# RD-E: 0201 Explicit Example

An imposed velocity is applied onto a shallow cylindrical roof at its midpoint. The analysis uses an explicit approach.

The purpose of this example is to study a snap-thru problem with a single
instability. Thus, a structure that will bend when under a load is used. The results
are compared to the references solution. ^{1}

## Options and Keywords Used

Node time histories do not indicate the pressure output. In order to obtain such
output at point C, a rigid body must be created at this point. Point C has a
constant imposed velocity of -0.01 ms^{-1}in the Z direction. Its
displacement is linked proportionally to time.

- Edge BC is fixed in an X translation, and in Y and Z rotations (symmetry conditions).
- Edge CD is fixed in a Y translation, and in X and Z rotations (Idem).
- Edge DA is fixed in X, Y, Z translations, and in X and Z rotations.
- Point C is fixed in X, Y translations, and in X, Y, Z rotations.

## Model Description

A shallow cylindrical roof, pinned along its straight edges upon which an imposed velocity is applied at its mid-point.

Units: mm, ms, g, N, MPa

- l
- 254 mm
- R
- 2540 mm
- Shell thickness
- t = 12.7 mm
- $\theta $
- 0.1 rad

**Material Properties****Value**- Initial density
- 7.85x10
^{-3}$\left[\frac{g}{m{m}^{3}}\right]$ - Young's modulus
- 3102.75 $\left[\mathrm{MPa}\right]$
- Poisson ratio
- 0.3

### Model Method

The structure is considered perfect, having no defects. To take account of the symmetries, only a quarter of the shell is modeled (surface ABCD).

**Shell Properties****Value**- Thickness
- 12.7 mm
- BT Elasto-plastic Hourglass formulation
`I`_{shell}= 3

## Results

### Curves and Animations

Only a quarter of the total load is applied due to the symmetry. Therefore, force Fz of the rigid body, as indicated in the Time History, must be multiplied by 4 in order to obtain force, P.

The displacement of point C is indicated in its absolute value. The curve illustrates the characteristic behavior of the instability of a snap-thru. Beyond the limit load, an infinite increase in load $\text{\Delta}Fz$ will cause a considerable increase in displacement $\text{\Delta}q$ due to the collapsing of the shell.

The first extreme defines the limit load =2208.5 N (displacement of point C = 10.5 mm).

The difference between the two curves is approximately 10% for reduced displacements (up to 5 mm) and slightly more (15%) for the higher nonlinear part of the curve (between 5 and 20 mm). For displacements exceeding 20 mm, the curves are shown much closer together.

Deformed Mesh (profile view) - Displacement Norm |
---|

Initial configuration |

Start of snap-thru |

Large motion phase |

Stable configuration |

Loading with a new structural rigidity |