Explicit Example

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

The purpose of this example is to study a snap-thru problem with a single instability. Thus, a structure that will bend when under a load is used. The results are compared to the references solution. 1

Options and Keywords Used

  • Explicit solver
  • T3 Shell
  • Elasticity and quasi-static analysis
  • Stability, snap-through problem, and limit load
  • Boundary conditions (/BCS)
  • Imposed velocity (/IMPVEL)
  • Rigid body (/RBODY)

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-1in 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.

rad_ex_fig_2-3
Figure 1. Boundary Conditions

Input Files

The input files used in this example include:
Explicit solvers
<install_directory>/hwsolvers/demos/radioss/examples/02_Snap-through/Explicit_solver/SNAP_EXP*

Model 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 Figure 2, with the following dimensions:
l
254 mm
R
2540 mm
Shell thickness
t = 12.7 mm
θ
0.1 rad

rad-ex-fig_2-6
Figure 2. Geometrical Data of the Problem
The material used follows a linear elastic law with the following characteristics:
Material Properties
Initial density
7.85x10-3 [ g m m 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada WcaaqaaiaadEgaaeaacaWGTbGaamyBamaaCaaaleqabaGaaG4maaaa aaaakiaawUfacaGLDbaaaaa@3BBC@
Young's modulus
3102.75 [ MPa ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaai Gac2eacaGGqbGaaiyyaaGaay5waiaaw2faaaaa@3BE6@
Poisson ratio
0.3

Model Method

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 (Figure 3)

rad_ex_fig_2-2
Figure 3. T3 Mesh
Shell Properties
Thickness
12.7 mm
BT Elasto-plastic Hourglass formulation
Ishell= 3

Results

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.

rad_ex_fig_2-4
Figure 4. Load P versus Displacement of Point C: Snap-thru Instability

The displacement of point C is indicated in its absolute value. The curve illustrates the characteristic behavior of the instability of a snap-thru. Beyond the limit load, an infinite increase in load Δ F z will cause a considerable increase in displacement Δ 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-thru, shows that the deformed configuration becomes more rigid.

rad_ex_fig_2-5
Figure 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.
Deformed Mesh (profile view) - Displacement Norm
rad_ex_initial_config

Initial configuration
rad_ex_start_of_snap-thru

Start of snap-thru
rad_ex_lg_motion

Large motion phase
rad_ex_stable_config

Stable configuration
rad_ex_struct_rigidity

Loading with a new structural rigidity