Issue 29

L. Cabras et alii, Frattura ed Integrità Strutturale, 29 (2014) 9-18; DOI: 10.3221/IGF-ESIS.29.02 13 (a) (b) (c) (d) Figure 5: Application of the Principle of Virtual Displacements. Lattice reinforced with longitudinal springs. (a) Simplified structured analyzed for the computation of the effective properties. The applied forces F N, F V , F T1 and F T2 correspond to macroscopic stresses components. (b) Disconnected statically determined structure introduced for the determination of the internal actions (M,N,S L ). (c-d) Statically determined structure adopted for the computation of the displacement of the points A and B. Such kinematic constraints must be restored imposing the kinematic compatibility equation that determines the values of the unknown X and uniquely defines the elastic solution of the problem, as follows:   cotγ 3+1α γ2 3cos +2α γ2 sin 1)+1(αVF+2)+2α γ2 sin+11)α -γ2 (3cos NF beam p 0 spring /2Lk L 1SL 1S )dξ EA 1N 1N EJ 1M 1 (M beam p 0 spring /2Lk L 1SL 0S )dξ EA 1N 0N EJ 1M 0 (M X                 (8) We note that for sufficiently slender beam structures, the contribution due to the shear deformation is negligible compared to that due to flexural and axial deformations and, therefore, it has been neglected. We reconstruct the distribution of internal actions by a linear combination of partial diagrams of N and M and of the spring forces S L , as functions of external forces F N , F T =(F T1 +F T2 )/2 and F V : N =N + X N 0 1 M=M + X M 0 1 L L L S = S + X S 0 1      (9) Applying a second time the PVW we calculate the displacement of the point A and B (center of the spring) as shown in Fig. 5c-5d. To do this we consider as kinematically admissible structure the real structure and as statically admissible structure, an isostatic structure subjected to horizontal and vertical forces of magnitude equal 1/4, respectively, so that the virtual external works coincide exactly with the horizontal and vertical displacement of the point B (u 1 and u 2 in Fig. 5c), and we calculate the displacement of the point A (u 3 in Fig. 5d ) considering a vertical forces of unitary magnitude, so that the virtual external works coincide exactly with the vertical displacement of the point A. In particular the PVW equations have the form: