Summary Optimised design of grooves in timber-concrete composite slabs Prof. Dr.-Ing. Ulrie Kuhlmann Institute of Structural Design/University of Stuttgart, Stuttgart, Germany Dipl.-Ing. Birgit Michelfelder Harrer Ingenieure/Karlsruhe former: Institute of Structural Design/University of Stuttgart, Germany The global structural and deformation behaviour of a timber-concrete composite slab strongly depends on the load-slip behaviour and the position of the connectors between the two components timber and concrete. So these factors are most important to lead to an economic fabrication of nail-laminated timber-concrete composite slabs. The presented paper deals with investigations on composite structures including grooves as connections to derive values of the ultimate strength and the stiffness of the connection, as well as to derive design and construction rules. Grooves as connectors proved to be easy and economically fabricated and show a high stiffness and strength. As grooves are usually placed in the outer areas of the composite slab in the sections of the maximum shear forces, the connectors are partially placed in higher distances to each other. In order to find out, if the existing methods of analysis are capable of taing various connection distances into account, comparisons of calculations with different methods are performed. 1. Introduction Originally timber-concrete composite slabs have been developed to strengthen existing timber joist floors with a concrete slab. But this construction technique is now also applied in new buildings, usually as a combination of nail-laminated board stacs and concrete. Timber-concretecomposite structures show many advantages compared to pure timber or pure concrete slabs. In timber concrete composite structures, the concrete is placed in the pressure zone and the timber in the tension zone. This leads to an increase of the stiffness and the load capacity and improves the sound and fire insulation of a pure timber slab. Compared to a pure concrete slab, the timber reduces the dead load and leads to a more ecologic building structure. 2. Ultimate strength and stiffness of grooves 2.1 General In order to find out the stiffness and the ultimate resistance of grooves, short-term shear tests were performed by varying different parameters, such as the material and the geometry of the grooves (see Fig 1 (a)). A total of 30 specimens were tested. Thereby 4 different failure modes were observed. In the timber section the grooves act similar to a carpenters notch, in the section of the concrete similar to a console. So the timber failed due to shearing and/or compression at the edge of the groove (see Fig 1 (b)). In the failure modes of concrete cracs formed due to shearing and compression at the edge of the notch (see Fig 1 (c) and (d)) occured. These failure modes and resulting ultimate loads are the bases for design equations to calculate the ultimate strength.
(a) Setup (b) Failure of timber (c)compression failure of concrete Fig 1 Setup and failure modes of the short-term shear tests (d) Shear failure of concrete 2.2 Model for calculation of the design resistance of grooves To visualise these failure mode and to allow for an estimation of the ultimate strength, [6] developed a strut and tie model especially for the concrete element (Fig 2), following a FE evaluation of the elastic study. Equations (1) to (5) give design loads for determining the resistance of the groove. For detailed information see [6]. 2 1 K timber Brettstapel (a) Fig 2 Points of verifications Beton concrete (b) K 3 5 60 4 Design timber shear resistance (1), see Fig 2 a): N = mod f b l Rd, H,1 γ red v, vh M (1) Design resistance of compression of the timber of the groove (2), see Fig 2 a): N mod Rd,H,2 = f c,0, b t γ M (2) incl. mod Modification factor according to [1] or [2] γ M Partial safety factor according to [1] or [2] f c,0, Compressive strength parallel to grain according to [1] or [2] f v, Shearing strength of the timber according to [1] or [2] red Reduction factor of the shearing strength [6] t K Depth of the groove b Relating width (normally b=1m)
Design resistance of the compression on the concrete notch (3), see Fig 2 b): N Rd,B,3 = t b 0,75 f cd (3) Design resistance of the strut in the concrete notch (4), see Fig 2 b): N Rd,B,4 = 0,75 f cd sin²θ t b Design resistance of the tie in the concrete notch (5), see Fig 2 b): N Rd, B,5 = f l b tanθ ctd eff (4) (5) incl. f ctd Tensile strength of the concrete according to [4] or [5] l eff Effective width of the tie f cd Compressive strength of the concrete according to [4] or [5] t K Depth of the groove b Relating width (normally b=1m) Θ=60 Angle of the strut The minimum value of the various equations should be taen as design strength of the groove. The factor red has been adapted to the statistical evaluation of the tests, however further investigations in order to verify the local timber and concrete strength have to be performed. 2.3 Probablilistic analysis concerning the stiffness of grooves The stiffness of the connection was determined by short term shear tests. To verify the test results and to investigate a wider range of parameters, these tests were simulated by a three-dimensional Finite Element model. Tests and simulation showed, that the natural scattering of the material properties of the timber, especially the modulus of elasticity, leads to a scattering of the connection stiffness. So, in addition to the deterministic analysis, also a probabilistic analysis have been performed. The aim was to identify a value for the connection stiffness to be used in the calculation of the composite slabs. Fig 3 Finite Element model for probabilistic analysis relative frequency 0,50 0,40 0,30 0,20 0,10 0,00 400 450 500 550 600 650 700 750 connections stiffness Ks [N/mm] Fig 4 Distribution of the connection stiffness Within the Finite Element model, the modulus of elasticity of the timber is considered by a Gaussian Distribution with a mean value of E mean =10500 N/mm² and a standard deviation of s= 2120 N/mm² in the probabilistic analysis (see Fig 3).A value of K ser = K u = 429 N/mm is determined for the shown geometry. Fig 4 shows the distribution of the connection stiffness for the geometry given in Fig 5, achieved by means of Monte Carlo simulation.
2 cm 25 cm 20 cm 90 Fig 5 Geometry of the regarded groove 8 cm 12 cm This value is valid for board stacs of fir wood and concrete C20/25 assembled with grooves as shear connectors. Due to the fact, that grooves with that geometry show a linear elastic behaviour up to the ultimate load, the connection stiffness follows the same value in the serviceability limit state and in the ultimate limit state. 3. Design rules 3.1 General As Finite Element calculation showed, there are also tensile stresses in the concrete parallel to the span in front of the loaded side of the notch. To be able to consider them in the calculation and design of a composite slab without FE, design rules have been developed by FE-analysis of various composite slabs. Based on an extensive parameter study with FE, a geometry was defined that leads to a maximum of effective bending stiffness of the slab, that only has maximum tensile stresses in the mid span and not in the critical zone of the notch. So the concrete can be assessed with the nown methods according to [4] and [5]. These studies lead to the following criteria. 3.2 Amount and arrangement of the grooves vh1 lk vh2 lk vh3 lk Fig 6 Arrangement of the grooves in the composite slab Tab 1 Amount and arrangement of the grooves span amount vh1 vh2 vh3 l = 5 m 2 25 cm 45 cm ---- l = 6 m 3 25 cm 25 cm 48 cm l = 7 m 3 25 cm 25 cm 63 cm l = 8 m 3 25 cm 25 cm 80 cm l = 9 m 3 25 cm 25 cm 99 cm l = 10 m 3 25 cm 25 cm 120 cm Fig 6 shows the parameters that have been varied, when developing the amount and arrangement of the grooves. Given in Table 1 are values of the distances between the grooves and the total amount of grooves per side for different spans. 3.3 Height ratio of board stac and concrete The studies also led to an optimised height ratio of the components timber and concrete. Height of the board stac Height of the concrete slab Total height h = 0.6 h t (6) h = 0.4 h c (7) h 20cm
4. Evaluation of existing design methods 4.1 General Normally the connectors are placed in accordance to the shear force along the beam, normally only in the areas of the supports, i.e. in the outer thirds of the slab. That means, that there is a strong gradation in stiffness in the joint. This gradation has to be considered in the calculation of such composite slabs to avoid neglecting locally critical stresses. With the different design methods, such as the γ-method according to [1] or [2], the frame model according to [3] or the shear analogy method according to [1], this gradation of stiffness could be. These methods are investigated by comparing the results of 78 different beam configurations to the results of numerical analyses verified by testing. The aim is to find out, how accurately the effective bending stiffness and the stresses in the connectors are calculated with the different methods. 4.2 γ-method For the determination of the internal forces and moments and the deformations of elastic composed beams, a simple commonly used design method was developed and is integrated into [1] and [2]. The flexibility of the connectors is considered by a γ-factor in the calculation of the effective bending stiffness. This method has been derived for single span beams with sinus-shaped loads and constant stiffness in the connection joint. According to [1] and [2], this method may also be used for single span beams with uniformly distributed loads and different connector spaces within a specific range. This means, that the maximum distance s max has to be lower or equal to 4 times the minimum distance s min (see Fig 7). For calculation an effective connectors distance s ef is defined, see equation (8). s ef = 0.75 s min + 0.25 s max (8) smin smax l/2 Fig 7 Definition of the connector distances The comparison of the results of the Finite Element analyses and the γ-method shows, that the effective bending stiffness is in general overestimated (with a mean value of 7% overestimation) by using the approach of equation (8), whereas the connectors forces are mostly calculated too high (mean value 130% overestimation). To improve the usability of the γ-method according to [1] and [2] also for composite systems with strongly gradated connector distance, a new approach was developed [6], that enables the calculation of the effective bending stiffness without overestimation. The basis of this new approach is a regression analysis that has led to the following equation of the effective connector distance to be considered in the γ-method. s ef = 1.14 s min + 3.14 s max / l (s max s min ) (9) Considering equation (9) when calculating the effective bending stiffness according to the γ- method leads to smaller effective stiffness values (with a mean value of 5% underestimation) and therefore higher calculated deformations than the FE does.
5,5 250 EI ef (γ-method) 5,0 4,5 4,0 [ 10 12 Nmm²] F [N] (γ-method) 200 150 100 50 3,5 3,5 4,0 4,5 5,0 5,5 EI ef (Finite Element analysis) 0 0 50 100 150 200 250 F [N] (Finite Element analysis) Fig 8 Comparison of the effective bending stiffness with the approach of equation (9) and FE-analysis Fig 9 Comparison of the force of the connector with the approach of equation (9) and FE-analysis However the maximum shear force determined even with the improved s ef distance of the groove is far too how compared to FE (mean value of 113% overestimation) and therefore leads to an uneconomical design. 4.3 Frame model As shown in Fig 10, timber-concrete composite structures may be modelled also as a frame [3]. gc + q concrete slab IC, AC groove 1 groove 2 groove 3 board stac It, At gt l/2 Fig 10 Truss model [3] The connectors are considered in their realistic position, so the shear force of the connectors is calculated with a high accuracy. Also the internal forces and moments needed for the assessment may be taen directly from the model. The lower flange of the frame-model represents the timber, whereas the upper flange corresponds the concrete slab. These girders are modelled by beam elements along the cross-section axis. Rigid hinged bars connecting the two beams induce the same deformations of both girders. At the position of a connector, vertical bars with a hinge in the connection joint are joined rigidly to the girders. Also the jumps of the normal forces caused by the transfer of the shear force and the jumps of the bending moment caused by the eccentric loading are modelled realistically. The elasticity of the connectors is considered by an effective bending stiffness of the bars representing the connectors. The effective bending stiffness is calculated by
K 3 3 h h EI* = ser/u c + t 3 2 2 (10) incl. K ser stiffness of the connection in the serviceability limit state K u stiffness of the connection in the ultimate limit state h c height of the concrete cross-section height of the timber cross-section h t The dead load of the single components is placed on the upper(concrete) and lower (timber) flange, all the service loads are placed at the upper flange. By using a computer aided frame calculation, the mid-span stresses and deformation and the forces of the grooves are determined very satisfying accuracy (see Fig 11 and Fig 12). EI ef (truss model) 5,5 5,0 4,5 4,0 [ 10 12 Nmm²] 3,5 3,5 4,0 4,5 5,0 5,5 EI ef (Finite Element analysis) Fig 11 Comparison of the effective bending stiffness of the frame model and FE-analysis F [N] (truss model) 100 80 60 40 20 0 0 20 40 60 80 100 F [N] (Finite Element analysis) Fig 12 Comparison of the connector forces of the frame model and FE-analysis 4.4 Shear analogy method Also included in [1] is the shear analogy method. The composite slab is thereby devided into two components of a beam system. These systems are coupled with stiff couple bars (see Fig 13 and equation (11) to (13)). By adjusting the shearing stiffness of beam B, a gradation of the stiffness in the connection joint may be taen into account. beam A a coupling beam B Fig 13 Beam system according to the shear analogy method [1] By a computer calculation of the beam system, the internal forces and moments of the two beams may be determined. Beam A represents the bending stiffness of the components, so the bending moments of timber and concrete and the shear force can be derived from the forces and moments of beam A. Beam B corresponds to the composite action, so the normal forces of timber and concrete, and the shear flow may be specified from the results of beam B. The bending stiffness l
and the internal forces and moments of the composite slab are calculated with the help of the shear analogy method satisfying precisely per length. The forces of the connectors could be determined slightly too high. For further information see [1] and [6]. EI = EI + EI A concrete timber (11) EA EA EI = a² concrete timber B EA + EA concrete timber (12) GA = a 2 B (13) incl.: = K / s ef Stiffness of the joint K Stiffness of the single connector Effective connector distance s ef 5. Discussion and conclusion With the design equations shown in chapter 2, a proposal for the determination of the design resistance of a connection is given. The stiffness of the connection is also derived for a typical design of a groove by a probabilistic study based on FE analysis. With the existing design methods, especially the frame model and the shear analogy method, the calculation of a timberconcrete composite slab with grooves can be performed with satisfying accuracy. Further investigations should be performed with regard to the long term behaviour, especially of the groove itself. The research presented was part of a research project performed at the Institute for Structural Design, University of Stuttgart. The Deutsche Gesellschaft für Holzforschung, e.v. (DGfH), the Arbeitsgemeinschaft Industrieller Forschungsvereinigungen Otto von Guerice e.v. (AiF), as well as all other supporters are gratefully acnowledged for financing and supporting the project. 6. References [1] Deutsches Institut für Normung: DIN 1052; Entwurf, Berechnung und Bemessung von Holzbauweren- Allgemeine Bemessungsregeln und Bemessungsregeln für den Hochbau; 2004. [2] CEN Comité Européen de Normalisation: DIN EN 1995-1-1, Bemessung und Konstrution von Holzbauten; Teil 1-1: Allgemeine Regeln und Regeln für den Hochbau; 2004 [3] Rautenstrauch, K.; Grosse, M.; Lehmann, S.; Hartnac, R.: Baupratische Dimensionierung von Holz-Beton-Verbunddecen; Institut für Konstrutiven Ingenieurbau; Bauhaus- Universität Weimar; 6. Informationstag des IKI; 2003. [4] DIN Deutsches Institut für Normung: DIN 1045-1; Tragwere aus Beton, Stahlbeton und Spannbeton, Teil 1: Bemessung und Konstrution; 2001. [5] CEN Comité Européen de Normalisation: Eurocode 2: Planung von Stahlbeton- und Spannbetontragweren; Teil 1: Grundlagen und Anwendungsregeln für den Hochbau; 1992. [6] Michelfelder, B.: Trag- und Verformungsverhalten von Kerven bei Brettstapel-Beton- Verbunddecen, Institut für Konstrution und Entwurf; University of Stuttgart; Mitteilungen, No. 2006-1, Dissertation submitted for accuracy.