# General Degenerated 4-Node Shell Formulations

^{1}and Q4 $\gamma $ 24 developed by Batoz and Dhatt

^{2}; they are suitable for both thin and thick shells and are applicable to linear and nonlinear problems. Their main feature is that a classical displacement method is used to interpolate the in-plane strains (membrane, bending), and a mix/collocation (or assumed strain) method is used to interpolate the out-plane strains (transverse shear). Certain conditions are also specified:

- They are based on the Reissner-Mindlin model,
- In-plane strains are linear, out-plane strains(transverse shear) are constant throughout the thickness,
- Thickness is constant in the element (the normal and the fiber directions are coincident),
- 5 DOF in the local system (that is, the nodal normal vectors are not constant from one element to another).

## Notational Conventions

- A bold letter denotes a vector or a tensor.
- An upper case index denotes a node number; a lower case index denotes a component of vector or tensor.
- The Einstein convention applies only for the repeated index where one is subscript and
another is superscript, e.g.:
${N}_{I}{x}^{I}={\displaystyle \sum {N}_{I}{x}^{I}}$

- {} denotes a vector and [ ] denotes a matrix.

## Geometry and Kinematics

with $z={\scriptscriptstyle \frac{a\zeta}{2}}$

- $a$
- Shell thickness

with $\left[{F}_{0}\right]=\left[\left\{{g}_{1}\right\}\left\{{g}_{2}\right\}{\scriptscriptstyle \frac{a}{2}}\left\{n\right\}\right];{g}_{1}\left(\xi ,\eta \right)={X}_{p,\xi}={N}_{I,\xi}{X}^{I},{g}_{2}\left(\xi ,\eta \right)={X}_{p,\eta}={N}_{I,\eta}{X}^{I}$

$F$ is the gradient tensor which is related to the Jacobian tensor $\left[J\right]={\left[F\right]}^{t}$ .

Where, $\beta $ and $\omega $ are the rotational velocity vectors of the normal: $\beta ={\beta}_{1}{t}_{1}+{\beta}_{2}{t}_{2}={\omega}_{2}{t}_{1}-{\omega}_{1}{t}_{2}$

and $\left({t}_{1},{t}_{2},n\right)$ is base of the local coordinate system.

Equation 5 can be written also by:

${v}_{i}={N}_{I}{v}_{i}^{I}+z\left(-{N}_{I}{\omega}_{1}^{I}{t}_{2i}^{I}+{N}_{I}{\omega}_{2}^{I}{t}_{1i}^{I}\right)\text{\hspace{0.17em}};\text{\hspace{1em}}i=1,3$

This velocity interpolation is expressed in the global system, but ${\omega}^{I}$ must be defined first in the local nodal coordinate system to ensure Mindlin's kinematic condition.

## Strain-Rate Construction

The in-plane rate-of-deformation is interpolated by the usual displacement method.

with $\left[Q\right]=\left[\left\{{t}_{1}\right\}\left\{{t}_{2}\right\}\left\{n\right\}\right]$ .

The Reissner-Mindlin conditions ${\epsilon}_{\zeta}=0$ and ${\sigma}_{\zeta}=0$ requires that the strain and stress tensors are computed in the local coordinate system (at each quadrature point).

After the linearization of ${L}_{t}$ with respect to $z$ , the in-plane rate-of-deformation terms are given by:

${\left[{L}_{t}\right]}_{2\times 2}=\left[{L}_{s0}\right]+z\left[{L}_{s1}\right]$

with the membrane terms:

$\left[{L}_{s0}\right]=\left[{C}_{1}\right]\left[{C}_{0}\right]$

The curvature-translation coupling is presented in the bending terms for a warped element (the first two terms in the last equation.)

The out-plane rate-of-deformation (transverse shear) is interpolated by the "assumed strain" method, which is based on the Hu-Washizu variation principle.

^{1}mix/collocation method has been proved very efficient in overcoming this problem. This method consists in interpolating the transverse shear from the values of the covariant components of the transverse shear strains at 4 mid-side points. That is:

## Special Case for One-point Quadrature and the Difficulties in Stabilization

The formulations described above are general for both the full integration and reduced integration schemes. For a one-point quadrature element, you have the following particularities:

Where, ${h}_{I}=\left(\begin{array}{cccc}1& -1& 1& -1\end{array}\right)$ .

This implies that all the terms computed at the quadrature point are the constant parts with respect to $\left(\xi ,\eta \right)$ , and the stabilizing terms (hourglass) are the non-constant parts.

It would be ideal if the integration term ${f}_{stab}^{\mathrm{int}}$ could be evaluated explicitly. But such is not the case, and the main obstacles are the following:

For a non-coplanar element, the normal varies at each point so that it is difficult to write the non-constant part of strains explicitly. For a physically nonlinear problem, the non-constant part of stress is not generally in an explicit form. Thus, simplification becomes necessary.

^{1}Dvorkin E. and Bathe K.J. “A continuum mechanics four-node shell element for 35 general nonlinear analysis”, Engrg Comput, 1:77-88, 1984.

^{2}Batoz J.L. and Dhatt G., “Modeling of Structures by finite element”, volume 3, Hermes, 1992.