As pointed out in the Preamble, analysis carried out within the framework of seismic design or assessment determines by calculation the effects of the design actions (including the seismic action) in terms of internal forces and deformations, for the purpose of dimensioning or assessing structural members. For concrete members, design action effects are used to verify the sizes of members and to dimension or assess the amount of reinforcement.
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Notes
- 1.
Data from Europe available at the time of drafting Eurocode 8 could not support dependence of the elastic spectrum on additional parameters.
- 2.
Under the Poisson assumption of earthquake occurrence (i.e. that the number of earthquakes in an interval of time depends only on the length of the interval in a time-invariant way), the return period, T R, of seismic events exceeding a certain threshold is related to the probability this threshold will be exceeded, P, in T L years as: T R = –T L/ln(1–P). So, for given T L (e.g., the conventional design life of T L = 50 years) the seismic action may equivalently be specified either via its mean return period, T R, or its probability of exceedance in T L years, P R.
- 3.
Low values of the multiplicative factor are applied for high values of the design ground acceleration and large ones for lower design ground accelerations.
- 4.
The commonly reported modal participation factors and effective modal masses along X, Y and Z are not so informative about torsion about the vertical.
- 5.
Dynamic DoFs are those contributing to the equation of motion with inertia terms.
- 6.
An eigenvalue analysis of the elastic structure should precede the nonlinear dynamic one anyway, for insight into the predominant features of the expected response.
- 7.
This is allowed in Eurocode 8 for structures regular in plan. Note than in a 2D model all nodes of a floor that may belong to different 2D frames have the same horizontal displacement, regardless of any static eccentricity between the floor centres of stiffness and mass.
- 8.
Strongly irregular arrangements, such as infills mainly along two adjacent faces of the building, cannot be taken into account in this simplified way, see Section 2.1.13.2.
- 9.
One way to achieve a non-pinned connection of a region modelled with 2D FEs to a 3D beam element in the same plane, is to provide the end of the 3D beam element with an almost rigid extension into the region modelled with 2D FEs, connecting the node at the physical end of the beam element with any FE node inside the 2D FE region.
- 10.
As a matter of fact, in the context of US codes (BSSC 2003, SEAOC 1999) a realistic stiffness for concrete members has little practical implication for strength-based member design, as the design base shear is not allowed to be less than 80% of the value computed from the design spectrum on the basis of empirical period formulas (SEAOC 1999), or less than that given by the design spectrum at a multiple between 1.4 and 1.7 of the empirical period (BSSC 2003). The reduction in lateral force demands due to concrete cracking may account partly for the high values of the force reduction factors R of US codes. The stiffness used for concrete members has implications mainly for the calculated interstorey drifts.
- 11.
The beam is dimensioned for the ULS in bending, or assessed on the basis of its ultimate deformation, for action effects about an axis parallel to the slab, using as beam depth the projection, hsinβ, of the actual depth h on the normal to the slab and as web width the value b w/sinβ (β is the angle of the web to the plane of the slab). Assessment of the beam and of its transverse reinforcement in shear can be based on the actual depth and width of the web, h and b w, but with a shear force equal to the shear V y from the analysis or capacity design in the direction of the normal to the slab, divided by sinβ.
- 12.
- 13.
If the slab is thick, thick-plate FEs, e.g. of the Midlin type, should preferably be used.
- 14.
Full lateral contact should develop over the effective embedment depth, capable of both sidewall friction and passive earth pressure. So, d cannot be more than the thickness of the footing and normally is less.
- 15.
In Eq. (4.63) ϕ y denotes the curvature from the analysis and index y signifies the cross-sectional axis about which ϕ y is defined. In this case index y has nothing to do with yielding.
- 16.
In this section M A and θ A denote the moment and the chord rotation from the analysis at member end node A. Index y signifies the cross-sectional axis about which M y A and θ y A are defined and has nothing to do with yielding.
- 17.
Note that the distribution of inelasticity along the member changes during the response. After plastic hinging, further flexural deformations take place mainly in the vicinity of the yielding end(s), spreading thereafter over the rest of the length with further loading. So, an interpolation function matrix B m(x) which is invariant during the response is against this physical reality.
- 18.
It is also very uncertain and difficult to model, as it very much depends on the width of the slab which is effective as a flange of the beam.
- 19.
Unlike the two-component element, which turns into a fully elastic model if p = 1, the one-component element can reproduce elastic overall behaviour only through very large values of the yield moments, M y A, M y B, at which the point hinges at the ends A and B are activated. As the rotational springs are in series with the elastic element in-between, setting p A = p B = 1 for them just increases the overall flexibility of the element (doubles it for skew-symmetric bending).
- 20.
In a beam indirectly supported on another beam at one end, plastic hinging can take place only at the other end and the beam’s shear span may be taken equal to the beam full clear span. In girders connected at intermediate points with cross-beams or girders, plastic hinging will develop only at the girder’s connection with vertical members. Then the shear span is determined on the basis of the girder clear span between columns into which the girder frames. Although the parts of a girder between joints with cross-beams may be modelled as individual beam elements, their effective elastic stiffness and hardening ratios should be taken the same, as established from the clear span of the overall girder.
- 21.
Recall, in this connection, that the ultimate deformation is conventionally identified with a drop in peak force resistance after ultimate strength equal to 20% of the ultimate strength value.
- 22.
By the same token, the tensile strength of concrete should be neglected in Fibre models.
- 23.
Residual deformations are very much affected by the details of the hysteresis rules. However, their estimation is even more influenced by the details of the ground motion. So, if estimation of residual deformations is indeed of interest, current rules about the minimum number of input motions and their conformity to 5%-damped elastic response spectrum should be revisited.
- 24.
Except in Saiidi and Sozen (1979), where this branch always heads towards the point on the primary loading at the maximum deformation ever reached in any of the two directions, even when this is first-time loading or real reloading.
- 25.
The pinching parameters of the model in Roufaiel and Meyer (1987) depend indeed on the shear span ratio, to reflect the more pronounced pinching of squat members.
- 26.
As a matter of fact, the universal value of 5% damping associated in codes with elastic response spectra is just a compromise between the lower values acknowledged for prestressed concrete and structural steel with bolted or welded connections on one hand and the higher ones for cracked concrete members.
- 27.
As there is coupling between the infill and the frame, these parameters cannot be given as if the infill panel were a stand-alone component, but depend in principle on the properties and sizes of the surrounding frame members.
- 28.
Recall that, even in linear analysis, the Elastic Modulus of concrete is derived from the best-estimate (mean) value of concrete strength, \( f_{\rm cm}\), and not from the nominal one, \( f_{\rm ck}\) (CEN 2004b).
- 29.
One of these bidrectional motions was used in all PsD tests of the SPEAR building and in the analyses of Sect. 4.10.5.2, but scaled to a different PGA.
- 30.
Except in the calculation of P-Δ effects, for which US codes use the displacements from the linear analysis with the 5%-damped elastic spectrum divided by R, without removing at all the effect of R (see Section 4.9.7).
- 31.
As a matter of fact, the value of this factor may be determined nationally within the range 2 to 3, as a Nationally Determined Parameter. 2.5 is the value recommended in Eurocode 8 (CEN 2005a).
- 32.
According to (ASCE 2007), cyclic degradation of strength and stiffness shoud be accounted for in the model of “secondary members” for a nonlinear analysis, while it may be disregarded for “primary members”.
- 33.
Note that, unless they are retrofitted, even the “primary members” of an existing substandard structure have larger cyclic degradation of strength and stiffness than a new member well designed and detailed for ductility.
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Fardis, M.N. (2009). Analysis and Modelling for Seismic Design or Assessment of Concrete Buildings. In: Seismic Design, Assessment and Retrofitting of Concrete Buildings. Geotechnical, Geological, and Earthquake Engineering, vol 8. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9842-0_4
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