Q2. What is the common type of locking in a shell?
Low order elements like quadrilaterals based on standard displacement interpolation are usually characterized by locking phenomena.
Q3. What is the bending stiffness of the arbitrary shaped shell?
For arbitrary shaped elements a transformation of the stiffness matrix, which considers the warping effects, leads to good results also for a non–flat geometry.
Q4. What is the main issue in the context of developing a finite shell model?
An important issue within the context of developing a finite shell model is the number and type of rotational parameters on the element.
Q5. What is the description of the shell?
shell behaviour is extremely sensitive to initial geometry and imperfections, thus a successful correlation between theory and analysis is achieved only after including specific details of these quantities.
Q6. What is the stabilization matrix for the Green–Lagrangean membrane strains?
The stabilization matrix is derived from the orthogonality between the constant part of the strain field and the non-constant part using 5 degrees of freedom at each node.
Q7. What is the method for nonlinear material behaviour?
For general nonlinear material behaviour a three field variational functional with independent displacements, stresses and strains is more appropriate.
Q8. What is the integrand in the patch test?
Furthermore the matrices H and G are introduced with B = [B1,B2,B3,B4] ,H = ∫(Ωe)STC−1S dA , G = ∫(Ωe)STB dA . (25)Since the integrand in (25)1 involves only polynomials of the coordinates ξ and η the integration can be carried out analytically.
Q9. What is the kinematic assumption for the deformation of a shell?
The position vector Φ of any point P ∈ B0 is associated with the global coordinate frame eiΦ(ξ1, ξ2, ξ3) = Φi ei = X + ξ 3 D(ξ1, ξ2)with |D(ξ1, ξ2)| = 1 and − h 2 ≤ ξ3 ≤ h 2(1)with the position vector X(ξ1, ξ2) of the shell mid–surface Ω, the shell thickness h, and ξi the convected coordinate system of the body.
Q10. What is the convergence behaviour of the investigated elements?
The convergence behaviour using distorted and warped elements is investigated with the hemispherical shell of the last section without the hole.
Q11. How many elements are used in the beam model?
The finite element calculations are performed with the developed shell element using a mesh with 530 nodes and 368 elements, see Fig. 13, and for comparison with 15 two– dimensional beam elements, see Fig. 17.
Q12. What is the bending stiffness of the shear–elastic shell?
The essential features and new contributions of the present formulation are summarized as follows:(i) Assuming linear elasticity the variational formulation of the shear–elastic shell is based on the Hellinger–Reissner principle.
Q13. What is the method to avoid transverse shear locking?
An effective method to avoid transverse shear locking is based on assumed shear strain fields first proposed in [15], and subsequently extended and reformulated in [16, 17, 18, 19].
Q14. What is the simplest way to approximate the shear strains?
Considering (4) and the finite element equations (10) - (15) the approximation of the strains is now obtained byεh = 4∑I=1BI vI , vI = [uI ,ϕI ] T , (16)withBI = NI ,1 X T,1 0NI ,2 X T,2 0NI ,1 X T,2 +NI ,2 X T,1 0NI ,1 D T,1 NI ,1 b T 1I NI ,2 D T,2 NI ,2 b T 2INI ,1 D T,2 +NI ,2 D T,1 NI ,1 b T 2I + NI ,2 b T 1IJ−1 NI ,ξ D T MNI ,η D T L J−1 NI ,ξ ξI b T MNI ,η ηI b T L (17)and bαI = DI ×X,α = WI X,α, bM = WI XM,ξ , bL = WI XL,η.
Q15. What are the nodal degrees of freedom?
the nodal degrees of freedom are: three global displacements components, three global rotations at nodes on intersections and two local rotations at other nodes.
Q16. what is the inverse of the shear strains?
An alternative three field variational formulation based on a Hu–Washizu principle for the shear part, which would be the appropriate variational formulation for an independent shear interpolation, leads to identical finite element matrices due to the fact that the shear stiffness matrix is diagonal.