The Quintessential Bending Element with Drilling Degrees of Freedom (QBD) is a membrane formulation for a quadrilateral shell element that contains three degrees of freedom at each node.

This formulation is based on a modified Hu-Washizu model with symmetric and asymmetric terms.

References

1. Kugler, S., Fotiu, P., & Murin, J. (2010). A Highly Efficient Membrane Element with Drilling Degrees of Freedom.

Degrees of Freedom

The QBD membrane formulation contains 3 degrees-of-freedom (DoFs) at each node:

1. x-translation
2. y-translation
3. z-rotation

There are four nodes defined for this formulation. These nodes are numbered from 1 to 4 in a counter-clockwise direction.

The local stiffness matrix is formulated with the following degree-of-freedom order:

$\begin{bmatrix}u_1 & u_2 & u_3 & u_4 & v_1 & v_2 & v_3 & v_4 & \theta_1 & \theta_2 & \theta_3 & \theta_4 \end{bmatrix}$

Where:

$u=$Local x-translation

$v=$Local y-translation

$\theta=$Rotation about the local z-axis

The subscript indicates the node number.

Stiffness Matrix Formulation

Shape Functions

$N_I=\frac{1}{4} (1+\xi_I \xi)(1+ \eta_I \eta)$ Note that there is no sum on $I$