RD-V: 0020 Cantilever Beam

Deflection of a cantilever beam modeled with different meshes and different element formulations.

The subject of this study is to analyze the quality of Radioss quasi-linear simulation using a simple use case. This should give an overview about the trade-off between quality and performance with respect to different modelling techniques. This example deals with the use of Radioss nonlinear solver.

Based on a well-known from literature small example, the set-up of a simple Radioss input deck for the linear-elastic application will be shown: the cantilever-beam.

The beam is clamped on one side and loaded with a concentrated force on the other side. The maximum deflection is used for results comparison. This problem is well known, and results can be easily compared with an analytical solution.

In this example, different mesh techniques are compared: beams, shells, hexahedron elements and tetrahedron (tetra4 and tetra10) elements, as well as different element sizes.

The results are extracted and compared between each other with respect to their mesh size and element formulation. As output, the maximum deflection, number of cycles, calculation time, element stress, and overall error are used. The maximum deflection is compared to the theoretical value.

Options and Keywords Used

/PROP/TYPE1 (SHELL)
/PROP/TYPE3 (BEAM)
/PROP/TYPE14 (SOLID)
/TETRA4
/TETRA10
/BEAM
/BRICK
/SHELL
/CLOAD
/ADYREL
Mesh density

Input Files

The following input file is used in this verification problem:

<install_directory>/hwsolvers/demos/radioss/verification/elements/0020_beam/

Model Description

This example's purpose is to compare different modeling methods for a simple cantilever beam in terms of quality and performance.

A cantilever beam is fixed on the left end and a concentrated load is applied on the right side of the beam.

The material used follows a linear elastic law (/MAT/LAW1) and has the following characteristics.

Initial density: 7.8 x 10^-9 [Mg/mm³]
Young's modulus: 210000 [MPa]
Poisson ratio: 0.3
Thickness: 10 [mm]
Length: 190 [mm]
Width: 10 [mm]
Load case: Fx = 0; Fy = -1000.0; Fz = 0

For the linear problem, the analytical solution gives:(1)

w = \frac{F L^{3}}{3 E I}

Where,

$F$: Force
$L$: Length of the beam
$E$: Young's modulus
$I$: Moment of inertia

The theoretical deflection is determined to w = 13.07 mm.

Simulation Iterations

The beam is modeled with five different elements:

/BEAM
/SHELL
/BRICK
/TETRA4
/TETRA10

Each formulation has particular properties (/PROP). /PROP/TYPE3 (BEAM) describes the beam property for torsion, bending, membrane or axial deformation. Beam elements use the default nonlinear strain formulation (I_smstr = 4). Furthermore, the following settings have to be defined.

Cross section: 100 mm²
Moment of Inertia (bending): 833.33333 mm⁴
Moment of Inertia (torsion): 1666.66666 mm⁴

For shell elements (/PROP/TYPE1 (SHELL)), several element formulations (Q4-shell with I_shell = 1, 2, 3, 4) are used. The results of modeling with QBAT-shell (I_shell = 12) and QEPH-shell (I_shell = 24) are compared.

For the solid mesh, the HA8 formulation (/PROP/TYPE14 (SOLID)) (I_solid = 14, 17, 18 and 24) are investigated. No reduced pressure integration is necessary for implicit computation, as the behavior is elastic (I_cpre = 0).

For Tetra4 elements the formulations I_tetra4 = 1, 1000 are used and for Tetra10 elements the formulations I_tetra10 = 2, 100 are used.

Even though the displacement is small and a linear solver could be used, the nonlinear explicit and implicit solvers are used. The nonlinear implicit solver can be activated by using /IMPL/NONLIN.

The auto-defined adaptive damping (/ADYREL) is used to damp out the vibrations which are typical when a quasi-static problem is solved using an explicit solution.

Results

The tables below provide an overview about maximum deflection in Z-direction compared to the theoretical result. The results are compared between each other with respect to their mesh size, total number of cycles, energy error and difference of deflection compared to the theoretical value.

Note: The calculation times should be considered approximate, since it was calculated on a desktop computer, instead of a dedicated computer server.

The displacement results in the table are reported at the main node of the rigid body attached to the end of the beam where the force is applied.

Explicit Solver

Table 1. Explicit Results of Deflection of a Cantilever Beam Modeled with Beams and Shell Elements
Element	Mesh Size [mm]	Element Formulation	Maximum Deflection in Z-direction [mm]	Calculation Time [s]	Total Number of Cycles	Difference to Theoretical Value [%]
Beam	10	BEAMN3	13.035	78.53	767445	0.2
Beam	5	BEAMN3	13.034	78.53	767445	0.2
Q4	10	`I`_shell = 3	13.032	80.87	607437	0.3
Q4	5	`I`_shell = 3	27.833	80.87	607437	113
QBAT	10	`I`_shell = 12	12.839	154.01	621035	1.7
QBAT	5	`I`_shell = 12	12.922	154.01	621035	1.1
QEPH	10	`I`_shell = 24	12.975	94.97	601316	0.7
QEPH	5	`I`_shell = 24	12.992	94.97	601316	0.6

Table 2. Explicit Results of Deflection of a Cantilever Beam Modeled with Tetra4 and Tetra10 Elements
Element	Mesh Size [mm]	Element Formulation	Maximum Deflection in Z-direction [mm]	Calculation Time [s]	Total Number of Cycles	Difference to Theoretical Value [%]
Tetra4	10	`I`_tetra4 = 1000	3.639	260.04	1034462	72
	5		5.24			60
	2.5		9.168			30
Tetra4	10	`I`_tetra4 = 1	10.353	825.89	1034019	21
	5		11.251			14
	2.5		12.472			5
Tetra10	10	`I`_tetra10 = 1000	13.381	3010.86	3207611	2
	5		13.840			6
	2.5		13.626			4
Tetra10	10	`I`_tetra10 = 2	13.376	1605.92	1500391	2
	5		13.842			6
	2.5		13.623			4

Table 3. Explicit Results of Deflection of a Cantilever Beam Modeled with Hexa8 Elements
Element	Mesh Size [mm]	Element Formulation	Maximum Deflection in Z-direction [mm]	Calculation Time [s]	Total Number of Cycles	Difference to Theoretical Value [%]
Hexa8	10	`I`_solid = 14	12.965	434.18	845372	0.8
	5		12.883			1.4
	2.5		12.934			1.0
Hexa8	10	`I`_solid = 17	15.69	306.27	845442	20
	5		9.915			24
	2.5		11.987			8
Hexa8	10	`I`_solid = 18	12.965	417.45	845373	0.8
	5		12.883			1.4
	2.5		12.934			1.0
Hexa8	10	`I`_solid = 24	13.074	161.43	846398	0.1
	5		12.941			0.9
	2.5		13.03			0.3

Implicit Solver

Table 4. Implicit Results of Deflection of a Cantilever Beam Modeled with Beams and Shell Elements
Element	Mesh Size [mm]	Element Formulation	Maximum Deflection in Z-direction [mm]	Calculation Time [s]	Total Number of Cycles	Difference to Theoretical Value [%]
Beam	10	BEAMN3	13.03	13.68	32	0.3
Beam	5	BEAMN3	13.03	13.68	32	0.3
Q4	10	`I`_shell = 1	13.007	5.05	75	0.4
Q4	5	`I`_shell = 1	13.024	5.05	75	0.3
Q4	10	`I`_shell = 2	13.011	17.07	33	0.4
Q4	5	`I`_shell = 2	13.037	17.07	33	0.2
Q4	10	`I`_shell = 3	13.012	27.72	38	0.4
Q4	5	`I`_shell = 3	13.041	27.72	38	0.2
Q4	10	`I`_shell = 4	13.011	17.64	33	0.4
Q4	5	`I`_shell = 4	13.037	17.64	33	0.2
QBAT	10	`I`_shell = 12	12.959	70.90	64	0.8
QBAT	5	`I`_shell = 12	12.981	70.90	64	0.6
QEPH	10	`I`_shell = 24	12.964	13.95	32	0.8
QEPH	5	`I`_shell = 24	12.979	13.95	32	0.7

Table 5. Implicit Results of Deflection of a Cantilever Beam Modeled with Tetra4 and Tetra10 Elements
Element	Mesh Size [mm]	Element Formulation	Maximum Deflection in Z-direction [mm]	Calculation Time [s]	Total Number of Cycles	Difference to Theoretical Value [%]
Tetra4	10	`I`_tetra4 = 1000	3.64	13.69	32	71
	5		5.24			59
	2.5		9.17			29
Tetra10	10	`I`_tetra10 = 1000	13.369	20.55	32	2
	5		13.84			6
	2.5		13.616			4

Table 6. Implicit Results of Deflection of a Cantilever Beam Modeled with Hexa8 Elements
Element	Mesh Size [mm]	Element Formulation	Maximum Deflection in Z-direction [mm]	Calculation Time [s]	Total Number of Cycles	Difference to Theoretical Value [%]
Hexa8	10	`I`_solid = 14	12.965	14.47	32	0.8
	5		12.89			1.3
	2.5		12.939			1.0
Hexa8	10	`I`_solid = 17	15.671	14.24	32	20
	5		9.907			24
	2.5		11.98			8
Hexa8	10	`I`_solid = 18	12.965	14.30	32	0.8
	5		12.89			1.3
	2.5		12.939			1.0
Hexa8	10	`I`_solid = 24	12.966	14.26	32	0.8
	5		12.89			1.3
	2.5		12.939			1.0

Conclusion

Explicit Solver
Beam: BEAM3N provides good results for the coarse and the fine mesh. The difference of deflection compared to the theoretical value is about 0.2%.
Shells: When the beam is modeled with shell elements, QEPH (I_shell = 24) will be the best element formulation. Compared to Q4-shells (I_shell = 3) and QBAT (I_shell = 12) it delivers the best results regarding mesh size, energy error, and very good precision versus cost. The QEPH formulation gives the same high quality results as the fully-integrated QBAT shell but at a much lower computational cost.
Tetras: I_tetra4 = 1000 (default) elements are too stiff and do not give reasonable results, unless a fine mesh is used. Although a higher computational cost, the I_tetra4 = 1 element formulation provides better results especially when a fine mesh is used.; Tetra10 elements provide good results, but with a higher calculation time compared to tetra4 elements. The I_tetra10 = 2 formulation gives the same results as the the I_tetra10 = 1000 formulation, but with half the computational cost.
Bricks: Hexa8 elements with I_solid = 24 element formulation provides the best results regarding mesh size, and calculation time. It also has very good precision versus cost ratio. The difference of deflection compared to the theoretical value is about 1%. Compared to the other element formulations calculation time is about 2 – 2.5 lower. When used with nonlinear materials, I_solid = 24 requires 3 or more elements through the thickness. For 1 or 2 elements through the thickness, a fully-integrated element like I_solid = 14 or 18 is recommended. The I_solid = 18 element automatically selects the best property settings depending on the material it is used with. The fully-integrated I_solid = 17 element suffers from shear locking which causes it to be too stiff and therefore not recommended.

Implicit Solver
Beam: BEAM3N provides good results for the coarse and the fine mesh. The difference of deflection compared to the theoretical value is about 0.3%.
Shells: Shell elements with different element formulations provide good results regarding mesh size and calculation time. The difference of deflection to the theoretical value is under 1%. The QEPH shell shows best performance vs. precision ratio and is recommended.
Tetras: Tetra10 elements provide good results in terms of quality compared to tetra4 elements, which behave too stiff. The difference of deflection compared to the theoretical value varies between 2 to 6% and depends on the mesh size. For Tetra4 elements, the deflection compared to the theoretical value is to low, but converges with finer mesh size. The I_tetra4 = 1 and I_tetra10 = 2 elements are not supported in implicit analysis.
Bricks: Hexa8 elements with different element formulations provide good results for the coarse and the fine mesh, except for element formulation I_solid = 17 which suffers from shear locking. The difference of deflection compared to the theoretical value is about 1 %. The Hexa8 elements with I_solid = 24 show the best performance regarding precision of results and calculation time.