Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

254

and the plane Poisson’s coefficient and the modulus of elasticity

E = 6 R

1 + ˜ R / R 3 + ˜ R / R

t

1 + ¯ h 2 3 + ¯ h 2

1 − ¯ h 2 3 + ¯ h 2

1 − ˜ R / R 3 + ˜ R / R

t ( b )

= 2 √ 3

E ( b )

ν =

(6.58)

=

,

based on well-known elasticity relations (e.g., [106]). Given the expression (6.58) 1 , it is evident that for slender beams ( h ( b ) ≪ L ( b ) ⇔ ¯ h ( b ) ≪ 1), shear bending loses importance and the value of the plane Poisson’s ratio approaches the upper limit of 1/3. Given the assumption of beam slenderness inherent in the Bernoulli-Euler formulation, the lower limit of the plane Poisson’s ratio is ≈ 0 . 2 [89]. Finally, it is easy to show that in the limit case of negligible shear bending stiffness ( ¯ h → 0), the plane-strain parameters (6.58) are reduced to their α model counterparts (6.37). The disadvantages of the Bernoulli-Euler beam lattices are: (i) the results are sensitive to the size of the beam elements and the direction of application of the load, (ii) the material response is excessively brittle (especially if ideally-brittle behavior is used for individual beams), (iii) the beam elements in the pressure zones could overlap, and (iv) exceptional computational effort is required at the structural level. All these shortcomings can be reduced in various ways. For example, the first one can be remedied by using an irregular geometry [94]. The second, by 3D modeling, using very small unit cells [95], as well as using a nonlocal approach in calculating the deformations of beam elements [94]. Triangular Timoshenko beam lattice When the lattice model contains short beams it is more appropriate to use lattice elements based on Timoshenko beam theory that takes into account shear deformation and rota tional bending effects. This formulation is presented herein in a much abbreviated form in comparison with Bernoulli-Euler beam presented in the preceding chapter. Unlike Bernoulli-Euler formulation, during the actual beam deformation, the cross sections of the beam remain neither perpendicular to the neutral line nor flat/straight (i.e., warping takes place). Timoshenko kept the assumption of a flat section, but introduced an (additional) shear-induced angular deformation, so that the cross section is no longer perpendicular to the neutral line. Thus, Timoshenko beam theory implies 3 remain unchanged. The dimensionless parameter (6.59) 2 is the key ingredient of Timoshenko beam theory. When the shear stiffness is dominant, ς T ≪ 1, the shear displacement is relatively small and Bernoulli-Euler beam theory is applicable. In contrast, at low shear stiffness, the shear displacement is no longer negligible and it is necessary to use Timoshenko beam theory. From the equivalence of strain energies (6.28), identical expressions for the com ponents of the elasticity tensor (6.56) as for the Bernoulli-Euler beam follow, with the difference that Q ( b ) = 12 E ( b ) I ( b ) ( 1 + ς T )( L ( b ) ) 3 L ( b ) ˜ γ ( b ) , ς T = = 12 E ( b ) I ( b ) G ( b ) A ( b ) ( L ( b ) ) 2 = E ( b ) G ( b ) ¯ h ( b ) (6.59) while the relations F ( b ) − γ ( b ) (6.50) 1 and M ( b ) − κ ( b ) (6.50)

=

h L

2

¯ h 2 1 + ς T

˜ R R

12 I L 2 A

1 1 + ς T

1 1 + ς T

(6.60)

=

=