denoted by Se, normalised by a , the design peak ground acceleration on type A ground. The spectra are plotted for an assumed structural damping ratio of 5 per cent. See EC8 Cl. 184.108.40.206 for mathematical definitions of these curves and EC8 Table 3.1 for fuller descriptions of ground types A-E.
As with the harmonic load case, there are three regimes of response. Very stiff, short period structures simply move with the ground. At intermediate periods there is dynamic amplification of the ground motion, though only by a factor of 2.5-3, and at long periods the structure moves less than the ground beneath it. In the region of the spectra between TB and TC the spectral acceleration is constant with period. The region bettveen TC and Td represents constant velocity and beyond TD is the constant displacement region.
It can be seen that in the high seismicity events (Type 1 spectra) the spectral amplifications tend to occur at longer periods, and over a wider period range, than in the moderate seismicity events. It is also noticeable that the different soil types give rise to varying levels of amplification of the bedrock motions, and affect the period range over which amplification occurs. The EC 8 values for TD have caused some controversy - it has been argued that the constant velocity region of the spectra should continue to higher periods, hich ould result in a ore onerous spectral acceleration for long-period (e.g. very tall) structures.
In a response spectrum analysis of an SDOF system, we generally wish to determine the force to which the structure is subjected, and its maximum displacement. We start by estimating the natural period, T, and damping ratio, X, The peak (spectral) acceleration, S , experienced by the mass can then be read directly from the response spectrum. Now the maximum acceleration in a vibrating system occurs when it is at its point of extreme displacement, at which instant the velocity (and therefore the damping force) is zero. The peak force is then just equal to the inertia force experienced by the mass:
This must be in dynamic equilibrium with the stiffness force developed within the structure. If we define the spectral displacement, SD, as the peak absolute displacement corresponding to the spectral acceleration, S , then we must have kSD = mSe, which, using the relationships between mass, stiffness and natural period given in Equation (3.8), leads to:
ote that, hile the force experienced depends on the ass, the spectral acceleration and displacement do not - they are functions only of the natural period and damping ratio.
It should be remembered that the spectral acceleration is absolute (i.e. it is the acceleration of the ass relative to the ground plus the ground acceleration, hence proportional to the inertia force experienced by the mass), but the spectral displacement is the displacement of the mass relative to the ground (and hence proportional to the spring force).
While elastic spectra are useful tools for design and assessment, they do not account for the inelasticity that ill occur during severe earthquakes. In practice, energy absorption and plastic redistribution can be used to reduce the design forces significantly. This is dealt with in EC8 by the modification of the elastic spectra to give design spectra Sd, as described in Section 3.4.2.
Not all structures can be realistically modelled as SDOF systems. Structures with distributed mass and stiffness may undergo significant deformations in several modes of vibration and therefore need to be analysed as multi-degree-of-freedom (MDOF) systems. These are not generally amenable to hand solution and so computer methods are widely used - see e.g. Hitchings (1992) or Petyt (1998) for details.
For a system with N degrees of freedom it is possible to write a set of equations of motion in matrix form, exactly analogous to Equation (3.4):
where m, c and k are the mass, damping and stiffness matrices (dimensions N X N), y is the relative displacement vector and I is an N X 1 influence vector containing ones corresponding to the Fs in the direction of the earthquake load, and zeroes elsewhere. k is derived in the same way as for a static analysis and is a banded atrix.
m is most simply derived by dividing the mass of each element between its nodes. This results in a lumped mass matrix, which contains only diagonal terms. To get a sufficiently detailed description of how the mass is distributed, it may be necessary to divide the structure into smaller elements than would be required for a static analysis. Alternatively, many finite element programs give the option of using a consistent mass matrix, which allows a ore accurate representation of the ass distribution ithout the need for substantial mesh refinement. A consistent mass matrix includes off-diagonal terms.
In practice c is very difficult to define accurately and is not usually formulated explicitly. Instead, damping is incorporated in a simplified form. e shall see ho this is done later.
As with SDOF systems, before attempting to solve Equation (3.15) it is helpful to consider the free vibration problem. Because it has little effect on free vibrations, we also omit the damping term, leaving:
he solution to this equation has the for y = ç sin at
where 9 is the mode shape, which is a function solely of position within the structure. Differentiating and substituting into Equation (3.16) gives:
This can be solved to give N circular natural frequencies w1, w2 ...w. ... WN, each associated with a mode shape 9, . Thus an N-DOF system is able to vibrate in N different modes, each having a distinct deformed shape and each occurring at a particular natural frequency (or period). he odes of vibration are system properties, independent of the external loading. Figure 3.10 shows the way modes of vibration of a four-storey shear-type building (i.e. one with relatively stiff floors, so that lateral deformations are dominated by shearing deformation between floors), with the modes numbered in order of ascending natural frequency (or descending period).
Often approximate formulae are used for estimating the fundamental natural period of multi-storey buildings. EC8 recommends the following formulae. For multi-storey frame buildings:
where T1 is measured in seconds, the building height, H, is measured in metres and the constant, C, equals 0.085 for steel moment-resisting frames, 0.075 for concrete moment-resisting frames or steel eccentrically braced frames, and 0.05 for other types of frame. For shear-wall type buildings:
where A is the total effective area of shear walls in the bottom storey, in m2
3.3.6 Multi-modal response spectrum analysis
Having determined the natural frequencies and mode shapes of our system, we can go on to analyse the response to an applied load. Equation (3.15) is a set of N coupled equations in terms of the N degrees of freedom. This can be most easily solved using the principle of modal superposition, which states that any set of displacements can be expressed as a linear combination of the mode shapes:
y = Y * + y2 <p2 + Y3 <P3 + + yN <pN = 2 Y V, (3.21)
The coefficients Y are known as the generalised or modal displacements. The modal displacements are functions only of time, while the mode shapes are functions only of position. Equation (3.21) allows us to transform the equations of motion into a set of equations in terms of the modal displacements rather than the original degrees of freedom:
where Y is the vector of modal displacements, and M, C and K are the modal mass, stiffness and damping matrices. Because of the orthogonality properties of the odes, it turns out that M, C and K are all diagonal atrices, so that the N equations in Equation (3.22) are uncoupled, i.e. each mode acts as an SDOF system and is independent of the responses in all other modes. Each line of Equation (3.22) has the form:
or, by analogy with Equation (3.11) for an SDOF system:
Here the subscript i refers to the mode shape and j to the degrees of freedom in the structure. So f.. is the value of mode shape i at DOF j. L. is an earthquake excitation factor, representing the extent to hich the earthquake tends to excite response in mode i. M. is called the modal mass. The dimensionless factor LJM. is the ratio of the response of a MDOF
structure in a particular mode to that of an SDOF system with the same mass and period.
Note that Equation (3.24) allows us to define the damping in each mode simply by specifying a damping ratio, X, without having to define the original damping matrix c.
While Equation (3.24) could be solved explicitly to give Y as a function of time for each mode, it is more normal to use the response spectrum approach. For each mode we can read off the spectral acceleration, S ., corresponding to that mode's natural period and damping - this is the peak response of an SDOF system with period, T, to the ground acceleration, xg . For our MDOF syste , the ay e have broken it do n into separate odes has resulted in the ground acceleration being scaled by the factor L JM.. Since the system is linear, the structural response ill be scaled by the sa e a ount. So the acceleration amplitude in mode i is (LJM).Sei and the maximum acceleration of DOF j in mode i is:
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