Issue 29

F. Tornabene et alii, Frattura ed Integrità Strutturale, 29 (2014) 251-265; DOI: 10.3221/IGF-ESIS.29.22 259 It is important to note that c is an arbitrary constant and can be set equal to 10 10 N c x    in order to obtain stable and accurate results. It must be pointed out that in the applications, the coordinate transformation from an interval   1 , k N c d       to another one   1 , k N x a x b x    should be defined, especially when different basis functions are used for the definitions of the weighting coefficients. Thus, the transformation of the GIQ weighting coefficients 1 N k w allows to switch from the interval   , c d to a generic one   , a b . Recalling Eq. (5), the following relation can be written           1 1 1 1 1 1 1 1 where b d N N k k k a c N N N N N N k k k k k k k k b a b a b a b a f x dx f c a d w f c a d c d c d c d c b a b a b a w f c a w f x w w d c d c d c                                                            (21) It is noted that 1 N k w  are the weighting coefficients in the shifted interval   , c d and the 1 N k w are the ones in the physical interval   , a b . A PPLICATIONS AND RESULTS n the present section the static and dynamic behaviors of 1D and 2D structural components, summarized in Tab. 3 and 4, are reported. In all the computations an isotropic material with elastic modulus 210 GPa E  , Poisson ratio 0.3   and density 3 7800 kg/m   has been considered. The one-dimensional problems (Rod, EB and Tim beams) have a length 2 m L  and a squared cross section / . 20 0 1 m    b h L . The two-dimensional applications (Membrane, KL and RM plates) are squared with . . , the membrane has a constant tension 1 N S  , the KL plate has a thickness 0.01 m h  and the RM plate has a thickness 0.08 m h  . In all the static calculations the uniform applied load for both 1D and 2D problems is taken equal to 100 Pa q  . The convergence, stability and reliability of the SFEM varying the basis functions, collocation, number of grid points and number of elements are discussed in the following. The spacing of floating point numbers, due to the machine in use, is 52 16 2 10 CPU      . The static and dynamic convergence and stability of Rods, Euler-Bernoulli (EB) and Timoshenko (Tim) beams are shown. For the present studies the basis functions, the grid point distributions, the number of points in each domain and the number of elements for the domain decomposition can be selected. The EB beam has the additional parameter given by the  -points location. The authors defined a new way of inserting the  -points other than the classic one [9, 12]. This new approach consists in adding the  -points as functions of the first and third (last and two-before the last) points. The  -points location is indicated in brackets “( )”. As far as the classic  -points technique is concerned, the extra points locations are directly reported as ( 5 10  ), whereas for the second approach the distance between the first and the third points is indicated. For instance, if (1/2) is used the  -points are the mid-points of the first and the third ones. In the following applications the GDQ method based on PDQ basis functions is considered. The reader can find the complete nomenclature of the grid point locations in Tab. 2. The one dimensional tests are divided into static and dynamic analysis. For the static benchmarks, the L2-norm of the absolute error of the displacement field is taken into account, since comparing a displacement of a single point of the domain can rise numerical instabilities. For the dynamic cases, the discrete spectra and the relative error on the first natural frequency will be shown in the following. Fig. 1a shows the L2-norm of the absolute error of the displacement for several structural elements composed of a single finite element varying the number of points N . The error given by the EB beam model is higher than the one exhibited by the Rod and Tim beam, because extra points have to be added in order to enforce the boundary conditions. The Che-Gau-Lob grid distribution is chosen within PDQ basis functions. For the EB beam model  -points are taken as (1/2), showing the best accuracy with respect to the classic value ( 5 10  ). The present method has been demonstrated to be stable even for 151 N  . Fig. 1b studies the accuracy of the domain decomposition by varying the number of elements with 7 N  in each element. The same basis functions and grid distribution of the previous case was used. Increasing the number of grid points or the elements, the same trends are obtained when Fig. 2a and 1b are compared. In the following the free vibration problems of the Rod, EB and Tim beams are reported in Fig. 2b, 3, 4a. In the computations d N represents the total number of degrees of freedom I