# Dynamic Analysis of Structures Under Blast Loads

### Technical Article

In this article, images of a blast scenario of a remote detonation in RF-DYNAM Pro - Forced Vibrations are displayed and the effects are compared in the linear time history analysis.

#### Basics

A structure must be designed and constructed in such a way that it withstands the possible actions and influences beyond its service life and also meets the required serviceability. In this regard, actions are classified according to their temporal change as follows:

- Permanent actions (for example self-weight)
- Variable actions (for example, live loads, snow and wind loads)
- Unusual actions (for example blast or vehicle impact)

This technical article deals with the extraordinary effects of a blast. An extraordinary action is of short duration and does not occur with any significant probability. However, it can have significant consequences for the stability of a structure.

"A blast is a" suddenly occurring, extremely rapid "oxidation or decay reaction with a sudden increase in temperature and pressure. This leads to a sudden volume expansion of gases and the release of large amounts of energy in a small space (...). The sudden volume expansion causes a pressure wave, which can be described by the model of the detonation wave in the case of an ideal blast (originating from a point source). " [1] In addition to the air blast loading, a blast is caused by other effects due to high temperatures and thrown objects (fragments, debris). accompanying. In this article, the load of a remote detonation is represented as a pure air blast load on a structure, but not other effects of the blast.

#### Air blast loading, remote detonation

The air blast loading can be represented schematically as a pressure-time curve (from [2] ).

The free air shock wave strikes the structure suddenly with peak overpressure. The course includes an overpressure phase that acts on the structure up to the time period t_{d and is} reduced over a negative pressure phase until the ambient air pressure is reached. This exponential approach is often simplified to the overpressure range. In this case, a virtual time t ^{~}_{d} (t ^{~}_{d} <t_{d} ) can be calculated, which describes the approach linearized with the same momentum in terms of amount, but completely neglects the negative pressure phase.

The governing input values for the calculation of the blast are the distance to the blast center R and the explosive mass as the TNT equivalent M_{TNT} . The following formulas refer to the load model developed in [2] . A scaled distance Z is determined from the two input values R and M_{TNT} .

$\mathrm{Z}=\frac{\mathrm{R}}{\sqrt[3]{{\mathrm{M}}_{\mathrm{TNT}}}}0,5\left(\mathrm{Ferndetonation}\right)$

Z. |
Scaled distance [m/kg ^{1/3} ] for Z> 2.8 |

[SCHOOL.SCHOOLORINSTITUTION] |
Distance to explosion center [m] |

M_{TNT} |
Mass of TNT equivalent [kg] |

In the following, the maximum peak overpressure, the positive specific impulse and the shape coefficient are calculated. The shape coefficient has a significant influence on the expression of the vacuum phase.

${\hat{\mathrm{p}}}_{10}={\mathrm{p}}_{0}\xb7\frac{808\xb7\left[1+{\left(\frac{\mathrm{Z}}{4,5}\right)}^{2}\right]}{\sqrt{1+{\left(\frac{\mathrm{Z}}{4,5}\right)}^{2}}\xb7\sqrt{1+{\left(\frac{\mathrm{Z}}{0,32}\right)}^{2}}\xb7\sqrt{1+{\left(\frac{\mathrm{Z}}{1,35}\right)}^{2}}}$

p_{10} |
Maximum peak pressure of remote explosion (Kinney & Graham) [kPa] |

p_{0} |
Ambient air pressure under normal conditions (101.3 [kPa]) |

Z. |
Scaled distance [m/kg ^{1/3} ] |

${\mathrm{i}}^{+}=2,1\xb7\frac{\mathrm{R}}{{\mathrm{Z}}^{2}}$

i ^{+} |
Positive specific impulse [kPa ms] |

[SCHOOL.SCHOOLORINSTITUTION] |
Distance to explosion center [m] |

Z. |
Scaled distance [m/kg ^{1/3} ] for Z> 2.8 |

$\mathrm{\alpha}=1,5\xb7{\mathrm{Z}}^{-0,38}$

α |
shape coefficient |

Z. |
Scaled distance [m/kg ^{1/3} ] for 0.1 <Z <30 |

In the next step, the time duration of the positive pressure action t_{d} as well as the virtual time duration of the positive pressure action t ^{~}_{d} can be calculated.

${\mathrm{t}}_{\mathrm{d}}=\frac{{\mathrm{i}}^{+}}{{\hat{\mathrm{p}}}_{10}}\xb7\left(\frac{{\mathrm{\alpha}}^{2}}{\mathrm{\alpha}-1+{\mathrm{e}}^{-\mathrm{\alpha}}}\right)$

t_{d} |
Duration of the positive pressure action |

i ^{+} |
Positive specific impulse [kPa ms] |

p_{10} |
Maximum peak pressure of remote explosion (Kinney & Graham) [kPa] |

α |
shape coefficient |

e |
Euler's number |

${\stackrel{~}{\mathrm{t}}}_{\mathrm{d}}=\hspace{0.17em}\frac{2\xb7{\mathrm{i}}^{+}}{{\hat{\mathrm{p}}}_{10}}$

t ^{~}_{d} |
Virtual duration of the positive pressure action |

i ^{+} |
Positive specific impulse [kPa ms] |

p_{10} |
Maximum peak pressure of remote explosion (Kinney & Graham) [kPa] |

To determine the reflected pressure-time curve, a reflection factor for the overpressure phase c_{r} and a reflection factor for the negative pressure phase c ^{-}_{r are} determined. The assumption is an infinitely perpendicular reflection surface. For details on the values, see [2] .

${\mathrm{c}}_{\mathrm{r}}=\frac{8\xb7{\hat{\mathrm{p}}}_{10}+14\xb7{\mathrm{p}}_{0}}{{\hat{\mathrm{p}}}_{10}+7\xb7{\mathrm{p}}_{0}}$

c_{r} |
Reflection factor overpressure |

p_{10} |
Maximum peak pressure of remote explosion (Kinney & Graham) [kPa] |

p_{0} |
Ambient air pressure under normal conditions (101.3 [kPa]) |

${\mathrm{c}}_{\mathrm{r}}^{-}=\frac{1,9\xb7\mathrm{Z}-0,45}{\mathrm{Z}}$

c_{r} ^{-} |
Reflection factor negative pressure |

Z. |
Scaled distance [m/kg ^{1/3} ] for Z> 0.5 |

From all determined values, you can then use the load model for the complete reflected pressure-time curve

${\mathrm{p}}_{\mathrm{r}0}\left(\mathrm{t}\right)=\left\{\begin{array}{ll}{\mathrm{c}}_{\mathrm{r}}\xb7{\hat{\mathrm{p}}}_{10}\xb7\mathrm{\phi}\left(\mathrm{t}\right)& \mathrm{t}\le {\mathrm{t}}_{\mathrm{d}}\\ {\mathrm{c}}_{\mathrm{r}}^{-}\xb7{\hat{\mathrm{p}}}_{10}\xb7\mathrm{\phi}\left(\mathrm{t}\right)& \mathrm{t}{\mathrm{t}}_{\mathrm{d}}\end{array}\right.$

p_{r0} (t) |
Load model for the complete reflected pressure-time curve |

c_{r} |
Reflection factor overpressure |

p_{10} |
Maximum peak pressure of remote explosion (Kinney & Graham) [kPa] |

φ (t) |
Load function (constant/linear/exponential approach) |

t_{d} |
Duration of the positive pressure action |

c_{r} ^{-} |
Reflection factor negative pressure |

and selected loading functions, the loading in RF-DYNAM Pro - Forced Vibrations is displayed as time diagrams (functions).

${\mathrm{p}}_{1}\left(\mathrm{t}\right)=\left\{\begin{array}{ll}{\hat{\mathrm{p}}}_{\mathrm{r}0}& \mathrm{t}\le {\mathrm{t}}_{\mathrm{d}}\\ 0& \mathrm{t}{\mathrm{t}}_{\mathrm{d}}\end{array}\right.\phantom{\rule{0ex}{0ex}}{\mathrm{p}}_{2}\left(\mathrm{t}\right)=\left\{\begin{array}{ll}{\hat{\mathrm{p}}}_{\mathrm{r}0}\xb7(1-\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{d}}})& \mathrm{t}\le {\mathrm{t}}_{\mathrm{d}}\\ 0& \mathrm{t}{\mathrm{t}}_{\mathrm{d}}\end{array}\right.\phantom{\rule{0ex}{0ex}}{\mathrm{p}}_{3}\left(\mathrm{t}\right)=\left\{\begin{array}{ll}{\hat{\mathrm{p}}}_{\mathrm{r}0}\xb7(1-\frac{\mathrm{t}}{{\stackrel{~}{\mathrm{t}}}_{\mathrm{d}}})& \mathrm{t}\le {\stackrel{~}{\mathrm{t}}}_{\mathrm{d}}\\ 0& \mathrm{t}{\stackrel{~}{\mathrm{t}}}_{\mathrm{d}}\end{array}\right.\phantom{\rule{0ex}{0ex}}{\mathrm{p}}_{4}\left(\mathrm{t}\right)={\hat{\mathrm{p}}}_{\mathrm{r}0}\xb7(1-\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{d}}})\xb7{\mathrm{e}}^{-\mathrm{\alpha}\xb7\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{d}}}}$

p_{1} (t) |
Load function of constant momentum |

p_{2} (t) |
Load function of linear momentum |

p_{3} (t) |
Load function of linear impulse with virtual time |

p_{4} (t) |
Load function exponential (Friedlander approach) |

t ^{~}_{d} |
Virtual duration of the positive pressure action |

t_{d} |
Duration of the positive pressure action |

e |
Euler's number |

α |
shape coefficient |

p_{r0} |
Fully reflected pressure-time curve |

#### Input in RF-DYNAM Pro - Forced Vibrations

The load functions can be entered as time diagrams in the add-on module. Time diagrams can be defined either transiently, periodically, or directly as a function. They excite the structure at a certain position. The position of the load is defined in static load cases. Almost any load type can be entered here. The static load cases are linked to the time diagrams. This is done in the dynamic load cases. The multiplier k is used to determine the final magnitude of the excitation force.

For the following calculations, a remote blast of M_{TNT} = 1 kg at a distance of R = 10 m is shown. This results in the following values when using the parameterized input.

In the parameter list stored in the RFEM model file, you only have to adjust the values for R and M_{TNT} . Insofar as they are within the value range for the scaled distance of 5 <Z <30, the calculation model presented in can be used.

With the values calculated in the parameter list, the entries for the four displayed time diagrams are made in the add-on module as follows. As in many numerical programs, the pressure is not applied directly at t = 0 s, but in our example from t = 0.01 s. It is useful to use nested If functions to represent the desired functions.

In order to compare the four functions in one file, four identical subsystems are analyzed in a dynamic load case. Each subsystem is assigned a load case that loads the front surface with 1 kN/m². Each subsystem is assigned a different time diagram, thus a different loading function.

Finally, the Rayleigh damping of the subsystems is entered, which can be determined from the two dominant mode shapes of the subsystems in the considered direction.

#### Results

After calculating and determining the results, you can compare the four load functions and their effects on the subsystems in the file. In this article, only acceleration and displacement in the global X-direction are briefly compared. The evaluation of the results is possible in the program interface in the Results navigator. Here, you can display various result values for the calculated time steps. In addition, after analyzing a dynamic load case, you can access the time history diagram by displaying and comparing further values of points. The values in the middle of the front surfaces are considered here.

The application of the constant momentum p_{1} (t) shows the greatest values as expected. The two linearized distributions p_{2} (t) and p_{3} (t) are very similar, with the values of p_{2} (t)> p_{3} (t) as expected. Ultimately, the distribution of p_{4} (t) shows that a consideration of the negative pressure phase should not be neglected and that larger values act on the structure compared to the common, linearized approach of p_{3} (t).

#### Conclusion

Displaying the real pressure-time curve of a remote blast using time diagrams in RF-DYNAM Pro - Forced Vibrations is an effective way to determine the effects of the overpressure and underpressure phases on the structure. The parameterization of the model makes it possible to display and compare different blast scenarios by adjusting R and M_{TNT} .

#### Keywords

Blast Pressure-time course Time diagram Friedlander Time history analysis

#### Reference

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