# Hourglass Modes

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).
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).

## 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 アワグラスモード. An hourglass normalized vector is defined as:(1)
$\Gamma =\left(1,\text{\hspace{0.17em}}-1,\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}-1\right)$
The hourglass velocity rate for the above vector is defined as:(2)
$\frac{\partial {q}_{i}}{\partial t}={\Gamma }_{I}{v}_{iI}={v}_{iI}-{v}_{i2}+{v}_{i3}-{v}_{i4}$
The hourglass resisting forces at node $I$ for in-plane modes are:(3)
${f}_{iI}^{hgr}=\frac{1}{4}\rho ct\sqrt{{h}_{m}}\frac{A}{2}\frac{\partial {q}_{i}}{\partial t}{\Gamma }_{I}$
For out of plane mode, the resisting forces are:(4)
${f}_{iI}^{hgr}=\frac{1}{4}\rho c{t}^{2}\sqrt{\frac{{h}_{f}}{10}}$
Where,
$i$
Direction index
$I$
Node index
$t$
Element thickness
$c$
Sound propagation speed
$A$
Element area
$\rho$
Material density
${h}_{m}$
Shell membrane hourglass coefficient
${h}_{f}$
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定式化. The hourglass resultant forces are defined as:(5)
${f}_{iI}^{hgr}={f}_{i}^{hgr}{\Gamma }_{I}$
For membrane modes:(6)
${f}_{i}^{hgr}\left(t+\text{Δ}t\right)={f}_{i}^{hgr}\left(t\right)+\frac{1}{8}{h}_{m}Et\frac{\partial {q}_{i}}{\partial t}\text{Δ}t$
For out of plane modes:(7)
${f}_{i}^{hgr}\left(t+\text{Δ}t\right)={f}_{i}^{hgr}\left(t\right)+\frac{1}{40}{h}_{f}E{t}^{3}\frac{\partial {q}_{i}}{\partial t}\text{Δ}t$
Where,
$t$
Element thickness
$\text{Δ}t$
Time step
$E$
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:(8)
$\frac{\partial {r}_{i}}{\partial t}={\Gamma }_{I}{\omega }_{iI}^{I}={\omega }_{i1}-{\omega }_{i2}+{\omega }_{i3}-{\omega }_{i4}$
The hourglass resisting moments at node $I$ are given by:(9)
${m}_{iI}^{hgr}=\frac{1}{50}\sqrt{\frac{{h}_{r}}{2}}\rho cA{t}^{2}\frac{\partial {r}_{i}}{\partial t}{\Gamma }_{I}$

Where, ${h}_{r}$ is the shell rotation hourglass coefficient.

1 Kosloff D. and Frazier G., 「Treatment of hourglass pattern in low order finite element code」, International Journal for Numerical and Analytical Methods in Geomechanics, 1978.
2 Flanagan D. and Belytschko T., 「A Uniform Strain Hexahedron and Quadrilateral with Orthogonal Hourglass Control」, Int. Journal Num.Methods in Engineering, 17 679-706, 1981.