Similarly for displacements, by analogy with Equation (3.14) M,. d '4k y-(max) = —-S .©...—'—r (3.28)
To find the horizontal force on mass j in mode i we simply multiply the acceleration by the mass:
and the total horizontal force on the structure (usually called the base shear) in mode i is found by summing all the storey forces to give:
The ratio L 2/M. is known as the effective modal mass. It can be thought of as the amount of mass participating in the structural response in a particular ode. If e su this quantity for all odes of vibration, the result is equal to the total ass of the structure.
o obtain the overall response of the structure, in theory e need to apply Equations (3.27) to (3.30) to each mode of vibration and then combine the results. Since there are as many modes as there are degrees of freedom, this could be an extremely long-winded process. In practice, however, the scaling factors L/M. and Lf/M. are small for the higher modes of vibration. It is therefore normally sufficient to consider only a subset of the modes. EC8 offers a variety of ays of assessing ho any odes need to be included in the response analysis. The normal approach is either to include sufficient modes that the sum of their effective modal masses is at least 90 per cent of the total structural mass, or to include all modes with an effective modal mass greater than 5 per cent of the total mass. If these conditions are difficult to satisfy, a permissible alternative is that the number of modes should be at least 3Vn where n is the number of storeys, and should include all modes with periods below 0.2 s.
Another potential problem is the combination of modal responses. Equations (3.27) to (3.30) give only the peak values in each mode, and it is unlikely that these peaks will all occur at the same point in time. Simple combination rules are used to give an estimate of the total response. Two methods are permitted by EC8. If the difference in natural period between any two modes is at least 10 per cent of the longer period, then the modes can be regarded as independent. In this case, the simple SRSS method can be used, in hich the peak overall response is taken as the Square oot of the Su of the Squares of the peak odal responses. If the independence condition is not met, then the SRSS approach may be non-conservative and a more sophisticated combination rule should be used. The most widely accepted alternative is the Complete Quadratic Combination, or CQC method ^ilson et al, 1981), which is based on calculating a correlation coefficient between two modes. Although it is more mathematically complex, the additional effort associated ith using this ore general and reliable method is likely to be minimal, since it is built into many dynamic analysis computer programs.
In conclusion, the ain steps of the ode superposition procedure can be summarised as follows:
a. Perform free vibration analysis to find natural periods, T, and corresponding mode shapes, 9 . Estimate damping ratio £.
b. ecide ho any odes need to be included in the analysis.
• compute the modal properties L and M, from Equation (3.25) and (3.26);
• read the spectral acceleration from the design spectrum;
• compute the desired response parameters using Equations (3.27) to (3.30).
d. Combine modal contributions to give estimates of total response. 3.3.7 Equivalent static analysis of MDOF systems
A logical extension of the process of including only a subset of the vibrational modes in the response calculation is that, in some cases, it may be possible to approximate the dynamic behaviour by considering only a single mode. It can be seen from Equation (3.29) that, for a single mode of vibration, the force at level j is proportional to the product of mass and mode shape at level j, the other terms being modal parameters that do not vary with position.
If the structure can reasonably be assumed to be dominated by a single (normally the fundamental) mode then a simple static analysis procedure can be used that involves only minimal consideration of the dynamic behaviour. For many years this approach has been a mainstay of earthquake design codes. In EC 8 the procedure is as follows.
Estimate the period of the fundamental mode, T1 - usually by some simplified approximate method rather than a detailed dynamic analysis (e.g. Equation (3.19)). It is then possible to check whether equivalent static analysis is permitted - this requires that T1 < 4TC where TC is the period at the end of the constant-acceleration part of the design response spectrum. The building must also satisfy the EC8 regularity criteria. If these two conditions are not met, the multi-modal response spectrum method outlined above ust be used.
For the calculated structural period, the spectral acceleration S can be obtained from the design response spectrum. The base shear is then calculated as:
where m is the total mass. This is analogous to Equation (3.30), with the ratio L2/M. replaced by Am. A takes the value 0.85 for buildings of more than two storeys with T1 < 2TC, and is 1.0 otherwise. The total horizontal load is then distributed over the height of the building in proportion to (ass ode shape). Normally this is done by making some simple assumption about the mode shape. For instance, for simple, regular buildings EC8 permits the assumption that the first mode shape is a straight line (i.e. displacement is directly proportional to height). This leads to a storey force at level k given by:
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