## Jph

(a) Semi-rigid connection

(b) Moment-rotation behaviour

Fig. 12.11. Different types of rotational stiffness, k, in a timber connection.

structures but to a lesser degree in timber structures, is to represent the rotational stiffness of each connection by a spring.

Where the rotational stiffness varies, an iteration procedure has to be incorporated into the analysis to take account of the non-linear behaviour. However, where the rotational stiffness can be considered to be linear this procedure is not required. As stated in 10.10 and 11.8, in EC5 the lateral stiffness at the serviceability limit states (SLS) and the ULS of a metal dowel type fastener or a connector are linear relationships. Consequently, the rotational stiffness will also be linear and, depending on the limit states being considered, will be based on either the slip modulus at the SLS, Kser, or at the ULS, Ku, and the rotational stiffness at these states is as indicated in Figure 12.11b.

Using the assumptions in 12.5.1, adopting a rotational stiffness based on Kser will realistically cover the behaviour of the fastener up to the SLS. Where the fastener is loaded beyond this state, its rotational stiffness will reduce and at the ULS it will be based on Ku. For connections formed using several fasteners, it is likely that at the ULS many fasteners will only be stressed to levels approximating the SLS condition, and in the following approach the rotational stiffness of all fasteners in the connection is assumed to be based on Kser.

Consider a single or double shear connection within a structure having a regular pattern of fasteners, as shown in Figure 12.12, in which Mi is the moment per shear plane to be taken by fastener i at the ULS design condition. As the distribution of forces in the structure will be affected by the rotational behaviour of the connection, as stated in 2.3.4.2, for the stress resultant analysis the stiffness properties must be based on final mean values adjusted to the load component causing the largest stress in relation to strength, and are given in equations (2.38), (2.39) and (2.40). Adopting the relationship in equation (2.40) for the stiffness of each fastener and assuming a rotation in the connection, the moment rotation relationship given in equation (12.5) can be expressed as:

Fastener i c o

(b) Fastener i

(a) Connection

Fig. 12.12. Semi-rigid connection subjected to a design moment, Md.

x where the functions are as previously defined, and kdef is the deformation factor discussed in 2.3.2 and 10.10 and f 2 is the factor for the quasi-permanent value of the action causing the largest stress in relation to the strength.

Summing the contribution of all fasteners in the connection the moment per shear plane, Msp, taken by the connection can be written as:

K si

where xi and yi are the coordinates of fastener i relative to the axes shown in Figure 12.12 and n is the number of fasteners per shear plane in the connection. The rotational stiffness of the connection per shear plane will be as follows:

For the ULS design condition,

For the SLS design condition,

K ser

kser

Here the symbols are as previously defined, and ku is the rotational stiffness of the connection per shear plane at the ULS design condition, kser is the rotational stiffness of the connection per shear plane at the SLS design condition, and Kser is the slip modulus per fastener per shear plane at the SLS, given in Chapter 10 for metal dowel type fasteners and in Chapter 11 for connectors.

When the connection is also subjected to vertical and lateral design forces, the lateral stiffness of fastener i at the ULS can be written as:

The lateral stiffness per shear plane of the connection at the ULS for horizontal displacement, KH,u, and for vertical displacement, KV,u will be:

where n is the number of fasteners per shear plane in the connection.

Using the above expressions the semi-rigid behaviour of connections can be taken into account in the structural analysis.

Where a computer-based frame analysis is to be used, lateral slip effects can be represented by the use of linear elastic spring elements at connection positions, as discussed in 10.11. To simulate rotational slip, linear elastic rotational springs can be used, each having the rotational stiffness of the connection being modelled. As stated in 10.11, when modelling these additional elements care has to be taken to ensure that stability of the structure is retained, that the shear and flexural properties of the elements properly represent the connection behaviour, and that the size of the elements used will not result in ill-conditioned equations.

Guidance on the loading and stiffness requirements to be used in this type of analysis is given in 2.3.4.1 and 2.3.4.2 respectively.

12.6.2 The analysis of beams with semi-rigid end connections

In the following analysis, the behaviour of beams with end connections that exhibit semi-rigid rotational behaviour and in which lateral displacement effects can be ignored is considered. This situation commonly arises in timber construction and the method of analysis is such that it can readily be undertaken using Mathcad or equivalent software.

Consider within a structure any prismatic member ab of length L, flexural rigidity EI and single or double shear connections at each end. The rotational spring stiffness at each connection is k1 and k2 per shear plane and each is of negligible length, as shown in Figure 12.13. Instability effects due to axial loading are ignored.

Ends 1 and 2 are the positions at which the members of the connection are rigidly attached to the structure and 1,a and b,2 are the rotational spring elements simulating the behaviour of the fixings between the connection members and the beam at each end. Under the action of end moments from the structure, the connection members will rotate and there will also be a relative rotation between these members and the beam ends, represented by rotation of the spring elements. The joint rotations at ends 1 and 2 are designated ^ and #2, respectively, and the relative rotation of the springs at each end will be #r1 and #r2 respectively.

The moment required to be applied to the end of a prismatic member to cause a unit rotation at that end when the other end is fixed in position is referred to as the member stiffness. For a member of length L, having a deformation factor kdef,m, a modulus of elasticity E = q+E^j ) and second moment of area about the axis of bending I, it

Coordinate system used

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