Seismic design buildings eurocode 8 pdf

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Wolfgang Elliot B. Johnston Carol Ritberger Ph. D John Maloof Dean A. Three regimes of structural response can be seen: a. When the ground motion period and natural period are similar, resonance occurs and there is a large dynamic ampliication of the motion.

In this region, the stiffness and inertia forces at any time are approximately equal and opposite, so that the main resistance to motion is provided by the damping of the system. Structural analysis 47 a 2 Downloaded By: If the ground motion is much faster than the natural oscillations of the structure, then the mass undergoes less motion than the ground, with the spring and damper acting as vibration absorbers.

Here the peak absolute displacement of the struc- ture X normalised by the peak ground displacement Xg is plotted against the ratio of the natural period Tn to the period of the sinusoidal loading T. The peak displacement at resonance is thus very sensitive to damping and is ininite for the theoretical case of zero damping.

For a more realistic damping ratio of 0. This illustrates the key principles of dynamic response, but it is worth noting here that the dynamic ampliications observed under real earthquake loading are rather lower than those discussed above, both because an earthquake time history is not a simple sinusoid, and because it has a inite usually quite short duration.

An earthquake can be measured and represented as the variation of ground acceleration with time in three orthogonal directions N-S, E-W and vertical. An exam- ple, recorded during the El Centro earthquake in California, is shown in Figure 3. Obviously, the exact nature of an earthquake time history is unknown in advance, will be different for every earthquake, and indeed will vary over the affected region due to factors such as local ground conditions, epicentral distance, etc.

Structural analysis 49 0. Although the method of evaluation is rather complex, the behaviour under a general dynamic load can be quite easily understood by comparison with the single-frequency, sinusoidal load case discussed in Section 3. In that case, we saw that large dynamic ampliications occur if the loading period is close to the natural period of the structure.

Irregular dynamic loading can be thought of as having many different components at dif- ferent periods. The structure will tend to pick up and amplify those components close to its own natural period just as it would with a simple sinusoid. The response will there- fore be dominated by vibration at or close to the natural period of the structure.

However, because the loading does not have constant amplitude and is likely to have only inite dura- tion, the ampliications achieved are likely to be much smaller than for the sinusoidal case. An example is shown in Figure 3.

The earthquake contains a wide band of frequency components, but it can be seen that the 0. The time-domain response of numerous SDOF systems having different natural periods is computed, and the maximum absolute displacement or accelera- tion, or velocity achieved is plotted as a function of the SDOF system period.

If desired, a range of curves can be plotted for SDOF systems having different damping ratios. So the response spectrum shows the peak response of a SDOF structure to a particular earthquake, as a function of the natural period and damping ratio of the structure.

For example, Figure 3. A key advantage of the response spectrum approach is that earthquakes that look quite different when represented in the time domain may actually contain similar frequency con- tents, and so result in broadly similar response spectra. This makes the response spectrum a useful design tool for dealing with a future earthquake whose precise nature is unknown. To create a design spectrum, it is normal to compute spectra for several different earthquakes, then envelope and smooth them, resulting in a single curve, which encapsulates the dynamic characteristics of a large number of possible earthquake accelerograms.

Within each category, spectra are given for ive different soil types: A — rock; B — very dense sand or gravel, or very stiff clay; C — dense sand or gravel, or stiff clay; D — loose-to-medium cohesionless soil, or soft-to-irm cohesive soil; E — soil proiles with a surface layer of alluvium of thickness 5 to 20 m.

The vertical axis is the peak, or spectral acceleration of the elastic structure, denoted by Se, normalised by ag, the design peak ground acceleration on type A ground. See EC8 Cl. 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 ampliication of the ground motion, though only by a factor of 2.

In the region of the spectra between 1 0. Structural analysis 51 a 5 E Downloaded By: T B and TC the spectra acceleration is constant with period. It can be seen that in the high seismicity events type 1 spectra , the spectral ampliica- tions 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 ampliication of the bedrock motions, and affect the period range over which ampliica- tion occurs. The EC8 values for T D have caused some controversy — it has been argued that the constant velocity region of the spectra should continue to higher periods, which would result in a more onerous spectral acceleration for long-period e.

The peak spectral acceleration Se experienced by the mass can then be read directly from the response spectrum. The peak force is then just equal to the inertia force experienced by the mass: Downloaded By: It should be remembered that the spectral acceleration is absolute i.

While elastic spectra are useful tools for design and assessment, they do not account for the inelasticity which will occur during severe earthquakes. In practice, energy absorption and plastic redistribution can be used to reduce the design forces signiicantly. This is dealt with in EC8 by the modiication of the elastic spectra to give design spectra Sd , as described in Section 3. Structures with distrib- uted mass and stiffness may undergo signiicant 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, for example, Hitchings or Petyt 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. This results in a lumped mass matrix, which contains only diagonal terms.

To get a suficiently detailed description of how the mass is distributed, it may be necessary to divide the struc- ture into smaller elements than would be required for a static analysis. Alternatively, many inite element programs give the option of using a consistent mass matrix, which allows a more accurate representation of the mass distribution without the need for substantial mesh reinement.

A consistent mass matrix includes off-diagonal terms. Structural analysis 53 In practice, c is very dificult to deine accurately and is not usually formulated explicitly.

Instead, damping is incorporated in a simpliied form. We shall see how this is done later. Differentiating and substituting into Equation 3. Thus an N-DOF system is able to vibrate in N different modes, each hav- ing a distinct deformed shape and each occurring at a particular natural frequency or period. The modes of vibration are system properties, independent of the external loading.

Often approximate formulae are used for estimating the fundamental natural period of multi-storey buildings. EC8 recommends the following formulae. For shear-wall type buildings: Downloaded By: The modal displacements are functions only of time, while the mode shapes are functions only of posi- tion.

Because of the orthogonality properties of the modes, it turns out that M, C and K are all diagonal matrices, so that the N equations in 3.

Each line of Equation 3. Li is an earthquake excitation factor, repre- senting the extent to which the earthquake tends to excite response in mode i. Mi is called Downloaded By: Note that Equation 3. While Equation 3. Since the system is linear, the structural response will be scaled by the same amount. It can be thought of as the amount of mass participating in the structural response in a particular mode.

If we sum this quantity for all modes of vibration, the result is equal to the total mass of the structure. To obtain the overall response of the structure, in theory we need to apply Equations 3. Since there are as many modes as there are degrees of freedom, this could be an extremely long-winded process. It is therefore normally suficient to consider only a subset of the modes.

EC8 offers a variety of ways of assessing how many modes need to be included in the response analysis. Another potential problem is the combination of modal responses. Equations 3. Simple combination rules are used to give an estimate of the total response. Two methods are permitted by EC8. In this case, the simple SRSS method can be used, in which the peak overall response is taken as the square root of the sum of the squares of the peak modal 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 alter- native is the complete quadratic combination CQC method Wilson et al. Although it is more mathematically complex, the additional effort associated with using this more general and reliable method is likely to be minimal, since it is built into many dynamic analysis computer programs.

In conclusion, the main steps of the mode superposition procedure can be summarised as follows: 1. Decide how many modes need to be included in the analysis. For each mode Compute the modal properties Li and Mi from Equations 3. Combine modal contributions to give estimates of total response. It can be seen from Equation 3.

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, which involves only minimal consideration of the dynamic behaviour.

For many years, this approach has been the mainstay of earthquake design codes. In EC8, the procedure is as follows. Estimate the period of the fundamental mode T1 — usually by some simpliied approxi- mate method rather than a detailed dynamic analysis e. The building must also satisfy the EC8 regularity criteria. If these two conditions are not met, the multi-modal response spectrum method outlined above must be used.

For the calculated structural period, the spectral acceleration Se can be obtained from the design response spectrum.

This is analogous to Equation 3. The total horizontal load is then distributed over the Downloaded By: 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 irst mode shape is a straight line i. Finally, the member forces and deformations can be calcu- lated by static analysis. A more economical design can be achieved by accepting some level of damage short of complete collapse, and making use of the ductility of the structure to reduce the force demands to acceptable levels.

Ductility is deined as the ability of a structure or member to withstand large deforma- tions beyond its yield point often over many cycles without fracture.