Hourglass modes are element distortions that have zero strain energy. The 4 node shell
element has 12 translational modes, 3 rigid body modes (1, 2, 9), 6 deformation modes (3, 4,
5, 6, 10, 11) and 3 hourglass modes (7, 8, 12).Figure 1. Translational Modes of Shell
Along with the translational modes, the 4 node shell has 12 rotational modes: 4 out of
plane rotation modes (1, 2, 3, 4), 2 deformation modes (5, 6), 2 rigid body or deformation
modes (7, 8) and 4 hourglass modes (9, 10, 11, 12).Figure 2. Rotational Modes of Shell
Hourglass Viscous Forces
Hourglass resistance forces are usually either viscous or stiffness related. The viscous
forces relate to the rate of displacement or velocity of the elemental nodes, as if the
material was a highly viscous fluid. The viscous formulation used by Radioss is the same as that outlined by Kosloff and Frasier 1. Refer to Hourglass Modes. An hourglass normalized vector is defined
as:
The hourglass velocity rate for the above vector is defined as:
The hourglass resisting forces at node for in-plane modes are:
For out of plane mode, the resisting forces are:
Where,
Direction index
Node index
Element thickness
Sound propagation speed
Element area
Material density
Shell membrane hourglass coefficient
Shell out of plane hourglass coefficient
Hourglass Elastic Stiffness
Forces
Radioss can apply a stiffness force to resist hourglass modes.
This acts in a similar fashion to the viscous resistance, but uses the elastic material
stiffness and node displacement to determine the size of the force. The formulation is the
same as that outlined by Flanagan et al. 2 Refer to Flanagan-Belytschko Formulation. The hourglass resultant forces are defined as:
For membrane modes:
For out of plane modes:
Where,
Element thickness
Time step
Young's modulus
Hourglass Viscous
Moments
This formulation is analogous to the hourglass viscous force scheme. The
hourglass angular velocity rate is defined for the main hourglass modes as:
The hourglass resisting moments at node
are given by:
Where, is the shell rotation hourglass coefficient.
1Kosloff D. and Frazier G., Treatment of hourglass pattern in low
order finite element code, International Journal for Numerical and
Analytical Methods in Geomechanics, 1978.
2Flanagan D. and Belytschko T., A Uniform Strain Hexahedron and
Quadrilateral with Orthogonal Hourglass Control, Int. Journal Num.
Methods in Engineering, 17 679-706, 1981.