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Example 38 - Buckling of L-Shaped Beam

Summary

This is a very famous example demonstrating the coupling of bending-torsion in lateral buckling during a nonlinear implicit analysis. The piloting techniques and some other hits are explained. Very accurate results are obtained with RADIOSS implicit.

Title

Buckling of a L-Shape beam

Number

38.1

Brief Description

The buckling of a thin L-shape beam is studied. This example is one of the most severe tests, due to the high dependence of the behavior to the torsion-bending coupling. The Eigenvalue computation method is used.

Keywords

• Linear and nonlinear implicit solver, timestep control by arc-length method, convergence

• Euler buckling modes computation, static linear analysis, critical loads

• Shells Q4, BATOZ formulation

RADIOSS Options

• Concentrated load (/CLOAD)

• Implicit options (/IMPL)

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• Euler buckling modes computation (/IMPL/BUCKL)

• Rigid body (/RBODY)

Compared to / Validation Method

• References (numerical simulations)

Input File

Nonlinear implicit computation:

Subcase_1: <install_directory>/demos/hwsolvers/radioss/38_Euler_Buckling_modes/L_Shaped_b

eam/Nonlinear_implicit/Px_positive/data/LAME_EQUERRE*

Subcase_2: <install_directory>/demos/hwsolvers/radioss/38_Euler_Buckling_modes/L_Shaped_b

eam/Nonlinear_implicit/Px_negative/data/LAME_EQUERRE*

Euler buckling modes computation:

Subcase_1: <install_directory>/demos/hwsolvers/radioss/38_Euler_Buckling_modes/L_Shaped_b

eam/Euler_modes/Px_positive/data/LAME_EQUERRE*

Subcase_2: <install_directory>/demos/hwsolvers/radioss/38_Euler_Buckling_modes/L_Shaped_b

eam/Euler_modes/Px_negative/data/LAME_EQUERRE*

RADIOSS Version

51h

Technical / Theoretical Level

Medium

Overview

Aim of the Problem

The purpose of this example is to study the buckling mode of a thin structure using the RADIOSS solver. A shell hypothesis is adopted. The simple test used is particularly severe for shell element, due to the torsion-bending coupling problem. It enables readers to appreciate the qualities/restrictions of the shell element formulations in RADIOSS. The results are compared to reference [1].

Two analyses are performed:

• A nonlinear implicit computation

• A linear stability analysis to determine buckling modes

In the nonlinear implicit approach, the arc-length method is used to overpass the lateral buckling point. The pre- and post-critical problems are considered. The predicted critical loads and buckling modes are obtained from a linear stability analysis.

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Physical Problem Description

The thin L-shaped beam is clamped at one end and subjected to a concentrated load Px in its plane. The two beams constituting the structure are stiff in one direction and very flexible in the other direction. A perturbation load, Py is applied in Y-direction in order to initialize buckling.

Units: mm, ms, g, N, MPa

Geometrical characteristics:

L = 240 mm

b = 30 mm

t = 0.6 mm

Loading:

Px = incremental load

PY = 0.001 Px: perturbation load

Fig 1: Definition of the problem.

The material undergoes a linear elastic law (/MAT/LAW1) with the following characteristics:

• Initial density: 2.8x10-3 g.mm-3

• Young modulus: 71240 MPa

• Poisson ratio: 0.3

Two subcases are considered:

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Analysis, Assumptions and Modeling Description

Modeling Methodology

The mesh is a regular shell mesh (Fig 2). The element size is about 10mm. There are 147 4-node shell elements. BATOZ formulation is used (type 1, Ishell = 12) with one integration point through the thickness, as the material is elastic. The properties are isotropic.

Fig 2: Mesh of the structure.

RADIOSS Options Used

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The clamping condition is imposed by fixations of the three components of displacement and rotation.

A rigid body with no mass is coupled to the four points at the low end in order to apply the load. An automatic master node is chosen. The center of gravity is computed using the master and slave node coordinates and the master node is moved to the center of gravity.

Two concentrated loads, Px and Py are applied on the master node of the rigid body for nonlinear analysis. The concentrated load, Px in X-direction is defined as the incremental load (Px increases and reaches a maximum value Px = 5N). The concentrated load, Py in Y-direction is introduced as perturbation load for initializing the buckling of the beam (Py = 0.001 x Px).

Fig 3: Boundary conditions and loading applied on structure

The loads are considered to be monotonous increasing in time. Loading is proportional to time.

Fig 4: Monotonous increasing time function relating to Px

The nonlinear implicit following parameters used are:

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Implicit type: Static nonlinear

Nonlinear solver: Modified Newton-Raphson

Tolerance: 10-3

Update of stiffness matrix: 5 iterations maximum

Time step control method: Arc-length

Initial time step: 2 ms

Minimum time step (stop): 10-3 ms

Maximum time step: 2 ms

Wished convergence iteration

number: 6

Maximum convergence iteration

number: 20

Decreasing time step scale factor: 0.8

Increasing time step scale factor: 1.006

For the linear solver the following parameters used are:

Linear solver: Direct Cholesky

Maximum iterations number: System dimension (NDOF)

Convergence criteria: Relative residual on energy

Tolerance for stop criteria: Machine precision

The input implicit options (/IMPL) set in the Engine file (0001.rad) are:

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Specific strategy used:

• Low scale factor for increasing time step (by default 1.1).

• Time step control using arc-length method.

• Relative residual computed on energy (by default on force).

The implicit Euler buckling modes computation is launched by the keyword /IMPL/BUCKL, defined in the

Engine file (0001.rad). The modes are sorted by increasing critical loads.

Number of computed modes: 10

Maximum number of Arnoldi

iterations: 300

Factor for compute initial subspace: 4

Maximum factor for subspace

dimension: 16

Solver for computation: Linear solver based on sparse LU decomposition

The computation of buckling modes follows a linear implicit computation.

The implicit method used is:

Implicit type: Static linear

Linear solver: Iterative

Precondition method: Factored Approximate Inverse

Stop criteria: Relative residual of preconditioned matrix

The implicit parameters used in the Engine file (0001.rad) are:

The critical loads are given in the Engine listing file (0001.lis). The critical loads are multiplying factors

of the loading.

Simulation Results and Conclusions

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Reference

The RADIOSS results of the nonlinear analysis and the linear buckling modes study are compared to the reference.

Table 1: Reference results

Subcase PX > 0 Subcase PX < 0

Critical load = 1.163 N Critical load = - 0.7005 N

Implicit Nonlinear Computation Results

Animations

The nonlinear results give the post-critical configurations of the structure for Px > 0 and Px < 0.

Fig 5: Post-critical behavior of the structure for Px positive and Px negative.

The critical loads are extracted from the load displacement curves. The low time step (10-2 - 10-3 ms) allow you to obtain, with accuracy, the nonlinear path including the pre-critical, the critical and the post-critical states.

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Fig 6: Y-displacement of the master node of the rigid body. Subcase Px > 0.

Fig 7: X-displacement of the master node of the rigid body. Subcase Px > 0.

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Fig 8: Y-displacement of the master node of the rigid body. Subcase Px < 0.

Fig 9: X-displacement of the master node of the rigid body. Subcase Px < 0.

Convergence and Timestep

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The buckling of the structure causes high nonlinearities, due to the large lateral displacement of the corner point. Therefore, a low timestep is required to pass the critical point. The convergence is insured by setting a low scale factor for increasing timestep during timestep control. The factor is set to 1.006.

Fig 10: Timestep control for the nonlinear computations.

In both subcases, the timestep remains constant at the maximum value set by you (2ms). Around the instability point, timestep decreases to reach a limit point, before increasing globally.

Euler Buckling Modes Computation Results

Static Linear Results (Preliminary Computation)

Fig 11: Trace stress superposed to displacement vectors.

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Euler Buckling Modes Computation

Critical loads are stored in the Engine listing file (0001.lis). Extractions are shown below:

Table 2: First critical load computed by RADIOSS

Subcase Px > 0 Subcase Px < 0

1.195 N - 0.7053 N

The first computed buckling mode is shown in the following figure for Px > 0 and Px < 0. Note that the amplitude of the deformed configuration is arbitrary.

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Fig 12: First buckling mode (critical loads: 1.195 N and 0.7053 N).

Conclusion

Table 3 compares results obtained by RADIOSS to reference [1].

Table 3: Simulation results compared to reference

Critical load (N)

Subcase Px > 0 Subcase Px < 0

Nonlinear implicit computation 1.17 N - 0.7 N

Linear Euler buckling computation

1.195 N - 0.7053 N

Reference 1 1.163 N - 0.7005 N

The linear and nonlinear approaches provide good results in this test and are in accordance with the reference. The Euler buckling modes computation allows you to obtain a fast and accurate solution about critical load values, due to the linear approach being faster than a nonlinear implicit computation. Both simulations are complementary for studying the critical and post-critical behaviors of a structure.

Reference

[1] Argyris, J.H.; Hilpert, O.; Melejannakis, G.A.; Scharpf, D.W.; "On the geometrical stiffness of the beam in space – A consistence V.W. Approach" CNAME, vol 20., P. 105-131, 1979.