## A

Stress

Elastic behaviour

Strain

Elastic behaviour

(b) Second-order analysis Fig. 2.1. First- and second-order linear elastic analysis.

• Elastic-perfectly plastic (Figure 2.2a) - linear elastic followed by pure plastic behaviour.

• Elasto-plastic (Figure 2.2b) - linear elastic followed by plastic behaviour with strain hardening.

• Rigid plastic (Figure 2.3) - pure plastic behaviour, using limit states analysis for the assessment of ultimate loading (e.g. for the behaviour of joints formed with metal dowel type fasteners in timber structures).

2.2.16.2 Structural analysis requirements for timber and wood-based structures

In regard to timber and wood-based structures, the structural analysis requirements are addressed in EC5, Section 5.

Because of the brittle nature of timber under tension-induced stress configurations, plastic analysis should not be used and EC5 requires that the forces in the elements

Stress

Strain

(a) Elastic-perfectly Plastic

Stress

Stress

Strain

Strain

(b) Elasto-plastic with strain (c) Continuously varying hardening

Fig. 2.2. Alternative stress-strain relationships commonly used in non-linear analysis.

Stress

Strain

Rigid plastic

### Fig. 2.3. Rigid plastic behaviour.

of the structure be determined using a linear elastic analysis. The effects of deviation from straightness of members have to be taken into account and this will be achieved by validation of the element strength using the design rules in EC5. Where, however, it is considered that a second-order linear analysis is necessary when dealing with plane frames or arches, this should be carried out in accordance with the requirements of EC5, 5.4.4.

Although all timber connections will exhibit semi-rigid behaviour to varying degrees, where their rotational deformation will have a negligible effect on the force distribution in the structure, EC5 states that the connections may be considered to be rigid and, where this is not the case, they may generally be assumed to be rotationally pinned (EC5,5.4.2(7)). As this is not a Principle in EC5, where it is felt that the effect of semirigid behaviour of connections should be taken into account in the analysis, provided it can be considered to be effectively linear and have adequate ductility, by incorporating the stiffness behaviour of the connections into the structural model a linear elastic analysis of the semi-rigid structure can still be undertaken.

When designing joints formed with metal dowel type fasteners the strength equations in EC5 assume that failure at the joint will be in accordance with the principles of plastic theory, as shown in Figure 2.3. In such situations, the joint forces will be derived from an elastic analysis of the structure at the ULS and the associated joint strengths determined from the application of the relevant EC5 strength equations, which are derived primarily from the assumption that rigid plastic behaviour will apply. This is one of the apparent anomalies between modelling to determine the action effect (i.e. the global analysis model) and modelling for strength verification where a different model can be assumed.

2.2.17 Verification by the partial factor method: general (EC0, 6.1)

For normal designs, the partial factor design method should be used for the design of the structure and its elements. In this method, the effects of actions are multiplied by partial factors to obtain the design value, Efd, and resistances, which are generally derived from material strengths, are divided by partial factors to obtain the design resistance, Rd, at the ULS and SLS. Verification is undertaken at the relevant state to demonstrate that Efd is less than or equal to the design resistance Rd, i.e.

Efd ULS

< RdULS and Ef

The values used for actions and material properties are the characteristic or other representative values and the values used for the partial factors vary depending on the limit state being considered and must be such that the level of reliability referred to in 2.2.3 for the structure at that limit state will be achieved.

2.2.18 Design values of actions (EC0, 6.3.1)

In general terms, the design value Fd of an action can be written as

where

In the above, Frep is the value to be taken into account in the relevant combination of actions. It can be the main representative value (i.e. the characteristic value, Fk), the combination value, f 0 Fk, the frequent value, f 1 Fk, or the quasi-permanent value, f 2Fk; Fk is the characteristic value of the action; f is either 1.00 or f 0, f 1 or f 2.

2.2.19 Design values of the effects of actions (EC0, 6.3.2)

The 'effects' of actions, Efd, are the response of the structure to the imposed actions and cover the internal stress resultants (e.g. moments, shear forces, axial forces, stress or strain) and the structural deformations (e.g. deflections and rotations).

Based on the content of EC0, 6.3.2(1) the design value of the effects of actions can be written in general terms as

where y sd is a partial factor taking account of uncertainties in modelling the effects of actions, y f,i is a partial factor for action i that takes account of the possibility of unfavourable deviations of the action values from the representative values, ad is the design value of the geometrical data (discussed in 2.2.22), and i is the number of representative actions.

For the design of timber and wood product structures in accordance with EC5, partial factors y f and y Sd are combined into one factor y F (i.e. Y F = Y f Y sd) simplifying equation (2.3) to

In the Eurocodes, when determining the design value of a permanent action, yF is defined as yg and when determining the design value of a variable action, it is defined as yq .

The values of yg and yq are dependent on the limit states being considered and this is addressed in 2.2.24 for ULS and 2.2.25 for SLS.

2.2.20 Design values of material or product properties (EC0, 6.3.3)

The design value Xd of a material or product property can be derived for the ULS and the SLS and reference to this value in timber design will invariably mean to the value used at the ULS. It is obtained from where Xk is the characteristic value of the property, n is the mean value of a conversion factor that takes into account volume and scale effects, the effects of moisture and temperature and any other relevant parameters, and ym is a partial factor that takes into account the possibility of the characteristic value of a material or product property (e.g. strength or stiffness) being less than the specified value and also the effect of scatter around the mean value of the conversion factor.

In EC5, n covers the effects of duration of load and variation in moisture content on the properties of timber and wood products and is referred to as the modification factor, kmod. Factors covering scale and volume effects are considered separately in EC5 and are discussed in 2.3.6.

The modification factor is extremely important in timber design and a brief overview of how load duration and moisture content effects are taken into account is given in the following sub-sections.

### 2.2.20.1 Load duration classes

When subjected to loading, the strength properties of members reduce and the longer the duration of the load the greater the reduction will be. In order to establish a common basis for design, load duration classes (see EC5, 2.3.1.2) have been defined to cover the range of durations likely to arise in practice and the duration associated with each class is given in Table 2.3, based on the content of NA.2.1 in the UKNA to EC5.

Table 2.3 Load-duration class definitions*

Examples given in NA.2.1 of the UKNA to

Class

Period of time

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