# Spring Failure

- Uni-directional failure, or
- Multi-directional failure

This is controlled by the option `I`_{fail}. If the `I`_{fail} flag is not set/present in the property, the default uni-directional failure is
considered for the spring. For example, in TYPE4 there is no option `I`_{fail}, so uni-directional failure is used.

`I`

_{fail2}. Similar to

`I`

_{fail}, if the

`I`

_{fail2}option is not set/present in the property, the displacement (or rotation) failure model is used.

Spring Type | Failure Criteria (Ifail) | Failure Model (Ifail2) | |||||
---|---|---|---|---|---|---|---|

Uni-directional Failure | Multi-directional Failure |
${\alpha}^{i},{\beta}^{i}$
in Multi-directional Failure |
Displacement (or Rotation) Criteria | Displacement (or Rotation) Criteria Consider Velocity Effect | Force (or Moment) Criteria | Internal Energy Criteria | |

TYPE4 | √ | √ | |||||

TYPE8 | √ | √ |
${\alpha}^{i}=1$
${\beta}^{i}=2$ |
√ | √ | √ | |

TYPE12 | √ | √ | |||||

TYPE13 | √ | √ | arbitrary
${\alpha}^{i},{\beta}^{i}$
(default ${\alpha}^{i}=1$ , ${\beta}^{i}=2$ ) |
√ | √ | √ | √ |

TYPE25 | √ | √ | arbitrary
${\alpha}^{i},{\beta}^{i}$
(default ${\alpha}^{i}=1$ , ${\beta}^{i}=2$ ) |
√ | √ | √ | √ |

## Failure Criteria

- Uni-directional (
`I`_{fail}= 0)If the criteria is uni-directional, the spring will fail as soon as the criteria is satisfied for one degree of freedom:- $\left|\frac{{\delta}^{i}}{{\delta}_{\mathrm{max}}^{i}}\right|\ge 1$ or $\left|\frac{{\delta}^{i}}{{\delta}_{\mathrm{min}}^{i}}\right|\ge 1$ with ${\delta}_{\mathrm{max}}^{i}$ and ${\delta}_{\mathrm{min}}^{i}$ being the failure limits in direction $i$ =1,2,3.
- $\left|\frac{{\theta}^{i}}{{\theta}_{\mathrm{max}}^{i}}\right|\ge 1$ or $\left|\frac{{\theta}^{i}}{{\theta}_{\mathrm{min}}^{i}}\right|\ge 1$ with ${\theta}_{\mathrm{max}}^{i}$ and ${\theta}_{\mathrm{min}}^{i}$ being the failure limits in direction $i$ =4,5,6.

Where, $i$ is any degree of freedom. Its property type dependent.

For property TYPE4, there is only $i$ =1, for translational X.

For property TYPE8, there are $i$ =1,2,3,4,5,6 for translational X, Y, Z and rotational X,Y,Z.

For property TYPE13, there are $i$ =1,2,3,4,5,6, but in this case, for tension/compression X, shear XY, shear XZ, torsion, bending Y, bending Z.

Examples of failure behaviors of uni-directional failure are:

If ${\delta}_{\mathrm{max}}^{1}=0.04\mathrm{m}$ in a tension only test, then there is spring failure, and the force goes to zero once the elongation reaches 0.04m.

The same is true for rotation, if ${\theta}_{\mathrm{max}}^{4}=0.035\mathrm{rad}$ , the spring fails and has zero force at 0.035rad.If a spring is subject to two load cases, for example, tension and torsion and`I`_{fail}= 0 (uni-directional failure) is in use, then spring failure occurs if either one of the failure criteria is reached.Here the rotation criteria is reached first (at Time=0.58s), then the force and moment fall to zero at the same time. - Multi-directional (
`I`_{fail}= 1)If the criteria is multi-directional, all degrees of freedom are coupled and failure occurs when:

$${{\displaystyle \sum _{i=1,2,3}{\alpha}^{i}\left(\frac{{\delta}^{i}}{{\delta}^{i}{}_{fail}}\right)}}^{{\beta}^{i}}+{{\displaystyle \sum _{i=4,5,6}{\alpha}^{i}\left(\frac{{\theta}^{i}}{{\theta}^{i}{}_{fail}}\right)}}^{{\beta}^{i}}\ge 1$$Where, ${\delta}^{i}{}_{fail}$ and ${\theta}^{i}{}_{fail}$ are failure criteria. Refer to Failure Criteria for more details.

For property TYPE8, ${\alpha}^{i}=1$ and ${\beta}^{i}=2$ (failure criteria shown as blue curve in Figure 4).

For properties TYPE13 and TYPE25, arbitrary ${\alpha}^{i},{\beta}^{i}$ may be input with ${\alpha}^{i}>0$ (default is ${\alpha}^{i}=1$ ). Figure 4 shows failure criteria with different ${\beta}^{i}$ .In a test case of two load cases, tension + torsion and with`I`_{fail}= 1.Comparing the failure value against the limit set in just one direction, it usually is smaller than the defined limit. In this example case, the tension limit is set at 0.04m and torsion limit set at 0.035rad. The spring fails at an elongation 0.0236 < 0.04 and rotation 0.02826 < 0.035. This is because the failure combination of tension and torsion reaches the failure circle (Figure 6) and; therefore, the spring failed (force and moment fall to zero).

## Failure Model

`I`

_{fail2}is available in properties TYPE8, TYPE13, and TYPE25.

- Displacement (or Rotation) Failure
Criteria (
`I`_{fail2}= 0)$${{\displaystyle \sum _{i=1,2,3}\left(\frac{{\delta}^{i}}{{\delta}^{i}{}_{fail}}\right)}}^{2}+{{\displaystyle \sum _{i=4,5,6}\left(\frac{{\theta}^{i}}{{\theta}^{i}{}_{fail}}\right)}}^{2}\ge 1$$With,

${\delta}^{i}{}_{fail}=\{\begin{array}{c}{\delta}_{\mathrm{max}}^{i},\text{}if\left({\delta}^{i}0\right)\\ {\delta}_{\mathrm{min}}^{i},\text{}if\left({\delta}^{i}\le 0\right)\end{array}$ and ${\theta}^{i}{}_{fail}=\{\begin{array}{c}{\theta}_{\mathrm{max}}^{i},\text{}if\left({\theta}^{i}0\right)\\ {\theta}_{\mathrm{min}}^{i},\text{}if\left({\theta}^{i}\le 0\right)\end{array}$

- Displacement (or Rotation) Failure
Criteria considering velocity effect
(
`I`_{fail2}= 1)This failure criteria will allow model velocity dependent failure limits, they are available with displacement, force and internal energy. Therefore, translational ${\delta}^{i}{}_{fail}$ and rotational ${\theta}^{i}{}_{fail}$ failure are modified to take into account velocity, as:

$${{\displaystyle \sum _{i=1,2,3}\left(\frac{{\delta}^{i}}{{\delta}^{i}{}_{fail}}\right)}}^{2}+{{\displaystyle \sum _{i=4,5,6}\left(\frac{{\theta}^{i}}{{\theta}^{i}{}_{fail}}\right)}}^{2}\ge 1$$${\delta}^{i}{}_{fail}=\{\begin{array}{c}{\delta}_{\mathrm{max}}^{i}+{c}_{i}\cdot {\left|\frac{{v}^{i}}{{v}_{0}}\right|}^{ni},\text{}if\left({\delta}^{i}0\right)\\ {\delta}_{\mathrm{min}}^{i}-{c}_{i}\cdot {\left|\frac{{v}^{i}}{{v}_{0}}\right|}^{ni},\text{}if\left({\delta}^{i}\le 0\right)\end{array}$ and ${\theta}^{i}{}_{fail}=\{\begin{array}{c}{\theta}_{\mathrm{max}}^{i}+{c}_{i}\cdot {\left|\frac{{\omega}^{i}}{{\omega}_{0}}\right|}^{ni},\text{}if\left({\theta}^{i}0\right)\\ {\theta}_{\mathrm{min}}^{i}-{c}_{i}\cdot {\left|\frac{{\omega}^{i}}{{\omega}_{0}}\right|}^{ni},\text{}if\left({\theta}^{i}\le 0\right)\end{array}$

The parameter ${c}_{i}$ is a scale of exponent function, and parameter $ni$ effects the failure as in Figure 7.The above formulas’ are valid for displacement/rotation criteria and are also force/moment and energy criteria.

- Force (or Moment) Criteria
(
`I`_{fail2}= 2) and Internal Energy Criteria (`I`_{fail2}= 3)Translational ${\delta}^{i}{}_{fail}$ and rotational ${\theta}^{i}{}_{fail}$ failure are:

$${{\displaystyle \sum _{i=1,2,3}\left(\frac{{\delta}^{i}}{{\delta}^{i}{}_{fail}}\right)}}^{2}+{{\displaystyle \sum _{i=4,5,6}\left(\frac{{\theta}^{i}}{{\theta}^{i}{}_{fail}}\right)}}^{2}\ge 1$$${\delta}^{i}{}_{fail}=\{\begin{array}{c}{\delta}_{\mathrm{max}}^{i}+{c}_{i}\cdot {\left|\frac{{v}^{i}}{{v}_{0}}\right|}^{ni},\text{}if\left({\delta}^{i}0\right)\\ {\delta}_{\mathrm{min}}^{i}-{c}_{i}\cdot {\left|\frac{{v}^{i}}{{v}_{0}}\right|}^{ni},\text{}if\left({\delta}^{i}\le 0\right)\end{array}$ and ${\theta}^{i}{}_{fail}=\{\begin{array}{c}{\theta}_{\mathrm{max}}^{i}+{c}_{i}\cdot {\left|\frac{{\omega}^{i}}{{\omega}_{0}}\right|}^{ni},\text{}if\left({\theta}^{i}0\right)\\ {\theta}_{\mathrm{min}}^{i}-{c}_{i}\cdot {\left|\frac{{\omega}^{i}}{{\omega}_{0}}\right|}^{ni},\text{}if\left({\theta}^{i}\le 0\right)\end{array}$

The above formulas’ are valid for displacement/rotation, force/moment and energy.

Here ${\delta}^{i}{}_{\mathrm{max}},{\delta}^{i}{}_{\mathrm{min}}$ ( ${\theta}^{i}{}_{\mathrm{max}},{\theta}^{i}{}_{\mathrm{min}}$ ) are not displacement (rotational angle) criterion, but maximum or minimum force(moment) for

`I`_{fail2}= 2 and internal energy for`I`_{fail2 }= 3.The influence of velocity is also taken into account and the relative velocity coefficient ${c}_{i}$ is relative to force/moment (

`I`_{fail2}= 2) or to internal energy (`I`_{fail2}= 3).