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Example 2 - Snap-through Roof (explicit / implicit)

Summary

A snap-through problem is studied on a shallow cylindrical roof upon which an imposed velocity is applied at its mid-point. The characteristic curve, caused by the limit load and achieved by simulation is compared to a reference. This example is considered a static problem.

Only one-quarter of the structure is taken into consideration and adequate boundary conditions are applied on the model sides.

The problem is solved using two different approaches:

• An analysis by an explicit solver

• An analysis by an implicit solver

The implicit strategy uses the arc-length method with a time step limitation. The RADIOSS implicit options are defined in the modeling description.

The simulations using explicit and implicit methods provide accurate results with a good evaluation of the limit load experimentally observed. A time step control with a low value is required in order to describe the nonlinear path of the load displacement curve. Both computations converge toward a single solution.

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EXPLICIT

Title

Snap Roof - Explicit

Number

2.1

Brief Description

An imposed velocity is applied onto a shallow cylindrical roof at its midpoint. The analysis uses an explicit approach.

Keywords

• Explicit solver

• T3 Shell

• Elasticity, quasi-static analysis

• Stability, snap-through problem, limit load

RADIOSS Options

• Boundary conditions (/BCS)

• Imposed velocity (/IMPVEL)

• Rigid body (/RBODY)

Compared to / Validation Method

• Experimental results

Input File

Explicit solver: <install_directory>/demos/hwsolvers/radioss/02_Snap-through/Explicit_solver/SNAP_EXP*

RADIOSS Version

51e

Technical / Theoretical Level

Beginner

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Overview

Aim of the Problem

The purpose of this example is to study a snap-through problem with a single instability. Thus, a structure that will bend when under a load is used. The results are compared to the references contained in: Finite Element Instability Analysis of Free Formed Shells. Report 77−2, 1977, Norwegian Institute Of Technology, Trondheim, HORRIGMOE G.

This static analysis is performed with an explicit approach.

Physical Problem Description

A shallow cylindrical roof, pinned along its straight edges upon which an imposed velocity is applied at its mid-point.

Units: mm, ms, g, N, MPa

Geometrical data are provided in Fig 1, with the following dimensions:

• l = 254 mm

• R = 2540 mm

• Shell thickness: t = 12.7 mm

• = 0.1 rad

Fig 1: Geometrical data of the problem

The material used follows a linear elastic law and has the following characteristics:

• Initial density: 7.85x10-3 g/mm3 • Young modulus: 3102.75 MPa • Poisson ratio: 0.3

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Analysis, Assumptions and Modeling Description

Modeling Methodology

The structure is considered perfect, having no defects. To take account of the symmetries, only a quarter of the shell is modeled (surface ABCD).

A regular mesh with a total of 72 3-node shells (Fig 2)

Fig 2: T3 mesh

The shells have the following properties:

• Thickness 12.7 mm

• BT Elasto-plastic Hourglass formulation (Ishell = 3).

RADIOSS Options Used

Node time histories do not indicate the pressure output. In order to obtain such output at point C, a rigid body must be created at this point. Point C has a constant imposed velocity of -0.01 ms-1 in the Z direction. Its displacement is linked proportionally to time.

Boundary conditions are:

• Edge BC is fixed in an X translation, and in Y and Z rotations (symmetry conditions).

• Edge CD is fixed in a Y translation, and in X and Z rotations (Idem).

• Edge DA is fixed in X, Y, Z translations, and in X and Z rotations.

• Point C is fixed in X, Y translations, and in X, Y, Z rotations.

Fig 3: Boundary conditions

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Simulation Results and Conclusions

Curves and Animations

Only a quarter of the total load is applied due to the symmetry. Therefore, force Fz of the rigid body, as indicated in the Time History, must be multiplied by 4 in order to obtain force, P.

Figure 4 represents a characteristic load displacement curve for a snap-through. This diagram plots the reaction at point C of the shell as a function of its vertical displacement.

Fig 4: Load P versus displacement of point C: snap-through instability.

The displacement of point C is indicated in its absolute value. The curve illustrates the characteristic behavior of a snap-through instability. Beyond the limit load, an infinite increase in load FZ will cause a considerable increase in displacement q due to the collapsing of the shell.

The first extreme defines the limit load =2208.5 N (displacement of point C = 10.5 mm).

The increase in the curve slope after the snap-through shows that the deformed configuration becomes more rigid.

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Fig 5: Comparison between a reference curve and a curve obtained using RADIOSS

The difference between the two curves is approximately 10% for reduced displacements (up to 5 mm) and slightly more (15%) for the higher nonlinear part of the curve (between 5 and 20 mm). For displacements exceeding 20 mm, the curves are shown much closer together.

The accuracy of the RADIOSS results in comparison to those obtained from the reference is ideal for this explicit approach.

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Deformed Mesh (profile view) – Displacement Norm

Initial configuration

Start of snap-through

Large motion phase

Stable configuration

Loading with a new structural rigidity

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IMPLICIT

Title

Snap Roof - Implicit

Number

2.2

Brief Description

A shallow cylindrical roof upon which an imposed velocity is applied at its mid-point. Analysis uses an implicit approach.

Keywords

• Implicit solver, time step control by arc-length method

• Static nonlinear analysis

• Stability, snap-through, limit load

• T3 Shell

RADIOSS Options

• Boundary conditions (/BCS)

• Implicit options (/IMPL)

• Imposed velocity (/IMPVEL)

• Rigid body (/RBODY)

Compared to / Validation Method

• Experimental results

Input File

Implicit solver: <install_directory>/demos/hwsolvers/radioss/02_Snap-through/Implicit_solver/SNAP_IMP*

RADIOSS Version

51e

Technical / Theoretical Level

Advanced

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Overview

Aim of the Problem

The purpose of this example is to study a snap-through problem with a single instability. Thus, a structure that will bend when under a load will be used. The results are compared to a reference solution [1]. This analysis is performed using an implicit approach. It illustrates an implicit strategy using an arc-length method.

Physical Problem Description

A shallow cylindrical roof, pinned along its straight edges, upon which an imposed velocity is applied at its mid-point.

Units: mm, ms, g, N, MPa

Geometrical data are indicated in Fig 6, with the following dimensions:

• l = 254 mm

• R = 2540 mm

• Shell thickness: t = 12.7 mm

• = 0.1 rad

Fig 6: Geometrical data of the problem

The material used follows a linear elastic law and has the following characteristics:

• Initial density: 7.85x10-3 g/mm3

• Young modulus: 3102.75 MPa

• Poisson ratio: 0.3

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Analysis, Assumptions and Modeling Description

Modeling Methodology

The modeling problem described in the explicit study remains unchanged.

The implicit computation requires specific implicit parameters that must be defined in the Engine file D01

using the options beginning with /IMPL.

Fig 7: Description of the problem (one quarter of the shell is modeled)

The imposed velocity is considered using the implicit method. Thus, the constant input curve is converted into an imposed displacement according to the computation time.

Fig 8: Imposed velocity curve

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RADIOSS Options Used

The limit point causes major nonlinearities. Therefore, a static nonlinear analysis is performed using the arc-length displacement strategy. The time step is determined by a displacement norm control. In order to exceed the limit point characterized by a null tangent on the load displacement curve and to describe the increasing and decreasing parts of the nonlinear path, a small time step is required, which is ensured by setting a maximum value.

The nonlinear implicit parameters used are:

Implicit type: Static nonlinear Nonlinear solver: Modified Newton Tolerance: 2x10-4 Update of stiffness matrix: 5 iterations maximum Time step control method: Arc-length Initial time step: 10 ms Minimum time step: 1 ms Maximum time step: 10 ms Desired convergence iteration number: 6 Maximum convergence iteration number: 20 Decreasing time step factor: 0.8 Maximum increasing time step scale factor: 1.1 Arc-length: Automatic computation Spring-back option: no

A solver method is required to resolve Ax=b in each iteration of a nonlinear cycle. It is defined in /IMPL/SOLVER. The linear implicit options used are:

Linear solver: Preconditioned Conjugate Gradient Precondition methods: Factored approximate Inverse Maximum iterations number: System dimension (NDOF) Stop criteria: Relative residual in force Tolerance for stop criteria: Machine precision

The input implicit options set in D01 are:

/IMPL/PRINT/NONL/-1

/IMPL/SOLVER/1 5 0 3 0.0 /IMPL/NONLIN 5 2 0.20e-3

/IMPL/DTINI 10 /IMPL/DT/STOP 1 10 /IMPL/DT/2 6 .0 20 0.8 1.1

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Please refer to RADIOSS Starter Input for more details about implicit options.

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Simulation Results and Conclusions

Curves and Animations

Only a quarter of the total load is applied due to the symmetry. Thus, force Fz of the rigid body as indicated in the Time History must be multiplied by 4 in order to obtain force, P.

Figure 9 represents the characteristic load displacement curve for a snap-through. This diagram plots the reaction at point C of the shell as the function of its vertical displacement. The implicit results are compared with the experimental data.

Fig 9: Load P versus displacement of point C.

For a time step equal to or less than 10 ms (maximum value set in the implicit /IMPL/DT/STOP option), agreement with RADIOSS is achieved, with good results obtained using the reference. Accuracy is improved by decreasing the maximum time step, even though the CPU time is increased.

Fig 10: Deformed configurations during the snap-through.

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COMPARISON BETWEEN IMPLICIT AND EXPLICIT RESULTS

The load displacement curves achieved through implicit computations (time step limit set to 10 ms) and explicit computations are very close. A maximum time step of 100 ms does not allow the nonlinear path of the load displacement curve to be described accurately. However, the final static solution is correct.

Fig 11: Load displacement curve obtained by implicit and explicit solvers.

Comparison of the computation time between the explicit and implicit (maximum time step set to 10 ms) approaches is shown in the table below:

Implicit solver Explicit solver

Normalized CPU 1 2.45

Cycles (normalized)

1 237

In comparison with the implicit computation, which uses a maximum time step of 10 ms, the saved CPU time using a maximum time step fixed at 100 ms, approximately corresponds to factor 4.

Reference

[1] Finite Element Instability Analysis of Free Formed Shells. Report 77−2, 1977, Norwegian Institute Of Technology, Trondheim, HORRIGMOE G.