and start to develop – following ductility definition based on the fracture energy value, the loading actions start to be re-distributed all over the surrounding structure following the stiffness conditions. As cracks develop, also the deflection under the loading point increases. Following the exponential crack opening law, crack width where no stress is resisted in the crack plane is given by formula (2).
wc: is the crack opening at the complete release of stress
Gf: fracture energy
ft.ef: the effective tensile strength derived from a failure function – reduced due to compression stress in the perpendicular direction
The maximal crack width, of cracks in direction parallel to the beams axes, was observed equal to 0.541 mm (see Figure 10). Following formula (2), crack width at the complete release of stress was evaluated to 0.382 mm.
Following the knowledge presented in this paper, the following conclusions have been drawn:
- pine wood shuttering layer may be neglected in numerical simulations due to its low bending stiffness contribution
- when stress-release crack width is not achieved, application of linear elastic material model of concrete represents the real structural behaviour within an acceptable range of error
- non-linear material properties of concrete given by standard, applied in numerical simulations, may not describe its real behaviour accurately in case of thin slabs loaded locally due to local defects and other phenomena such as tri-axial stress state, non-homogeneity in aggregate size, local loading concentration
- application of homogenous aggregate size with maximal sieve-fraction size related to the slab thickness may prevent local defects and so increase the global structural resistance
- initial crack evolution under the loaded area influences a further crack pattern evolution and so the global loading distribution across the slab in the horizontal direction
- application of high ductility concretes, e.g. fibre concrete, is an effective solution for thin low reinforced or not reinforced concrete slabs
This publication was supported by the European social fund within the framework of realizing the project „Support of inter-sectoral mobility and quality enhancement of research teams at Czech Technical University in Prague“, CZ.1.07/2.3.00/30.0034. Period of the project´s realization 1.12.2012 – 30.6.2015.
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