S General requirements for nonlinear modelling

Modelling for the purposes of non-linear analysis should be an extension of that used for linear methods, to include the post-elastic behaviour of members beyond their yield strength. Put differently, as a non-linear analysis degenerates into a linear one if member yield strength is not attained during the seismic response, in the linear range of behaviour, modelling for non-linear analysis should be consistent with that used for linear analysis. Consistency does not imply that the level of discretization and the modelling of elastic stiffness needs to be identical to that used in linear analysis: as non-linear analysis is done mainly for the purposes of evaluation of a design, its modelling is not bound by the fact that present-day seismic dimensioning and detailing rules address the member as a whole and hence point in the direction of member-by-member modelling in linear analysis. However, all things considered - including the consistency with linear analysis and the computational and modelling effort required for non-linear finite-element modelling - the member-by-member type of modelling, with every beam, column, bracing or part of a wall between successive floors modelled as a non-linear 3D beam element, is the most appropriate option for non-linear analysis.

In principle, only the stiffness properties of members are of interest for linear elastic analysis. As emphasized in Section 4.6.4, to reflect the requirement that the elastic global stiffness corresponds to the stiffness of the elastic branch of a bi-linear global force-deformation response in monotonic loading, the elastic stiffness of a bilinear monotonic force-deformation relation in a member model should be the secant stiffness to the yield


point. Member models to be used in non-linear analysis should also include the yield strength of the member, as this is governed by the most critical (i.e. weakest) mechanism of force transfer in the member, and the post-yield branch in monotonic loading thereafter.

The bilinear force-deformation relationship advocated here for the monotonic force-deformation relation in a non-linear member model is a minimum requirement according to the relevant clause of Eurocode 8. For concrete and masonry, the elastic stiffness of such a bilinear force-deformation relation should be that of the cracked concrete section according to Section 4.6.4. If it is taken equal to the default value of 50% of the uncracked gross section stiffness for consistency with the linear analysis, storey drifts and member deformation demands are seriously underestimated. In case the response is evaluated by comparing member deformation demands to (realistic) deformation capacities, such as those given in Annex A of Part 3 of Eurocode 8,52 then demands should also be realistically estimated by using as the effective elastic stiffness a representative value of the member secant stiffness to incipient yielding (also given in Annex A of EN 1998-3,52 after Biskinis and Fardis54).

If the monotonic behaviour exhibits strain hardening after yielding (as in concrete members Clause in bending and in steel or composite members in bending or shear, or in tension) a constant (3) hardening ratio (e.g. 5%) may be considered for the post-yield stiffness. Alternatively, positive strain hardening may be neglected and a zero post-yield stiffness may be conservatively adopted. However, elements exhibiting post-elastic strength degradation, e.g. (unreinforced) masonry walls in shear or steel braces in compression, should be modelled with a negative slope of their post-elastic monotonic force-deformation relationship. It should be pointed out that the ductile mechanisms of force transfer also exhibit significant strength degradation when they approach their ultimate deformation. However, as in new designs the deformation demands in ductile members due to the design seismic action stay well below their ultimate deformation, there is no need to introduce a negative slope anywhere along their monotonic force-deformation relationship.

Gravity loads included in the seismic design situation according to Section 4.4.2 should be Clauses taken to act on the relevant elements of the model as in linear analysis. Eurocode 8 requires (5), taking into account the value of the axial force due to these gravity loads, when determining (6) the force-deformation relations for structural elements. This means that the effect of the fluctuation of axial load during the seismic response may be neglected. In fact, this fluctuation is significant only in vertical elements on the perimeter of the building and in the individual walls of coupled wall systems. Most element models can take into account - be it only approximately - the effect of the fluctuation of axial load on the force-deformation relations of vertical elements. Examples are the fibre models, as well as any simple lumped inelasticity (point hinge) model with parameters (e.g. yield strength and effective elastic stiffness) which are explicitly given in terms of the current value of the axial load.

For simplicity, Eurocode 8 allows the bending moments in vertical members due to gravity loads to be neglected, unless they are significant with respect to the flexural capacity of the member.

Non-linear models should be based on mean values of material strengths, which are higher Clause than the corresponding nominal values. For an existing building the mean strength of a . (4) specific material is the one inferred from in situ measurements, laboratory tests of samples and other relevant sources of information. For the mean strength of materials to be incorporated in the future in a new building, Eurocode 8 makes reference to the material Eurocodes. However, only the mean strength of concrete is given there: Eurocode 2 gives the mean strength as 8 MPa greater than the characteristic strength, fA. Statistics drawn from all over Europe suggest a mean value of the yield strength of steel about 15% higher than the characteristic or nominal value, fyk. Locally applicable data should be used for the reinforcing steel, if known. Similarly for structural steel, for which the relatively small number of manufacturers serving most parts of Europe points towards the most likely supplier of the steel to be used as the source of relevant statistics.

Other than the use of the mean value of material strengths instead of the design values, member strengths (resistances) to be used in the non-linear member models may be computed as for the relevant force-based verifications.

It is noteworthy that the use of mean material properties is not specific to non-linear analysis: linear analysis is based on mean values of elastic moduli, which are the only material properties used for the calculation of (the effective) elastic stiffness.

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  • luigina
    What is "seismic dimensioning"?
    8 years ago

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