7.3 Plates and Shells

Plate or shell structures with a thickness that is relatively small compared to the in-plane dimensions and with out-of-plane loadings, may be defined with plate bending or shell elements. These elements may be located anywhere in three-dimensional space. Three different types of plate and shell elements can be distinguished:

• Plane bending elements

  Plate bending elements must fulfill the following conditions with respect to shape and loading [Fig.7.49].

  Figure 7.49: Plate bending elements, characteristics

  
  They must be plane, i.e., the coordinates of the element nodes must be in one flat plane, the xy plane of the element. The thickness t must be small in relation to the dimensions b in the plane of the element. Force loading F must act perpendicular to the element plane, moment loading M must act around an axis which is in the element plane.

  Plate bending elements are characterized by the following facts. The direct stress component perpendicular to the face is zero, ( $\sigma_{{zz}}^{}$ = 0 )this means that the plane stress condition is fulfilled. The normals of the element plane remain straight after the deformation, but by definition, they do not have to be perpendicular to the element plane. The displacement perpendicular to the plane does not vary along the thickness.

  DIANA offers two classes of plate bending elements: the first based on the Discrete Kirchhoff theory and therefore called Discrete Kirchhoff plate elements, the second based on a Mindlin-Reissner theory and simply called Mindlin plate elements. Both classes of plate bending elements are numerically integrated.

  Typical applications of plate bending elements are the analysis of floors and other two-dimensional structural parts which are not subjected to in-plane forces.

  Plate elements implemented according to the Discrete Kirchhoff theory, simply called `Kirchhoff plates', are based on the principle that the condition of zero transverse shear strain is satisfied at some discrete points in the element. The displacement and rotation field is expanded by introducing some shear constraints. The effect of shear deformation is included which makes Kirchhoff plate elements suitable for application in both thin and thick plates.

  In the Mindlin-Reissner plate theory the transverse displacements and rotations of the mid surface normals are independent and obtained by employing an isoparametric interpolation respectively from the translations and rotations in the nodes. This technique includes transverse shear deformation. Elements implemented according to this theory are simply called `Mindlin plate elements'.

  In their standard form these elements are sensitive for shear locking which results in a excessively stiff behaviour. To overcome this difficulty for the linear and quadratic elements, DIANA modifies the transverse shear strain field.

• Flat shell elements

  Flat shell elements basically are a combination of plane stress elements and plate bending elements. But unlike the plane stress elements, the basic variables are forces rather than Cauchy stresses. Flat shell elements must fulfill the following conditions with respect to shape and loading [Fig.7.50].

  Figure 7.50: Flat shell elements, characteristics

  
  They must be plane, i.e., the coordinates of the element nodes must be in one flat plane, the xy plane of the element, otherwise the curved shell elements must be used. They must be thin, i.e., the thickness t must be small in relation to the dimensions b in the plane of the element. Force loads F may act in any direction between perpendicular to the plane and in the plane. Moment loads M must act in the plane of the element.

  Flat shell elements are characterized by the following facts. The normals of the element plane remain straight after the deformation. The displacement perpendicular to the plane does not vary in the direction of the thickness.

  The flat shell elements in DIANA basically are combinations of a plane stress element and a plate bending element and there is no coupling between membrane and bending behaviour. Generally, the membrane behaviour is conform its corresponding plane stress element except the primary stresses which are defined in terms of moments and forces rather than Cauchy stresses. The bending behaviour is based on the Mindlin-Reissner theory and is conform its corresponding Mindlin plate bending element. For all flat shell elements the numerically integration is only performed in the reference surface.

  DIANA offers two classes of flat shell elements: regular elements, elements with drilling rotation. The regular elements have three translations and two in-plane rotation degrees of freedom in each node. The elements with drilling rotation have an additional rotation $\phi_{{z}}^{}$ normal to the plane in each node, the so-called `drilling rotation'. This drilling rotation may avoid an ill-condition of the global stiffness matrix in some cases.

  Typical applications of flat shell elements are the analysis of tunnels and other two-and-a-half dimensional structural parts like box girders.

• Curved shell elements

  The curved shell elements in DIANA are based on isoparametric degenerated-solid approach by introducing two shell hypotheses:

  Straight-normals

  -- assumes that normals remain straight, but not necessarily normal to the reference surface. Transverse shear deformation is included according to the Mindlin-Reissner theory.

  Zero-normal-stress

  -- assumes that the normal stress component in the normal direction of a lamina basis is forced to zero: $\sigma_{{{zz}_{l(\xi,\eta,z)}}}^{}$ = 0 . The element tangent plane is spanned by a lamina basis which corresponds to a local Cartesian coordinate system (x_l, y_l) defined at each point of the shell with x_l and y_l tangent to the $\xi$,$\eta$ plane and z_l perpendicular to it.

  Figure 7.51: Curved shell elements, characteristics

  
  The in-plane lamina strains $\varepsilon_{{xx}}^{}$ , $\varepsilon_{{yy}}^{}$ and $\gamma_{{xy}}^{}$ vary linearly in the thickness direction. The transverse shear strains $\gamma_{{xz}}^{}$ and $\gamma_{{yz}}^{}$ are forced to be constant in the thickness direction. Since the actual transverse shearing stresses and strains vary parabolically over the thickness, the shearing strains are an equivalent constant strain on a corresponding area. A shear correction factor is applied using the condition that a constant transverse shear stress yields approximately the same shear strain energy as the actual shearing stress.

  Five degrees of freedom have been defined in each element node: three translations and two rotations. Further characteristics of curved shells are the following [Fig.7.51]. They must be thin, i.e., the thickness t must be small in relation to the dimensions b in the plane of the element. Force loads F may act in any direction between perpendicular to the surface and in the surface. Moment loads M should act around an axis which is in the element face.

  Also a set of curved shell elements with drilling rotation is available. These curved shell elements have six degrees of freedom in each element node: three translations and three rotations, because an additional rotation $\phi_{{z}}^{}$ , the drilling rotation, has been added to the basic variables of the regular curved shell elements. In applications where the elements are nearly co-planar in the nodes, the use of shell elements with drilling rotation is very attractive because they avoid an ill-condition of the assembled global stiffness matrix. Like for regular curved shell elements, the basic variables of regular curved shell elements with drilling rotation are the translations u and the rotations $\phi$ . The derived variables are the strains, the Cauchy stresses and the generalized moments and forces.

  Typical applications of curved shell elements are the analysis of curved structures like shell roofs, storage tanks and ship or aircraft hulls.