4.5.3.1. Modal analysis and its results Clause 4.3.3.3 Unlike linear static analysis, designers may not be so familiar with linear dynamic analysis of the modal response spectrum type. Moreover, some commercial computer programs with modal response spectrum analysis capability may not perform such an analysis in accordance with the relevant requirements of Eurocode 8. For instance, along the line of other seismic design codes (e.g. some US codes), a program may use the modal response spectrum method just to estimate peak inertia forces at storey levels, and to then apply these forces as static forces and calculate the static response to them as in the lateral force method. For these reasons, an overview is given below of how modal response spectrum analysis should be performed to fulfil the letter and spirit of EN 1998-1.
The first step in a modal response spectrum analysis is the determination of the 3D modal shapes and natural frequencies of vibration (eigenmodes and eigenvalues). Today, this task can be performed very reliably and efficiently by many computer programs dedicated to seismic response analysis for the purposes of earthquake-resistant design.
Even when the building qualifies for two separate 2D analyses in two orthogonal horizontal directions, X and Y, it is preferable to do the modal response spectrum analysis on a full 3D structural model. Then, each mode shape, represented by vector <&n for mode n, will in general have displacement and rotation components in all three directions, X, Y and Z. In other words, vector <i>„ will in general include all degrees of freedom of the structural model (unless the solution of the eigenvalue problem has been based on a few degrees of freedom, with the rest condensed out, statically or dynamically - see below).
If the origin of the global coordinate system, X-Y-Z, is far from the masses of the structure, the accuracy of an eigenmode-eigenvalue analysis in 3D might be adversely affected. Although most widely used computer programs take this into account, the designer should ideally choose the origin of the axes to be inside the volume of the structure.
The outcome of the eigenmode-eigenvalue analysis necessary for the subsequent estimation of the peak elastic response on the basis of the response spectra in the three directions, X, Y and Z, comprises for each normal mode, n:
• The natural circular frequency, wn, and the corresponding natural period, Tn = 2ix/u>n. ' The mode shape, represented by vector
• The modal participation factors rXn, FYn, FZn in response to the component of the seismic action in directionX, For Z, computed as rY„ =
i where i denotes the nodes of the structure associated with dynamic degrees of freedom, M is the mass matrix, Ix is a vector with elements equal to 1 for the translational degrees of degrees of freedom parallel to direction X and with all other elements equal to 0, cpXj n is the element of corresponding to the translational degree of freedom of node i parallel to direction X and mxi is the associated element of the mass matrix (similarly for <pn „, tpZii ,„ mYi and mZi). If M contains rotational mass moments of inertia, lm, ImY, Im, the associated terms also appear in the sum of the denominator. rYn, rZn are defined similarly.
The effective modal masses in directions X, Y and Z, MXn, MYn, and MZn, respectively, computed as
i and similarly for MYn, MZn These are essentially base-shear-effective modal masses, because the reaction force (base shear) in direction X, YorZ due to mode n are equal to Fbx, n = (,Tn)MXn, FbYi „ = 5a(r„)My„ and FbZ „ = S,(Tn)MZn, respectively. The sum of the effective modal masses mX, Yor Z over all modes of the structure (not just the N modes taken into account) is equal to the total mass of the structure.
The first objective of a modal response spectrum analysis is to determine the peak value of any seismic action effect of interest (be it a global effect, such as the base shear, or local ones, such as member internal forces, or even intermediate ones, such as interstorey drifts) in every one of the N modes considered due to the seismic action component in direction X, Y or Z. This may be accomplished through different approaches in different computer programs. A simple and efficient approach is the following:
® For each normal mode n, the spectral displacement, SdX(Tn), is calculated from the design (pseudo-)acceleration spectrum of the seismic action component of interest, say X, as (TJ2^2SJTn).
• The nodal displacement vector of the structure in mode n due to the seismic action component of interest, say in direction X, U^, is computed as the product of the spectral displacement, S^TJ, the participation factor of mode n to the response to the seismic action component of interest, TXn for the component in direction^, and the eigenvector, i>„, of the mode: UXn = (TJ2-K)2SJTn)r^n.
' Peak modal values of the effects of the seismic action component of interest are computed from the modal displacement vector determined in the previous step: deformations of members (e.g. chord rotations) or of storeys (e.g. interstorey drifts) are computed directly from the nodal displacement vector of the mode n; member modal end forces are computed by multiplying the member modal deformations (e.g. chord rotations) by the member stiffness matrix, as in the back-substitution phase of the solution of a static analysis problem; modal storey shears or overturning moments, etc., are determined from modal member shears, moments, axial forces, etc., through equilibrium, etc.
The peak modal responses obtained as above are exact. However, they can only be combined approximately, as they occur at different instances of the response. Appropriate rules for the combination of peak modal responses are described in Section 4.5.3.3. Rules for taking into account, at different levels of approximation, the simultaneous occurrence of the seismic action components are given in Section 4.9.
For buildings with horizontal slabs considered to act as rigid diaphragms, and provided that the vertical component of the seismic action is not of interest or importance for the design, static and dynamic condensation techniques are sometimes applied to reduce the number of static degrees of freedom to just three dynamic degrees of freedom per floor (two horizontal translations and one rotation about the vertical axis). Dynamic condensation profits from the small inertia forces normally associated with vertical translations and nodal rotations about the horizontal axis due to the horizontal components of the seismic action. The reduced dynamic model in 3D has just 3nsl normal modes, where nst is the number of storeys. For each normal mode n, the response spectrum is entered with the natural period Tn of the mode, to determine the corresponding spectral acceleration SJTJ. Then, for each one of the two horizontal components of the seismic action, two horizontal forces and one torque component with respect to the vertical axis are computed for normal mode n and at each floor level i: FXl „, FYi n and Mi n, where the indexes X and Y now denote the direction of the two forces and not that of the seismic action component (which may be either X or Y). These forces and moments are computed as the product of:
• the participation factor of the normal mode n to the response to the seismic action component of interest, say rXn for the seismic action component in direction X
• the mass associated with the corresponding floor degrees of freedom - floor mass mXl = mYi and floor rotational mass moment of inertia, la
• the corresponding component of the modal eigenvector, <pXi n, (p,n n, ip0i n
For each mode n and separately for the two horizontal components of the seismic action, a static analysis of the full static model in 3D of the structure is then performed, under static forces and moments FXi n, FYi n and Mi>n, applied to the corresponding dynamic degrees of freedom of each floor i. Peak modal response quantities, like nodal displacements, member internal forces, member deformations (chord rotations) or interstorey drifts, etc., are computed separately for each mode and combined for all modes according to the rules in Section 4.5.3.3 for each horizontal component Xor Yof the seismic action.
The approach of the previous paragraph is not feasible for structures without rigid diaphragms at storey levels, and cannot be used when the vertical component of the seismic action is of interest. Moreover, with today's hardware the savings in computer time and memory are not worth the complexity in analysis software for the reduction of the static degrees of freedom into a much smaller number of dynamic degrees of freedom at floor levels.
In closing this relatively long account of modal analysis, it is noted that modal participation factors and effective modal masses are more than mathematical quantities internally used in the procedure: they convey a certain physical meaning, which is essential for the understanding of the nature and relative importance of each mode. For instance, the relative magnitude of the modal participation factors or of the effective modal masses determines the predominant direction of the mode: the inclination of this direction to the horizontal direction X is equal to rXn/rYn, etc.; the predominant direction of the mode with the largest modal base shear, along with the orthogonal direction, is a good choice for the often ill-defined 'principal' or 'main' directions of the structure in plan, along which the horizontal components of the seismic action should be taken to act. Unfortunately, the presence of torsion in a mode cannot be appreciated on the basis of modal participation factors and effective modal masses defined along the three directions, X, Y and Z: participation factors and modal masses for rotation about these axes would be necessary for that purpose, and such quantities are normally not reported in the output of computer programs. The importance of torsion in a mode may be judged, instead, on the basis of the modal reaction forces and moments. Last but not least, irrespective of the qualitative criteria for regularity in plan, a good measure of such regularity is the lack of significant rotation about the vertical axis (and global reaction torque with respect to that axis) in the (few) lower modes.
4.5.3.2. Minimum number of modes to be taken into account Clauses All modes of vibration that contribute significantly to the response quantities of interest
4.3.3.3.1 (2), should normally be taken into account. However, as the number of modes to be considered 4.3.3.3.1(3), should be specified as input to the eigenvalue analysis, a generally applicable and simple 4.3.3.3.1 (4) criterion should be adopted. Such a criterion can only be based on global response quantities.
The most commonly used criterion, adopted by Eurocode 8, requires that the N modes taken into account provide together a total effective modal mass along any one of the seismic action components,^, Y ox even Z, considered in design, of at least equal to 90% of the total mass of the structure.
As an alternative, in case the criterion above turns out to be difficult to satisfy, the eigenvalue analysis should take into account all modes with effective modal mass along any individual seismic action component, X, Yor Z, considered in design, of greater than 5% of the total. It is obvious, though, that this criterion is hard to apply, as it refers to modes that have not been captured so far by the eigenvalue analysis. Clause As a third alternative for very difficult cases (e.g. in buildings with a significant contribution
4.3.3.3.1(5) from torsional modes, or when the seismic action components in the vertical direction, Z, should be considered in the design), the minimum number N of modes to be taken into account should be at least equal to 3\'«sl (where nst is the number of storeys above the foundation or the top of a rigid basement) and should be such that the shortest natural period captured does not exceed 0.2 s. It is clear from the wording of the code that recourse to the third alternative can be made only if it is demonstrated that it is not feasible to meet any of the two criteria above.
The most commonly used criterion, requiring a sum of effective modal masses along each individual seismic action component, X, Y or Z, considered in design, of at least 90% of the total mass, addresses only the magnitude of the base shear captured by the modes taken into account, and even that only partly: modal shears are equal to the product of the effective modal mass and the spectral acceleration at the natural period of the mode; so, if the fundamental period is well down the tail of the (pseudo-)acceleration spectrum and higher mode periods are in the constant (pseudo-)acceleration plateau, the effective modal mass alone underplays the importance of higher modes for the base shear. Other global response quantities, such as the overturning moment at the base and the top displacement, are even less sensitive to the number of modes than the base shear. However, estimation of global response quantities is less sensitive to the number of modes considered than that of local measures, such as the interstorey drift, the shear at an upper storey, or the member internal forces. As important steps of the seismic design process, such as member dimensioning for the ultimate limit state are based on seismic action effects from the analysis at the local (i.e. member) level, the modes considered in the eigenvalue analysis should preferably account for much more than 90% of the total mass (close to 100%), to approximate with sufficient accuracy the peak dynamic response at that level.
There exist techniques to approximately account for the missing mass due to truncation of higher modes (e.g. by adding static response). Unlike some other codes, including EN 1998-2 (on bridges), EN 1998-1 does not require such measures.
Within the response spectrum method of analysis, the elastic responses to two different Clause vibration modes are often taken as independent of each other. The magnitude of the 4.3.3.3.2(1) correlation between modes i and j is expressed through the correlation coefficient of these two modes, r^:42,43
where p = TJTj, and and are the viscous damping ratios assigned to modes i and j, respectively. If two vibration modes have closely spaced natural periods (i.e. if p is close to unity), the value of the correlation coefficient is also close to unity, and the responses in these two modes cannot be taken as independent of each other. For buildings, EN 1998-1 considers that two modes i and j cannot be taken as being independent of each other if the ratio of the minimum to the maximum of their periods, p, is between 0.9 and 1/0.9; for the two extreme values of this range of p and = § = 0.05, equation (D4.8) gives rtj = 0.47. (EN 1998-2 on bridges is more restrictive, considering that two modes i and j are not independent if the value of the ratio p of their periods is between 1 + 10(£,-£-)1/2 and 0.1/[0.1 + (£4)1/2]; for = = 0.05 and p equal to these limit values, equation (D4.8) gives r,j = 0.05.) It is noted that in buildings with similar structural configuration and earthquake resistance in two horizontal directions, X and Y, pairs of natural modes with very similar natural periods at about 90° in plan (often not in the two horizontal directions, X and Y) are quite common; the two modes in each pair are not independent but closely correlated.
If all relevant modal responses may be regarded as independent of each other, then the Clause most likely maximum value Ee of a seismic action effect may be taken equal to the square 4.3.3.3.2(2) root of the sum of squares of the modal responses (SRSS rule44):
where the summation extends over the TV modes taken into account and Eu is the peak value of this seismic action effect due to vibration mode i.
If the response in any two vibration modes i and; cannot be taken as independent of each other, Eurocode 8 requires that more accurate procedures for the combination of modal maximum responses are used, giving the complete quadratic combination (CQC rule42) as an
Clause
Clauses
example. According to this rule, the most likely maximum value EB of a seismic action effect may be taken as equal to where rtj is the correlation coefficient of modes i and j given by equation (D4.8) and EZi and EEj are the peak values of the seismic action effect due to vibration modes i orrespectively. Comparison with the results of response-history analyses has demonstrated the accuracy of the CQC rule, in cases where the SRSS rule has been found to be unconservative due to mode correlation.
The SRSS rule, equation (D4.9), is a special case of equation (D4.10) for rtJ = 0 if i^j (obviously ri} = 1 for i = ;'). As in computer programs with capabilities of eigenvalue and response spectrum analysis the additional complexity of equation (D4.10) is not an issue, there is no reason to implement in such a program the simpler equation (D4.9) instead of the more general and always accurate and acceptable one, equation (D4.10).
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