06-12-2010, 10:36 AM

Every physical system vibrates (the so-called free vibration). This is due to the fact that the atoms and molecules that make up that system are always losing and or gaining energy as such always changing energy level-thus in motion. This vibration is not noticeable due to the natural damping that accompanies that system. As that system vibrates, it does so at particular frequencies –known as the natural frequencies. To locate these frequencies, we may have to resort to the technique of modal Analysis. For simple systems, simple modal analysis could yield the desired result, but if we have to contend with complex systems, finite element analysis will have to be performed in order to determine the dynamic response. The natural vibration of the system does not constitute any problem per say. Problems usually arise with the input of energy into the system (the forced vibration). This is usually verified when the energy input pulsates (non steady state) like is witness in the event of wind loading on structures. Like was said previously, one of the main problems that had to be confronted and resolved in the realization of this structure is that of wind loading. So how is it to be solved (assuming that we are making our own independent preliminary design or assessment of the structure)? We could adopt the criteria as defined in the codes that specified the minimum gust duration to be employed in the analysis. In the alternative, we decide on the reference period or the life span of the structure (Vn). We could also adopt the standard 50years return period for wind loading for building structures. Based on importance factor (If) and the use coefficient (Cu), we calculate the design period Vr (Vr = If * Cu *Vn ), then calculate the return period Tr (Tr = Vr/-In (1- pr), if we do not want to use the 50years return period or if it is not justified to use the 50years return period. Note that in the above equation for the calculation of the return period, Pr is the probability of exceeding the design load in the life span of the structure). With any of the approaches and the physical conditions of the site, we chose the basic wind speed for our design (say V = 40m/s). Then calculate the characteristic wind pressure Wk = 0.625S^2V^2; S depends on the exposure condition and the form of our structure. (particularly the plan area and height of our structure).Since we are building inside a town and our structure has its plan dimension greater than 50m and height greater than 50m (remember that it is in excess of 800m height), then our S is about 2 (we are using the British standard and the euro codes that did not include such a height (800m), as such we had to do some interpolation from the included height-which though may not represent the actual fact but which is enough for this stage of our design) for which our Wk = 0.625(2)^2(40)^2 = 4000 N/m^2. We approximate our structure to a triangular one and assume that wind attack face is on one of the sides. The height to breadth ratio is over 40 (as such we adopt the infinity ratio), so the total force coefficient is 2.1 for which the modified Wk = 2.1 x 4000 N/m^2 = 8400 N/m^2 or 8.4 KN/m^2. If we assume a rectangular face of 134m height on each set-back with varying widths (as the structure tappers from the bottom to the top) for each structural lift then, the surface area of each lift = 134m multiplied by 90% of the width of the lift directly underneath, except for the ground floor for which the full 100% width has to be used in the calculation (we assume that the structure is split into 6 lifts of 800/6 = 134m). (We will come back to this later).

The effect of the wind loading on the structure, apart from the fact that it causes torsional and bending stresses, is that it makes none regular force inputs into the structural system. These force inputs on the structure force it to deflect and to take up differing modal shapes. Since the forces are not at steady states but varying with time (due to none steady force input as already mentioned, vortex shading etc) the structure oscillates (vibrates). The maximum vibration is given by Dmax = P/[K^2(1-N^2/n^2)^2 + 4 pi^2N^2u^2]^0.5. K is the spring constant (if we assume that the masses of the structure are concentrated on the beams and that the beams are supported on columns built in at the ends, then K = 24EI/L^3), n is the natural frequency of the structure, N is the frequency of applied wind loading (P). The natural frequency of the structure is given by n =1/2pi (K/M)^0.5, M being the mass of the structure and u being the damping coefficient of the structure. An inspection of the vibration equation will show that if N =n, then the term n (the natural frequency of the structure) will disappear from the equation as such, apart from damping, the structure will be taken over by the wind-thus attains resonance. In such a situation, the structure vibrates wildly with the wind and the deflection tends to infinity (theoretically) except for damping. If this persists, a phenomenon known as “lock-in” could develop. In that case, the vibration is self propagating and eventually, the vibration will overcome damping and the structure will vibrate indefinitely and collapse. So, we have to demonstrate (through our calculation) that this condition will not be reached in the design by showing that the natural frequency of the structure is always below (in worst case, above the range of expected frequencies due to wind loading. One of the problems is that the critical velocity at which this resonance could occur may be below or above our chosen or calculated wind design speed as such we could be looking for that point outside the design wind speed range. If this is the case, then the above calculation could only serve for the bending and torsional stress analysis-thus, the base shear and bending designs; as such, we have to calculate the natural frequency of the structure and match it to the expected range of wind loading that could be imposed on the structure in its service life.

But how do we calculate the frequency of the wind loading at which our structure gets into trouble as to compare it with the natural frequency of the structure, given that in this case, we are not looking for it at a given point but over what could turn out to be a wide range?

Regards

Teddy