Seismic design by the response spectrum method: A new interpretation of elliptical response envelopes and a novel equivalent static method based on probable linear combinations of modes

• Proposal of the notion of ellipsoid of the coefficients of linear combinations of modes (α-ellipsoid).
• Proposal of a new interpretation of the so-called “elliptical response envelopes”.
• Proposal of a variant of the classical approach for the “elliptical response envelope” discretization.
• Proposal of a new “equivalent static method”, taking into account all the structural modes.
• Case study: design of a nuclear building using the seismic analysis methods described in the article.

At the design calculation phase, the seismic analysis of a nuclear building is usually based on the assumption of structural linearity. Moreover, design codes impose to consider three-component ground motions. In this context, the response spectrum method is likely the most used seismic design approach. Different variants of the response spectrum method exist, the main differences among them being related to the way of superposing the contributions of the three ground motion components and to the treatment of the simultaneity of physically distinct responses. A set of nr ≥ 2 simultaneous values of distinct responses can be called response vector. The basic variants of the response spectrum method provide the peak value of each distinct response, allowing the definition of a parallelepiped envelope for the response vectors: this envelope is conservative since the peaks of distinct responses, in general, do not occur simultaneously. Conversely, the so-called elliptical response envelopes, which are hyper-ellipsoids of dimension nr, correctly take into account response simultaneity. In the first part of this paper, a new interpretation of the response envelopes is presented, based on the notion of probable linear combinations of modes. The practical use of response envelopes is not straightforward when nr > 3, like e.g. in the case of shell finite elements with three membrane forces and three moments. A standard procedure consists in approximating the response envelope by an enveloping polyhedron, whose vertices define a set of response vectors. The definition of a refined polyhedron, closer to the surface of the response envelope, is proposed in the second part of the paper.Another commonly used approach for the seismic structural design is the so-called equivalent static method, where the seismic action is represented by a static force field applied to the structure. The main difficulty in the practical application of this method is the definition of a static force field taking into account all the ground motion components and the contribution of all the structural modes. A novel procedure for the definition of equivalent static loads is described in the third part of this article, based on the use of pseudo-inertia forces and on the notion of probable linear combinations of modes.In the last part of the paper, both the response envelope approach and the proposed procedure for the equivalent static load definition are applied to the seismic design of a representative reinforced concrete nuclear building. The results are compared with those obtained by several classical variants of the response spectrum method.