Linear and Nonlinear Analyses

General

In SeismoBuild all the analytical methods (both linear and nonlinear)that are proposed by the different Standards have been programmed, namely (i) the Linear Static Procedure LSP, (ii) the Linear Dynamic Procedure LDP,(ii) the Nonlinear Static Procedure NSP and (iv) the Nonlinear Dynamic Procedure NDP.

In General, the nonlinear methods are considered numerically more advanced and more accurate in the representation of the earthquake loading. They take explicitly into account the concentration of damage at the weakest locations of the building and the redistribution of forces upon the formation of plastic hinges, considering both material inelasticity and geometric nonlinearities. Furthermore, the nonlinear dynamic method (although more complicated in its application) is considered to be the most accurate method of analysis, since it manages to better represent the dynamic nature of seismic loading with respect to its static counterparts. Consequently, the nonlinear methods they are the ones that are mainly employed for the assessment and strengthening of existing reinforced concrete buildings.  

Linear Static Procedure

With the Linear Static Procedure (Lateral Force Method with the EC8 naming conventions) a triangular,lateral, pseudo-seismic force distribution that is assumed to approximate the earthquake loading is applied to a linear elastic structural model, in order to calculate the internal forces and the system displacements. These action effects are then compared against the members' capacities for the selected performance level, always in terms of forces, and, if the capacities are larger than the demands, the structure is considered safe.

The fundamental period of vibration of the building for lateral motion in the direction considered is calculated by eigenvalue analysis or with more approximate empirical methods, from which the ordinate of the response spectrum Sa is calculated. The total lateral force is proportional to the spectral acceleration Sa and the building weight W:

C1, C2, Cm and l are different easily calculated modification factors that are related to higher mode effects, and parameters, such as the expected maximum inelastic displacements,the effect of pinched hysteresis shapes, the stiffness and strength deterioration. This total force is then distributed at each floor level, according to the mass distribution of the building, and the modal shape of the fundamental mode (in EC8) or an inverted triangular distribution (in both ASCE-41 and EC8).

Because of its approximate nature,the linear static procedure is permitted only in cases of very regular,low-rise constructions that sustain limited damage and do not undergo large inelastic deformations. In particular:

  1. The demand to capacity ratios DCR should be small for all structural members. For the brittle failure types, they should be below unity.
  2. There should be no in-plane strength or stiffness discontinuities & irregularities.
  3. There should be no out-of-plane strength or stiffness discontinuity & irregularities.
  4. There should be no weak storey strength or stiffness irregularities.
  5. There should be no torsional strength or stiffness irregularities.
  6. The fundamental period should not be large.

Linear Dynamic Procedure

The Linear Dynamic Procedure (Modal Response Spectrum Analysis, according to the EC8 naming conventions) is similar to the LSP, at least as regards the modelling approach. The model is again elastic and there is no stiffness degradation during the analysis. However, the method is somehow more sophisticated, since the profile of the lateral forces is not arbitrary anymore, but rather it is calculated as a combination of the modal contributions of the different modes of vibration of the structure. The action effects of the structural members are again compared against the capacities for the selected performance level in terms of forces, and, if the capacities are larger than the demands, the structure is considered safe. The Linear Dynamic Procedure is based on the well-known response-spectrum analysis (RSA) [e.g. Rosenblueth, 1951; Chopra, 1995] and it is the method of analysis that is typically employed for the design of new structures.

The response-spectrum analysis is a pseudo-dynamic method, which is capable of providing the peak values of response quantities, such as forces and deformations, of a structure under seismic excitation with a series of static analyses, rather than time-history dynamic analysis. In this context, the time–acceleration history imposed to the supports of the structure is replaced by the equivalent static forces, which are distributed to the free DOFs of the structure and represent the contribution from each natural mode of vibration. These equivalent forces are derived for each mode of vibration separately as the product of two quantities: (i) the modal inertia force distribution (thus eigenvalue analysis is needed), and (ii) the pseudo-acceleration response per mode (obtained from the 5% damped response spectrum). For each mode of interest, a static analysis is conducted, and then every final peak response quantity is derived by the superposition of the quantities corresponding to the modes.

A sufficient number of modes has to be considered, so that to capture at least 90% of the participating mass of the building in each of two orthogonal principal horizontal directions of the building, thus neglecting only the less significant ways of vibrating in terms of participant mass. EC8 also requires that all modes with more than 5% of the participating mass in any direction should be considered.

Because the peaks in the responses of each mode generally occur at different time instants and rigorous time-history analysis has not be conducted, it is not possible to determine the exact peak values of the response quantities. Therefore, approximations need to be introduced by implementing one of the modal combination (statistical) rules, such as the absolute sum (ABSSUM), square-root-of-sum-of-squares (SRSS) and the complete quadratic combination (CQC). CQC is suggested when periods are closely spaced, with cross-correlation between the modal shapes. SRSS can be used when the periods differ by more than 10%, whilst ABSSUM offers a very safe, upper limit of response.

The same procedure is repeated for each desired seismic direction EX, EY and EZ by using different or the same response spectra. It is usually requested that two or three seismic loading directions (EX, EY, EZ) are to be considered simultaneously, together with the gravity static loads (G+Q) of the structure (the vertical component EZ is mandatory only for the elements, where the vertical vibration is considered critical, e.g. large cantilevers). The seismic loading directions may be combined linearly (E = ±EX±EY±EZ) with different factors fEX, fEY, fEZ per direction (usually fEX=fEY=fEZ=1.00 or 0.30) or by the SRSS rule (E = ± ). The gravity and live loads are defined and added algebraically. Because the seismic loads are taken into account with both signs for every direction, the results of RSA loading combinations in terms of any response quantity are presented as envelopes.

Contrary to the Linear Static Procedure, the Linear Dynamic Procedure is suitable for buildings with larger fundamental period, where higher-mode effects are important. Apart from this, all the recommendations and limitations described for the LSP apply for the LDP as well.

  1. The demand to capacity ratios DCR should be small for all structural members. For the brittle failure types, they should be below unity.
  2. There should be no in-plane strength or stiffness discontinuities & irregularities.
  3. There should be no out-of-plane strength or stiffness discontinuity & irregularities.
  4. There should be no weak storey strength or stiffness irregularities.
  5. There should be no torsional strength or stiffness irregularities.

Nonlinear Static Procedure

Conventional (non-adaptive) pushover analysis is employed in the estimation of the horizontal capacity of structures implying a dynamic response that is not significantly affected by the levels of deformation incurred (i.e. the shape of the horizontal load pattern, which aims at simulating dynamic response, can be assumed as constant).

The introduced vertical loads applied to the 3D model, in addition to the incremental loads, are equal to CgG+CqQ, where Cg and Cq are the permanent and live loads coefficients, respectively, defined in the Static Actions tab. It is noted that the snow load is also introduced when it is required, i.e. CgG+CqQ+CsS for ASCE 41-23 and TBDY. The self weight of the beam and column elements is automatically computed according to the materials’ specific weight and sections’ geometry. The slabs’ additional gravity and live loads are automatically introduced as beams’ additional mass. Nonlinear static analysis may be applied with two vertical distributions of loads:

(i) a “uniform pattern”, which attempts to simulate an inelastic response dominated by a soft-storey mechanism (development of plastic hinges at both top and bottom ends of all columns of a storey, in general the ground floor, which is subjected to highest lateral forces);

(ii) a “modal pattern”, proportional to the fundamental elastic translational mode shape.

The incremental loads may be applied in both positive and negative directions. Furthermore, the incremental loads applied in X and in Y direction, may be taken as acting simultaneously by employing both of the following combinations:

  1. ±Fx ± 0.30Fy
  2. ±0.30Fx ± Fy

With Fx and Fy representing the incremental loads applied in the X and Y direction of the structure, respectively.

Finally, in order to account for uncertainties in the location of masses and in the spatial variation of the seismic motion, the calculated centre of mass at each floor may be considered as being displaced from its nominal location in each direction by an accidental eccentricity equal to 5% of the floor-dimension perpendicular to the direction of the seismic action.

The applied incremental load P is kept proportional to the pattern of nominal loads (P°) defined by default by the program according to the Code requirements: P = λ(P°). The load factor λ is automatically increased by the program until a Code-defined limit, or numerical failure, is reached. For the incrementation of the loading factor, a displacement control strategy is employed, which refers to direct incrementation of the global displacement of the control node and the calculation of the loading factor that corresponds to this displacement.

Nonlinear Dynamic Procedure

The Nonlinear Dynamic Procedure constitutes a sophisticated approach for examining the inelastic demands produced on a structure by a specific suite of ground motion acceleration time-histories. Being the numerically more advanced method of analysis, it is the most accurate in the representation of the dynamic nature of seismic loading. As nonlinear dynamic analysis involves fewer assumptions than the nonlinear static procedure, it is subject to fewer limitations than nonlinear static procedure. It automatically accounts for higher-mode effects and shifts in inertial load patterns as structural softening occurs. In addition, it provides reliable results even for highly irregular structures, or with irregular seismic action (e.g. near-fault ground motion or loading in 2 or 3 directions simultaneously). As a result, the NDP is the only method that can be used for any structural configuration and any type of loading. In practice, we can analyse with adequate accuracy any structural configuration subjected to any type of seismic action.

Similarly to pushover analysis, the introduced vertical loads applied to the 3D model are equal to CgG+CqQ (or CgG+CqQ+CsS for ASCE 41-23 and TBDY). The coefficients Cg, Cq and Cs are the permanent, live and snow loads coefficients defined in the Static Actions tab. The self weight of the beam and column elements is automatically computed according to the materials’ specific weight and sections’ geometry. The slabs’ additional gravity and live loads are automatically introduced as beams’ additional mass.

Nonlinear dynamic analysis is performed by applying at the foundation of the building sets of acceleration time-histories. In SeismoBuild the ground motions consist of pairs of orthogonal horizontal ground motion components. Both components are artificial records compatible (for the selected seismic hazard level) with the given target spectrum or user-defined records. In EC8, NTC-18 and KANEPE, when 7 or more record pairs are specified, the average response should be considered; instead, when fewer records are considered, the most unfavourable value of the response quantity among the analyses should be used in the verification checks. Similarly, according to ASCE 41 and TBDY a suite of not less than 11 ground motions shall be selected for each target spectrum and the mean response is checked.