Response of beam elements to a planar wave

This example illustrates the use of Abaqus/Standard and Abaqus/Explicit to model the response of beam elements with circular cross-sections to a planar linearly increasing wave. The results obtained using Abaqus are compared with those determined theoretically using the equations of Hicks.

This page discusses:

Problem description
Results and discussion
Input files
References
Tables

ProductsAbaqus/StandardAbaqus/Explicit

Problem description

To validate the incident wave loading feature, several beams of increasing diameter are subjected to the same incident wave load, athwartships. From Hicks we have expressions describing the actual load on the beam in terms of the cross-sectional area of the beam and the response as a function of the beam's structural mass and the entrained fluid mass. Therefore, the acceleration of a rigid mass is

\ddot{u} = - μ (r_{0}) \nabla P \frac{1}{ρ_{f}},

where

μ (r_{0}) \equiv \frac{A}{B} \equiv \frac{(C + \bar{C}) π r_{0}^{2} ρ_{f}}{m + c_{0} π r_{0}^{2} ρ_{f}} .

In these expressions, P is the incident fluid pressure, $ρ_{f}$ is the fluid mass density, $c_{0}$ is the fluid speed of sound, C and $\bar{C}$ are area coefficients for the cross-section, $r_{0}$ is the (equivalent) circular radius of the cross-section, and m is the mass of the beam structure. For a beam with a circular cross-section $C \equiv \bar{C} \equiv 0$ .

The loading of immersed beams in Abaqus is achieved using incident wave loading. In this test an unconnected array of eight beam elements is subjected to a plane wave incident at 90° from the plane of the array. The amplitude of the wave is linearly increasing, providing a uniform pressure gradient. The beams have identical structural properties, but their wetted cross-sections vary from 0.3 m to 35.0 m. Consequently, the loads generated on the elements due to the incident wave will vary, and the responses will vary due to the different entrained fluid masses. Both Abaqus/Standard and Abaqus/Explicit are tested.

Results and discussion

The acceleration results from Abaqus/Standard and Abaqus/Explicit are summarized in Table 1. The results agree closely with the theory.

Input files

iw_bfi_bmk_std.inp: Abaqus/Standard analysis.
iw_bfi_bmk_xpl.inp: Abaqus/Explicit analysis.

References

Hicks, A. N., “The Theory of Explosion Induced Hull Whipping,” Naval Construction Research Establishment, Dunfermline, Fife, Scotland, Report NCRE/R579, March 1972.

Tables

Table 1. Finite element results.
$r_{0}$	A	B	$μ (r_{0})$	Theoretical Acceleration (Component)	Abaqus/Standard Acceleration (Component)	Abaqus/Explicit Acceleration (Component)
0.3	579.6	3.262e4	1.777e−2	8.17e−9	8.17e−9	8.17e−9
1.0	6440.0	3.555e4	1.811e−1	8.33e−8	8.33e−8	8.33e−8
3.0	5.796e4	6.131e4	9.453e−1	4.35e−8	4.35e−8	4.35e−8
5.0	1.610e5	1.128e5	1.427	6.56e−7	6.56e−7	6.56e−7
7.0	3.155e5	1.901e5	1.660	7.63e−7	7.63e−7	7.63e−7
9.0	5.216e5	2.931e5	1.779	8.18e−7	8.18e−7	8.18e−7
17.0	1.861e6	9.629e5	1.933	8.89e−7	8.89e−7	8.89e−7
35.0	7.889e6	3.977e5	1.9837	9.12e−7	9.12e−7	9.12e−7