Deflections of Thin-Walled Storage Tanks with Roof due to ...

w n = (H/R) n 2 u n [1/(1+α I ratio )] (3)Where α is function of the wave number n, the thickness t and of the Poisson’s relationν, andI ratio is the ratio between the circumferential bending stiffness of a ring stiffener on the top andthe bending stiffness of the cylindrical shell. Jonaidi and Ansourian (1998) argued that errors inthe range of 10-18% are obtained from the use of this membrane model.More refined analysis have used finite element models for the shell and assumed harmonicsettlement (Jonaidi and Ansourian 1998) including more realistic features such as tapering wallthickness and the influence of the top ring stiffener (wind girder). These authors used ABAQUS(Hibbitt, Karlsson and Sorensen 1997) to evaluate out-of-plane displacements, bending andmembrane stress resultants as a function of the wave number n. Their numerical results showedthat there is a critical value of n (close to 8) for which the displacements w n reach a maximumvalue.Experiments on small scale models have been done by D’Orazio et al. (1989) for an opencylinder supported on eight points around the perimeter, and the settlement in the laboratorymodel are related to wall movements in real tanks by:w real = (H/R) real K s (w R/H) model (4)In this equation w is the change in radius and K s is a factor of scale. Note that these expressionsare independent of the slenderness of the shell and represent a linear relation.Jonaidi and Ansourian (1998) reported tests on steel open cylinders with variable thicknessand a simulation of the top ring. The mean wall-slenderness is R/t = 375 and R/H = 1.88. Thetests were performed at low amplitude settlements, consistent with the linear shell theory, andalso for large deflection, the main purpose being the evaluation of stress mechanisms. Moststudies refer to open tanks. “Little data and few analyses exist to set a criterion for the validity ofconed-roof tanks” (Marr et al. 1982).EXPERIMENTAL WORK ON A SMALL-SCALE MODELA small-scale elastic model was tested to observe the main features of the behavior of theshell as the supports settle. The tank was constructed using an acetate sheet, which was curvedto form a cylinder, and a circular plate of acetate was cut to make the roof. The experiments didnot use similitude theory so that only the phenomenological behavior was identified from thetest. The dimensions were H = 90 mm, 2R = 226 mm and t = 0.2 mm, leading to ratios R/H =565 and R/t = 2.5. This model is about 135 times smaller than several tanks that have beenconstructed in Puerto Rico. Because of the difficulties in constructing a conical roof, only casesof flat roof were tested. The object of the test was to induce a vertical displacement at the base(over an arc covering 30º of the circumference) in order to measure the out-of-planedisplacements of the shell and to obtain the general pattern of displacements.The mechanical properties of the material were approximately E = 0.464 GPa and ν = 0.3.The cylinder was fixed to a rigid wooden base (with dimensions 600 × 400 × 12 mm) by meansof a ring with an “L” cross section in order to maintain the circular shape of the tank at the base,but this ring was discontinued over a central angle of 30º to allow for the vertical displacements.Small holes were drilled on the cylinder at the base in the region chosen for the settlement, so as