## Bending Stiffness Of Builtup Columns

In Chapter 5 it is shown that when dealing with solid section columns, for normal design situations the design strength will generally be determined by the buckling strength of the member. This is also the case for columns made from built-up sections. With built-up sections, however, the determination of the buckling strength is a more involved exercise, because of the following reasons:

• Materials having different properties can be combined.

• There will be slip within the section when mechanical fasteners are used.

• There is the added risk of local instability effects with certain sections, e.g.

### T- or certain I-sections.

If the interfaces of the members in a built-up column are connected by mechanical fasteners, when subjected to flexure there will be slip between adjacent elements leading to a discontinuity in strain at these positions and the curvature of the individual elements will differ. In these situations, conventional bending theory cannot be used to determine the bending stiffness of the section and to analyse this condition the effect of the slip has to be taken into account.

This can be achieved by fully modelling the behaviour of each fastener in the connection and analysing the built-up section using a finite element analysis approach. An alternative, slightly less accurate, method, but suitable for normal design purposes, is to apply conventional bending theory to each element in the built-up column, assume compatibility in the curvature and displacement of adjacent column elements at each interface, and simulate the slip effect at these interfaces by assuming that the fastener resistance in these zones can be represented by linear spring elements.

From this type of analysis a reasonable estimate of the bending stiffness of the composite section will be obtained. When the stiffness of the springs is set equal to infinity, the bending stiffness of a built-up section having glued interfaces will be obtained, and if it is zero, the bending stiffness will equate to a section in which the members are not connected.

An example of the modelling applied to a built-up T-section connected by fasteners is shown in Figure 8.3. For a displacement of the section in the z-z direction, as shown in Figure 8.3a, the resistance offered by the fasteners will equate to the fastener stiffness multiplied by the relative slip between the elements at the interface as shown in Figure 8.3c. In the analysis, a spring arrangement having the same stiffness per unit length as the fastener stiffness is fitted at the interface between the beam and the flange as shown in Figure 8.3d. By applying simple bending theory to the model and using the principles of linear elastic theory, the structure can be analysed and a reasonable estimate of the bending stiffness of the composite column section can be obtained. The method is used in EC5 where the stiffness is referred to as the effective bending stiffness and is defined as (EI)ef.

When designing columns, the design state will be the ultimate limit state (ULS), and where members are connected using mechanical fasteners, the fastener stiffness per unit length is taken to be the ULS slip modulus per shear plane, Ku, divided by the fastener spacing. The ULS slip modulus per shear plane is referred to in 10.10 and is obtained from: 2

. Displaced shape of column

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