## Modelling of beams columns and bracings

Beams, columns and bracings are normally modelled as prismatic 3D beam elements, characterized by their cross-sectional area, moments of inertia, Iy and I7, with respect to the principal axesy andz of the cross-section, shear areas^ andAz along these local axes (for shear flexibility, which is important in members with low length-to-cross-sectional-depth ratio) and torsional moment of inertia, C or Ix for St Venant torsion about the member centroidal axis x.

Members with a cross-section consisting of more than one rectangular part (e.g. L-, T- and C-shaped sections) are always dimensioned for internal forces (moments and shears) parallel to the sides of the cross-section. So, the analysis should provide action effects referring to centroidal axes parallel to the sides. In columns, walls or bracings with non-symmetrical cross-section (e.g. L- and T-shaped sections, etc.), these axes normally deviate from the principal axes of the cross-section. When this deviation is large and the difference in flexural rigidity between the two actual principal directions of bending is significant (e.g. in L-shaped sections), and if it is considered important that the bending moments from the analysis reflect this difference (e.g. for consistency with the different flexural capacities in these two directions), then, along with the easily computed moments of inertia with respect to centroidal axes y and z parallel to the sides of the cross-section, its product of inertia Iyz should also be specified (alternatively, the orientation of the principal axes y and z with respect to the global coordinate system, and principal moments of inertia should be given). For the same type of section, shear areas in the two directions parallel to the sides may be taken as equal to the full area of the rectangle(s) with the long sides parallel to the direction of interest and projected on the principal centroidal axes, to find the shear areas and^4z in these directions.

Concrete or composite beams connected with a concrete slab are considered to have a T, L, etc., cross-section, with the effective flange width considered constant throughout the span. The effective slab width, taken for convenience to be the same as for gravity loads, is specified in the material Eurocodes as a fraction of the distance between successive points of inflection of the beam. In long girders supporting at intermediate points secondary joist beams or even vertically interrupted ('cut-off) columns and modelled as a series of sub-beams, the effective flange width of all these sub-beams should be taken to be the same, and established on the basis of the overall span of the girder between supports on vertical elements. In contrast, the effective flange width of secondary joist beams will depend on their shorter spans between girders.

At variance with the statement in the second paragraph of the present subsection on columns, walls or bracings with an L, T or other non-symmetrical cross-section, beams with a concrete flange connected to a floor slab should be assigned local y and z axes normal and parallel to the plane of the slab, respectively, even when their webs are not normal to the plane of the slab (e.g. horizontal beams supporting an inclined roof). The moment of inertia I, is computed for the T or L section on the basis of the effective flange width and the shear area A is that of the beam web alone. If the slab to which the beam is connected is considered as a rigid diaphragm, the values of A, Iy and Az are immaterial; if this is not the case, these properties may have to be determined to model the flexibility of the diaphragm.

According to Section 4 of EN 1998-1 the structural model should also account for the Clause 4.3.2(2) contribution of joint regions (e.g. end zones in beams or columns of frames) to the deformability of the structure. To this end, the length of the 3D beam element which falls within the physical region of a joint with another member is often considered as rigid. If this is done for all members framing into a joint, the overall structural stiffness is overestimated, as significant shear deformation takes place in the joint panel zone (there is also slippage and partial pull-out of longitudinal bars from concrete joints). It is recommended, therefore, that only the part within the physical joint of the less bulky and stiff elements framing into it, e.g. normally of the beams, is considered to be rigid. There are two ways of modelling the end region(s) of a member as rigid:

(1) to consider the clear length of the element, say of a beam, as its real 'elastic' length and use a (6 x 6) transfer matrix to express the rigid-body-motion kinematic constraint between the degrees of freedom at the real end of the member at the face of the column and those of the mathematical node, where the mathematical elements are interconnected

(2) to insert fictitious, nearly infinitely rigid, short elements between the real ends of the 'elastic' member and the corresponding mathematical nodes.

Apart from the increased computational burden due to the additional elements and nodes, approach 1 may produce ill-conditioning, due to the very large difference in stiffness between the connected elements, real and fictitious. If this approach is used due to lack of computational capability for approach 2, the sensitivity of the results to the stiffness of the fictitious members should be checked, e.g. by ensuring that they remain almost the same when the stiffness of the fictitious elements changes by an order of magnitude.

If the end regions of a member, e.g. of a beam, within the joints are modelled as rigid, member stress resultants at member ends, routinely given in the output of the analysis, can be used directly for dimensioning the member end sections at the column faces. If no such rigid ends are specified, as recommended above for columns, then either the stress resultants at the top and the soffit of the beam will be separately calculated on the basis of the beam depth, etc., or dimensioning of the column will be conservatively performed on the basis of the stress resultants at the mathematical nodes.

If the centroidal axes of connected members do not intersect, the mathematical node should be placed on the centroidal axis of one of the connected members, typically a vertical one, and the ends of the other members should be connected to that node at an eccentricity. The eccentricity of the connection will be readily incorporated in the modelling of the beam end regions within the joint as rigid: the rigid end will not be collinear with the beam axis but at an angle.

Distributed loads specified on a member with rigid ends are often considered by the analysis program to act only on the 'elastic' part of the member between the rigid ends. The part of the load which is unaccounted for as falling outside the 'elastic' member length should be specified separately as concentrated forces at the nodes.

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