Failure criteria for built-up beams

5.2.3.1 Recommendations regarding the stiffness properties to be used

For ordinary beams designed under the assumption of full composite action we need only to account for the difference in elastic modulus and creep properties of the parts making up the cross-section, as explained in Section 5.1.1.1. It is recommended to perform the analysis for partial composite action in the same way, that is using a transformed cross-section such that all parts can be assumed to have the same modulus of elasticity. It may also be necessary to make relevant checks at “instantaneous” and “final” conditions in order to cope with the creep effects.

There is one additional complication for partial composite beams due to the creep properties of the shear interface. Mechanical connections in timber materials are definitely sensitive to creep. Usually, the connection is more prone to creep than the timber itself, mainly due to relatively high stress levels in the small timber volumes surrounding the connector. In Eurocode 5 it is recommended to use a creep factor kdef for the slip modulus of the connector that is twice as large as for the timber itself, that is to use 2 kdef. Creep usually plays a sub stan- tial role even if a thick shear interface layer (core) is used without any mechanical con nectors, because such materials are often sensitive to creep.

Here, we will concentrate on mechanical connectors in built-up timber beams. Their slip modulus k is in Eurocode 5 characterised by either Kser or Ku = 2/3 × Kser. Kser is the gradient of a straight line going through a point on the load-slip relationship corresponding to approxi- mately 40 percent of the connector’s ultimate resistance, while Ku corresponds to the 70 percent level.

How will the internal forces and stresses change in the composite beam if the slip modulus increases or decreases? The answers are that a decreasing slip modulus will give:

• A smaller axial force N1 but the bending moments M1 and M2 will increase such that the net outcome is greater stress at the outermost edge of each layer.

• A smaller connector force F, i.e. the shear flow H = –N'1 is decreasing and more of the load is carried by individual bending of the layers. We will have no composite action at all for a slip modulus equal to zero.

• A greater deflection.

An increasing slip modulus will give the opposite effects to those listed above. Luckily a large change in the slip modulus will produce a much smaller change in member forces, member stresses and deflection. Reducing k to half its value or doubling it will change these values in the order of 10 %, under typical conditions. But note that the connector force will typically change by the same order as k.

connectors, hence subjected to a smaller load than actually calculated. Therefore, Ku is still a good choice, also noting that many connectors along the beam are not fully used. Of course, if one really wants to be on the safe side, Kser might be used when the connectors are the weakest part of the beam. If the connectors are much stronger than the timber, then a value closer to Kser may be beneficial when checking the timber resistance.

In the ULS it may still be necessary to distinguish between “instantaneous” and “final” conditions due to the creep properties of the timber itself. Here the key recommendation is to check the members most prone to creep at instantaneous conditions, while members less sensitive to creep are checked at final conditions.

In the serviceability limit state (SLS) it is recommended that Kser is used when determining all the cross-sectional constants.

Following the recommendations given in Eurocode 5 as close as possible, the slip modulus and the modulus of elasticity may for the instantaneous and final conditions be taken from Table 5.4.

Table 5.4 Recommended values for the slip and elastic modulus of composite timber beams for which mechanical connectors are used and partial composite action prevails.

Instantaneous conditions, inst Final conditions, fin

Serviceability limit state, SLS

Ultimate limit state, ULS

* A note on determination of final deformations. Cross-sectional constants,that is Ifca, C, ω, CMb and CM of transformed cross-sections,

should be based on ESLS,fin and kSLS,fin for the instantaneous condition before the effect of creep is added. But all moduli beyond these

fictitious cross-sectional constants should keep their instantaneous values ESLS,inst = E0,mean and kSLS,inst = Kser, because they will at a later

stage be multiplied by either (1 + kdef) or (1 + 2 kdef) when going from instantaneous to final deformation as given by Equation 5.86. Note, that the relations of Table 5.4 should be used to calculate all cross-sectional constants of a transformed cross-section when determining the distribution of internal forces between members of different materials having different creep properties. After that the final SLS- deformation is estimated using Equation 5.86, the method outlined here will slightly over- estimate the final creep deformation in that all the cross-sectional constants are determined using the creep factor for permanent loading. In the ULS ψ2 shall be determined for the load causing the greatest stress, which effectively is a way of calculating a kind of weighted mean value of the cross-sectional constants that will give reasonable estimations of internal forces and stresses.

Equations 5.59 and 5.84 are made up of two terms: the first is called wfca as it is the deflection in case of full composite action, while the second is called wslip as it is the additional deflection caused by the connector slip. In wfca the modulus of elasticity E is controlling the deflection, while the slip modulus k (kcore) is controlling the deflection in wslip. Of course these moduli are also hidden within Ifca, C and ω, but are then purely relational and determine the distribution of internal forces between the members. Suppose now that we have permanent loading denoted G and variable loading denoted Q1 for the leading variable load and Qi for the other variable loads, i is 2, 3, etc. Following the recommendations in Eurocode 5 to use the char- acteristic combination in Eurocode 0 for determining SLS-displacements, the instantaneous deflection may formally be expressed as:

and the final deflection as:

where ψ2 is a reduction factor giving the quasi permanent load value, while ψ0 gives the combination value of the load.

In Equations 5.85 and 5.86, make sure to use E0,mean in all wfca and Kser in all wslip. Note also that shear deflection caused by shearing of each timber member is neglected, which is justified by the fact that its contribution usually is small compared to wslip.

5.2.3.2 Failure criteria in the ULS

Any failure criterion must be adjusted to meet the requirements of each particular material used as members and connectors. Stresses in each member must in general be checked, remembering that shear lag must be considered in very wide flanges and in wide compression flanges local buckling as well. The effective width can be determined using the approach given in Section 5.1.2.1. The integrity of the shear interface/interfaces must also be checked, that is shear stresses in a continuous core material or connector forces if discrete mechanical fasteners are used. Here, the failure criteria of a beam made of timber members joined by mechanical connectors will be discussed.

If the flanges are made of structural timber, glulam, LVL or alike, then the interaction formulas in Section 3.1.3 (or if the bending part in a member is small the criteria in Equation 5.6 or

Eq. 5.85

If the connectors are made of nails, staples, screws, dowels or bolts, then the failure modes according to Chapter 4 must be checked. Note, however, that bolts (and maybe also dowels) are quite unsuitable to use as connectors in this kind of composite beam, because the large initial slip caused by the clearance between bolt and hole will give large deflections before any shear force can be transmitted.

5.3 References

Elsander (1999): Elementarfall för tvåskikts samverkansbalkar med linjärt elastiskt beteende. Royal Institute of Technology, div. of Steel Structures, Master of Science Thesis 125.

(In Swedish)

Foschi (1970): Rolling shear failure of plywood in structural components. Forest Products Laboratory, Madison. Information report VP-x-67.

Granholm, H. (1949): Om sammansatta balkar och pelare med särskild hänsyn till spikade

träkonstruktioner. (On composite beams and columns with particular regard to nailed timber

structures). Transactions, Chalmers University of Technology, Rep. No. 88. (In Swedish). Höglund (1990): Design of timber beams with webs of plywood or structural board. Royal Institute of Technology, div. of Steel Structures, Report.

Larsen, Riberholt (2005): Träkonstruktioner, Beregning. Statens Byggeforskningsinstitut. SBI-anvisning 210, Hørsholm. (In Danish)

Lekhnitsky (1968): Anisotropic plates. Gordon & Breach Science Publishers, New York. Newmark, N.M, Siess, C.P, Viest, I.M. (1951): Tests and analysis of composite beams with incomplete interaction. Proceedings, Society for Experimental Stress Analysis 9(1): 75–92. Oduyemi, T.O.S., Wright, H.D. (1991): Partial interaction analysis of double skin composite beams. Journal of Constructional Steel Research 19(4): 253–283.

SS-EN 1990:2002/A1:2005/AC:2009/AC:2010 Eurocode 0: Basis of Structural Design. SIS Förlag AB.

SS-EN 1993-1-5:2006/AC:2009 Eurocode 3 – Design of steel structures – Part 1–5:

Plated structural elements. SIS Förlag AB.

SS-EN 1995-1-1:2004/AC:2006/A1:2008/A2:2014 Eurocode 5: Design of timber structures

– Part 1–1: General – Common rules and rules for buildings. SIS Förlag AB.

StBK-N5 (1980): Tunnplåtsnorm. AB Svensk Byggtjänst, Stockholm. (In Swedish). Stüssi, F. (1947): Zusammengesetzte vollvandträger. (Built-up girders). International

Association for Bridge and Structural Engineering, IABSE, Vol. 8, pp. 249–269. (In German). Thielgard, Larsen (1978): Limede I-bjælker af træ. Aalborg University, inst. for

6 Horizontal stabilization ...201

6.1 General considerations ...201

6.2 The importance of bracing...205

6.3 Fundamental statics ...206

6.4 Strength and stiffness requirements for bracing systems ...209

6.4.1 Perfectly straight column ...209

6.4.2 Column with initial imperfections ...212

6.4.3 Beam, truss or column systems ...215

6.5 Typical bracing systems for heavy timber structures...219

6.6 Special topics ...223

6.6.1 Forces arising due to the slope of the roof ...223

6.6.2 Stability of the upper chord of low-arch or low-truss bridges ...223

6.6.3 Bracing of continuous beams at intermediary supports ...224

6.7 Bracing of timber framed buildings ...225

6.7.1 Floor and roof diaphragms ...226

6.7.2 The in-plane resistance of shear walls under horizontal loading ...227

6.8 References ...230

Chapter 6

Horizontal stabilization