Analysis of timber-concrete composite girders

Timber-concrete composite girders are often used in the renovation of high-rise structures, and they can also be used in bridges. These systems are relatively complex to analyse as two materials presenting different stiffness and rheological properties participate in ensuring an appropriate bearing capacity. While the analysis of instantaneous deformations is clearly defined in Eurocode 5, the analysis of longterm deformations, which are often relevant, is not clearly defined. The objective of this paper is to present the analysis of a timber-concrete composite girder, taking into account instantaneous and long-term deformations in a relatively simple way, suitable for practical engineers.


Introduction
The principle of joining together different materials is based on the idea that a material that is highly resistant to tensile stress (e.g.steel, timber, etc.) should be placed in the tensile zone of crosssection, while a material that is highly resistant to compressive stress (concrete being the most frequent one) should be placed in the compressive zone of cross-section.The effective cross-section has a high load bearing capacity and stiffness, and a joint effect of two materials connected in this way is greater than the sum of their individual effects.Lightweight concrete i.e. concrete made with low-weight aggregate (rendered light by addition of expanded polystyrene granules) can also be used in floor structures in order to increase thermal and insulating properties of such structures.Lightweight-aggregate concretes are also considered appropriate because their elastic modulus is similar to that of timber.When using lightweight-aggregate concrete, a special attention must be paid to the selection of connectors (continuous composite actions are more favourable as in case of discrete action the connector can fail at the contact between the connector and concrete much before the relevant edge stress is achieved) [1].Poorer mechanical properties of such concretes also play a significant role in this respect [1].The thickness of slabs used does not usually exceed 8 cm, especially in case of traditional concrete, so as to avoid significance increase in self-weight of the structure as a whole.The analysis of timberconcrete composite girders is defined in Eurocode 5: Design of timber structures -Part 1-1: General -Common rules and rules for buildings [2] and in Eurocode 5: Design of timber structures -Part 2: Bridges -National annex [3].Here it should clearly be emphasized that the analysis made in the mentioned standards is actually the analysis relating only to short-term effects toward the final ultimate state.The issue of calculating long-term deformation of such composite girders is considered in paper [4].The current situation in this area regarding such influences is presented in this paper through review of relevant literature.The aim of this paper is to present analysis of a timber-concrete composite girder taking at that into account long-term deformations, all this in a relatively simple way that can be useful to practical engineers.Symbols/designations used in the paper do not necessarily correspond to those given in Eurocode 5 (e.g.elastic modulus of concrete is designated as E C rather than as E 1 as given in Eurocode 5).The authors consider that the designations used in the paper are more favourable for clarity reasons.Eurocode 5 provisions are given for a general composite girder and, as a timberconcrete composite girder is considered in this paper, the mentioned designations are used.

Analysis of concrete slab and timber beam composite structure with flexible connectors
These systems are dimensioned using the y-procedure defined in Eurocode 5 [2].Figure 2 shows a concrete slab and timber beam composite structure with flexible connectors, where s is the distance between connectors. Figure 3 shows the cross section and the corresponding composite-girder geometrical properties that will be explained further on in the paper.Effective bending stiffness of shear-flexible composite beam is defined as follows: where: E c -secant modulus of elasticity for concrete (E cm ) E t -modulus of elasticity for timber I c -moment of inertia for concrete part of cross-section I t -moment of inertia for timber part of cross-section A c -area of concrete part of cross-section A t -area of timber part of cross-section γ c -composite action coefficient for concrete γ t -composite action coefficient for timber a c -eccentricity of centre of the concrete part of cross-section a t -eccentricity of centre of the timber part of cross-section Eccentricity of centre of the timber (a t ) and concrete (a c ) parts of cross-section: (2) (3) where: Analysis of timber-concrete composite girders h c -height of concrete part of cross section h t -height of timber part of cross section The sliding coefficient for shear-flexible composite T-beam with neutral axis in timber web is determined as follows: where: s -distance between connectors K -sliding modulus L -span.
Longitudinal normal stress in centres of individual parts of the shear-flexible composite T-beam: (5) Moduli of elasticity are: where: E cm -mean modulus of elasticity in bending, for concrete E 0,mean -mean modulus of elasticity for timber in the direction of fibres Stresses in the shear-flexible T-section (Figure 4) are due to joint action of a pair of longitudinal forces in centres of individual parts of cross-section (resulting from sliding) and bending (M Ed ) where: σ t -stress caused by compressive force in timber part of cross section σ c -stress caused by compressive force in concrete part of cross sectio σ m,t -bending stress in timber part of cross-section σ m,c -bending stress in concrete part of cross-section M d -design bending moment (EI) eff -effective bending stiffness.

Assumptions for the analysis:
connectors are positioned at the design spacing of s i = s, along the length of the element L liding modulus, K i (N/mm) is experimentally defined from shear test diagrams or push-out tests [7], and its values are: for SLS (13) Dean Čizmar, Maja Vrančić where: K ser -initial (useful) sliding modulus K u -effective sliding modulus.
If the sliding modulus can not be determined experimentally, as shown in papers [1,7], then expressions presented in Eurocode 5 [2] can be used, as has been done in Section 3.
In addition, considering the level of shear and yield of connectors, tension can occur in the bottom part of concrete cross-section, i.e. the neutral axis of cross-section can be in the flange of cross-section.This case is not favourable and so attempts are made to prevent this situation by providing a sufficient number of connectors, i.e. to try to reach the situation in which the concrete cross-section is fully in compression.This shows that the limit stress in the bottom part of concrete cross-section must be lower compared to the compressive strength of concrete (as presented in expressions 15 and 16).However, if tension occurs, then the stress must be lower than the tensile strength of concrete f ctm .Stress values at the top, σ c,g and bottom edges of concrete slab of shear-flexible composite T-section, σ c,d must comply with the following equations: where: σ c -stress caused by compressive force in concrete part of cross section σ m,c -bending stress in concrete part of cross-section f c,d -design compressive strength of concrete.
Bearing capacity verification for timber beam of shear-flexible composite cross-section: If necessary, the bearing capacity proof (17) can be written at the level of forces as follows: -Design longitudinal force in timber element: -Design resistance to longitudinal force: -Design bending moment in timber element: where W t is the resistance moment for timber cross-section.
-The bearing capacity proof is written as follows: Proof load as follows: (23)

Verifications for serviceability limit state: Verification of momentary deflections
The following K i value has been adopted: Eccentricities of centres of the timber and concrete parts of cross-section: Effective flexural stiffness of shear-flexible composite beam: Total deflection due to permanent load: (28) Analysis of timber-concrete composite girders Total deflection due to variable load: The following conditions must be met for momentary deflection: The following condition must be met for total deflection: (34)

Analysis of long-term effects
According to [4], the analysis of such systems with regard to long-term load is much more challenging and complex as mechanical changes in timber, concrete, and steel have to be taken into account due to changes in moisture, temperature, and load over time.Therefore, the following is taken into account during analysis of long-term effects: creep of concrete, creep of timber element, and connector slip (displacement).The analysis presented in this paper is based on paper [6].

Verifications for ultimate limit state:
Total deformation ε cσ (t,t 0 ) is generally defined as follows: The effective elastic modulus for the long-term load of concrete is defined as follows: (35) where: Final mean value of elastic modulus of concrete in the direction of fibres (36) where: Ψ -factor for combined value k def -deformation factor.
Sliding modulus: (37) Flexural stiffness parameters over the life span of the structure: -Sliding coefficient for concrete: (38) -Eccentricity of centres of the timber (a t ) and concrete (a c ) parts of cross-section: -Effective flexural stiffness: -Quasi-constant combination: -Design bending moment: (43) -Design transverse force: (44) The following conditions must be met for the timber part of cross-section: -Longitudinal stress in timber caused by longitudinal force, due to load combination q Sd,1 -Longitudinal stress in timber by bending moment due to load combination q Sd,1 Verification of bearing capacity for timber beam having shearflexible composite cross-section: (47) It should be noted that the results obtained by this method are satisfactory if both materials are situated in the linear-elastic area.A general deficiency of this method lies in the fact that it does not take into account ductility of the connection.Other than this method, the use can be made of the fixed shear force method.The method assumes an elastoplastic loading and load relaxation relationship and thus partly takes into account ductility of the connection.The assumption that all connecting devices yield simultaneously leads to an estimation error, and so this method can be considered relatively conservative.The third calculation method, i.e. the elastoplastic method, is considered to be suitable for the ultimate limit state analysis.
The method assumes a perfectly stiff connection, i.e. a perfectly plastic strain to yield relationship.As most connecting devices do not provide a perfectly stiff connection, we obtain a bearing capacity that exceeds a realistic one when calculating structural behaviour under service-life load.The elastoplastic method is favourable for obtaining the final effective stiffness and the bearing capacity of the structure itself, but it overestimates the initial effective stiffness in the elastic area.Based on the above considerations, and as this method has been defined in Eurocode 5, the example will be presented using the γ method.
Usual beam dimensions and floor and soffit layers are given in the example (Figure 5).The condition after renovation (Figure 6) implies removal of wood debris and formwork with lime plaster so as to achieve properties compliant with present-day requirements.Dimension of elements in figures below have been taken from paper [5].Both permanent and service loads are taken into account.It is assumed that the class of use (moisture) will be 1 and so the factor of change and factor of deformation were assumed to be k mod = 0.9 and K def = 0.6, respectively.The factor for quasi-permanent variable action was defined in accordance with category A: houses, residential buildings.

Verification for ultimate limit state
This evaluation was carried out as follows: Eccentricity of centre of the timber (a t ) and concrete (a c ) arts of cross-section: Effective flexural stiffness of the shear-flexible composite beam: Design load:

Longitudinal normal stress in centres of individual parts of a shear-flexible composite T beam:
The stress of the top, σ c,g and bottom, σ c,d edges of the concrete slab having a shear-flexible composite T section must comply with the following equations: Verification of bearing capacity for the timber beam having a shear-flexible composite section: Verification of shear resistance of timber part of cross-sectionfully assumes maximum transverse force, V d : Analysis of timber-concrete composite girders τ = 0,67 N/mm 2 ≤ 1,73 N/mm 2

Verifications for serviceability limit statexerification of momentary deflections
The following value is adopted for the sliding modulus K i : Sliding coefficients: Eccentricity of centres of the timber (a t ) and concrete (a c ) parts of cross-section: Effective flexural stiffness of shear-flexible composite beam: The following conditions must be met for momentary deflection: 6,39 mm ≤ 20mm 4,10 mm ≤ 20mm.

Verification for ultimate limit state
Effective elastic modulus of concrete for long-term load: The following parameters were used in Figure 7 for calculation of creep coefficient: concrete class C25/30, N curve in diagram (for N class of cement), and start of system load after 10 days (t 0 = 10 days).j(t,t 0 ) = 3,5 -Quasi-permanent combination: -Design bending moment: -Design transverse force: The following requirements must be med for the timber part of cross-section: -Longitudinal stress in timber caused by longitudinal force, from load combination q Sd,1 -Longitudinal stress in timber due to bending moment, from load combination q Sd,1 -Bearing capacity verification for timber beam of shearflexible composite cross-section: 0,32 ≤ 1 Shear stress in timber caused by load combination q Sd,1 τ max(qSd,1) = 0,35 N/mm 2 ≤ 1,73 N/mm 2 Verification of pressure perpendicular to fibres on the support, for load combination q Sd,1 The following conditions must be met for the concrete part of cross-section: -Longitudinal stress in concrete caused by longitudinal force, from load combination q Sd,1 -Longitudinal stress in concrete by bending moment, from load combination q Sd,1 -Total stress at top edge of concrete: Analysis of timber-concrete composite girders

Verification for serviceability limit state:
The effective elastic modulus for the long-term load of concrete is defined as follows: Final mean elastic modulus of timber: Sliding modulus: Sliding coefficients: Eccentricity of centres of the timber (a t ) and concrete (a c ) parts of cross-section: Effective flexural stiffness of shear-flexible composite beam:: The following conditions must be met for final deflection: 11,84 mm ≤ 30 mm 7,61 mm ≤ 30 mm The following conditions must be met for final deflection due to permanent and variable load: 11,84 + 7,61 mm ≤ 30 mm 19,45 mm ≤ 30 mm

Analysis of required reinforcement
The concrete slab must be strengthened with minimum reinforcement so as to ensure proper ductility of cross-section, as well as a lower influence of creep and shrinkage of concrete.This analysis is clearly defined in Eurocode 2 [9].

Analysis of connectors
The analysis of bearing capacity of composite beam connectors is not defined in Eurocode 5.In this paper, the analysis will be made according to [7,8].

Parameters for connectors:
-Compressive strength of timber along periphery of hole for load in the direction of fibres: -Typical tensile strength of steel for the construction of connectors S 275:

Figure 1 .
Figure 1.Schematic presentation of the system

Figure 3 .
Figure 3.View of cross-section and longitudinal normal stresses in centres of individual parts of shear-flexible composite T-beam, [1]

Figure 4
Figure 4 Stress in shear-flexible T-section (17) where: σ t,d -design tensile stress σ m,d -design bending stress f t,o,d -design tensile strength parallel to fibres f m,d -design bending strength.Verification of shear bearing capacity of the timber part of cross section -fully assumes maximum transverse force,V d : (18) b eff = k cr • b = 0,67 • b where: V d -design transverse force b eff -design width of timber element k cr -cracking factor for shear resistance b -width of timber cross-section τ -shear stress f v,d -design shear strength.

Figure 7 .
Figure 7. Determination of creep coefficient j(∞,t 0 ) for concrete under normal ambient conditions