Steady-state transport analysis

This example verifies the steady-state transport analysis capability in Abaqus. The verification concentrates on frictional effects, inertia effects, and material convection. Frictional effects are verified by comparing results obtained with Abaqus to results published by Faria (1989). Inertia effects are verified by comparing special cases of steady-state transport analyses with results obtained from an Abaqus analysis where centrifugal loads are applied using a distributed load with load type CENT. Material convection is verified by comparison with a transient Lagrangian analysis.

This page discusses:

Frictional effects
Inertia effects
Material convection effects

ProductsAbaqus/Standard

Frictional effects

Problem description

In this series of tests the free rolling angular velocity, $ω$ , of a circular disk in contact with a flat rigid surface is calculated for different disk geometries, contact pressures, friction coefficients, material models, and element types. The ground velocity is specified as either a straight-line translational velocity of $v_{0}$ = 2.0 or as a cornering angular velocity of $Ω$ = 0.02. By specifying a large cornering radius, $R_{c}$ = 100.0, straight line rolling with velocity $v_{0} \approx Ω R_{c}$ = 2.0 is recovered. The results obtained with Abaqus are compared to numerical results published by Faria (1989).

The model consists of a ring with outer radius $R_{o}$ = 2.0 and variable inner radius $R_{i}$ . Three different geometries ( $R_{i}$ = 0.2, 1.0, 1.7) are considered. The model is fully fixed on the inside, and plane strain boundary conditions are imposed along the axial direction.

Two material models are considered: a linear elastic material with E = 800.0 and $ν$ = 0.3 and an incompressible hyperelastic material with $C_{1}$ = 80.0 and $C_{2}$ = 20.0. The friction coefficients considered are $μ$ = 0.02 and $μ$ = 0.2. The first analysis step is a static analysis where the rigid surface is displaced a distance $Δ$ = 0.05 or $Δ$ = 0.1 to establish a contact pressure. The friction coefficient during this step is held constant at zero. This step is followed by a steady-state transport analysis where the ground traveling velocity and spinning angular velocity are applied and the friction coefficient is ramped to its final value.

The problem is discretized with different types of three-dimensional elements. The models that are discretized with first-order elements use 34 element divisions along the circumference and 5 element divisions in the radial direction. The second-order and cylindrical element models use 18 elements along the circumference and 3 elements in the radial direction. All the models are discretized with one element in the axial direction. A first-order finite element mesh for the case $R_{i}$ = 1.0 is shown in Figure 1.

Results and discussion

Table 1 and Table 2 compare the free rolling angular velocity, $ω$ , obtained from the Abaqus simulation with the reference solution. The results presented in Table 2 are obtained using C3D8RH elements.

Additional frictional tests

Additional verification tests are performed to verify contact between a spinning deformable body and a spinning rigid body. In all these tests the deformable body uses the properties and discretization described earlier. The rotating rigid body is in contact either with the inside surface of the deformable body (such as in the case where a tire is mounted on a rigid rim) or with the outside surface of the deformable body (such as in the case where a tire is in contact with a rotating drum). No reference solutions are available for the case where the rigid body is in contact with the inside surface of the deformable body. By specifying a large radius for the rigid body in the case where a rigid spinning drum is in contact with the outside surface of the deformable body, straight line rolling is recovered. We selected a rigid body radius of $R_{r}$ = 1000.0 and an angular velocity of $Ω$ = 0.002, which corresponds to straight line rolling with a velocity $v_{0} \approx Ω R_{r}$ = 2.0.

Another verification test is performed to verify contact between a rolling gear-like thick cylinder with an outer radius of 8.5 and a flat rigid surface. The model is generated by revolving a single three-dimensional 15° sector about the symmetry axis. The gear-like cylinder travels at a ground velocity of 2.7778 with an angular velocity varying from 0.2 to 0.5. The results are compared to those obtained from a transient Lagrangian analysis.

Input files

pstca4shhfa.inp: Axisymmetric model using CAX4H elements and a hyperelastic material.

pstc38shhfs.inp: C3D8H elements, hyperelastic material, $R_{i}$ = 1.0, $Δ$ = 0.1, $μ$ = 0.02, straight line rolling with $v_{0}$ = 2.0 (requires two-dimensional input file pstca4shhfa.inp).
pstca4syhfa.inp: Axisymmetric model using CAX4RH elements and a hyperelastic material.
pstc38syhfs.inp: C3D8RH elements, hyperelastic material, $R_{i}$ = 1.0, $Δ$ = 0.05, $μ$ = 0.2, straight line rolling with $v_{0}$ = 2.0 (requires two-dimensional input file pstca4syhfa.inp).
pstca8sfefa.inp: Axisymmetric model using CAX8H elements and an elastic material.
pstc3ksfefs.inp: C3D20 elements, elastic material, $R_{i}$ = 0.2, $Δ$ = 0.10, $μ$ = 0.02, straight line rolling with $v_{0}$ = 2.0 (requires two-dimensional input file pstca8sfefa.inp).
pstca8srefa.inp: Axisymmetric model using CAX8R elements and an elastic material.
pstc3ksrefs.inp: C3D20R elements, elastic material, $R_{i}$ = 1.7, $Δ$ = 0.05, $μ$ = 0.02, straight line rolling with $v_{0}$ = 2.0 (requires two-dimensional input file pstca8srefa.inp).
pstca4siefa.inp: Axisymmetric model using CAX4I elements and an elastic material.
pstc38siefc.inp: C3D8I elements, elastic material, $R_{i}$ = 10.2, $Δ$ = 0.05, $μ$ = 0.02, cornering with $Ω$ = 0.02 and $R_{c}$ = 100.0 (requires two-dimensional input file pstca4siefa.inp).
pstca3shhfa.inp: Axisymmetric model using CAX3H elements and a hyperelastic material.
pstc36shhfc.inp: C3D6H elements, hyperelastic material, $R_{i}$ = 1.0, $Δ$ = 0.10, $μ$ = 0.02, cornering with $Ω$ = 0.02 and $R_{c}$ = 100.0 (requires two-dimensional input file pstca3shhfa.inp).

Additional frictional tests:

pstc38shhfd.inp: Contact between a rigid drum and the outside surface of a deformable body, C3D8H elements; similar to pstc38shhfs.inp (requires two-dimensional input file pstca4shhfa.inp).
pstc38syhfd.inp: Contact between a rigid drum and the outside surface of a deformable body, C3D8RH elements; similar to pstc38syhfs.inp (requires two-dimensional input file pstca4syhfa.inp).
pstca3shhfr.inp: Axisymmetric model using CAX3H elements and a hyperelastic material.
pstc36shhfr.inp: Contact between a rigid “rim” and the inside surface of a deformable body and contact between a flat rigid foundation and the outside surface of a deformable body, C3D6H elements; similar to pstc36shhfc.inp (requires two-dimensional input file pstca3shhfr.inp).
pstcc12shhfs.inp: CCL12H elements, hyperelastic material, $R_{i}$ = 1.0, $Δ$ = 0.1, $μ$ = 0.02, straight line rolling with $v_{0}$ = 2.0 (requires two-dimensional input file pstca4shhfa.inp).
sstransp_per_hyper_preload.inp: 3D sector model using C3D8H elements and a hyperelastic material with viscosity. Preloaded with an internal pressure.
sstransp_per_hyper_rolling.inp: Contact between a flat rigid surface and a gear-like deformable cylinder (requires three-dimensional input file sstransp_per_hyper_preload.inp).

References

Faria, L. O., Tire Modeling by Finite Elements, Ph.D. dissertation, The University of Texas at Austin, 1989.

Tables

Table 1. Comparison of Abaqus results with reference solutions for the free rolling angular velocity.
Input file	Reference solution	Abaqus	% Difference
pstc38shhfs.inp	0.95009	0.94635	0.39
pstc38syhfs.inp	0.98006	0.98213	0.21
pstc3ksfefs.inp	1.02970	1.02726	0.24
pstc3ksrefs.inp	1.00297	1.00283	0.01
pstc38siefc.inp	1.02180	1.00674	1.47
pstc36shhfc.inp	0.95195	0.94568	0.66

Table 2. C3D8RH results.
Material type	$Δ$	$R_{i}$	$μ$	Reference solution	Abaqus	% Difference
Hyperelastic	0.05	0.2	0.02	0.99349	0.99219	0.13
		1.0	0.02	0.97977	0.98053	0.08
		1.7	0.02	0.87183	0.84974	2.53
		1.0	0.20	0.98066	0.98212	0.15
	0.10	0.2	0.02	0.98558	0.98422	0.14
		1.0	0.02	0.95009	0.95059	0.05
		1.7	0.02	0.73057	0.65790	9.95
		1.0	0.20	0.95195	0.95100	0.10
Linear elastic	0.05	0.2	0.02	1.02180	1.02332	0.15
		1.0	0.02	1.02415	1.02574	0.16
		1.7	0.02	1.00297	1.00263	0.03
	0.10	0.2	0.02	1.02970	1.03410	0.43
		1.0	0.02	1.02810	1.02872	0.06
		1.7	0.02	0.99156	0.99542	0.39

Figures

Inertia effects

Problem description

In this series of tests the effects of inertia on a free spinning and/or cornering structure are verified for different element types and angular velocities. An incompressible hyperelastic material with $C_{1}$ = 80.0, $C_{2}$ = 20.0, and $ρ$ = 0.036 is used. The model consists of a ring with outer radius $R_{o}$ = 2.0 and inner radius $R_{i}$ = 1.0. Each input file contains two models with identical geometry: one model is loaded using a distributed centrifugal load (load type CENT) and serves as the reference solution; the loading on the other model is caused by steady-state rolling inertia effects.

The problem is discretized with different types of three-dimensional elements. The models that are discretized with first-order elements use 24 element divisions along the circumference and 2 element divisions in the radial direction. The second-order and cylindrical element models use 12 elements along the circumference and 1 element in the radial direction. All the models are discretized with 1 element in the axial direction.

The models are fully fixed on the inside. Plane strain boundary conditions are imposed along the axial direction in the first step. Since the material is incompressible, the loading does not give rise to any deformation.

Results and discussion

For the straight line rolling tests, the results match the reference solution. For the cornering tests without free spinning, the results match the reference solution. For the cornering tests with free spinning, the results do not match the reference solution because the steady-state inertia loading includes Coriolis effects due to a spinning wheel in a rotating reference frame; these effects are not accounted for in the reference solution.

Input files

pstc36shhic.inp: C3D6H elements; requires two-dimensional input file pstca3shhia.inp. Cornering only; no free spinning.
pstca4syhia.inp: Axisymmetric model using CAX4RH elements and a hyperelastic material.
pstc38syhic.inp: C3D8RH elements; requires two-dimensional input file pstca4syhia.inp. Cornering only; no free spinning.
pstc38shhic.inp: C3D8RH elements; requires two-dimensional input file pstca4shhia.inp. Cornering and free spinning.
pstc3kshhic.inp: C3D20H elements; requires two-dimensional input file pstca8shhia.inp. Cornering and free spinning.
pstca8yhhia.inp: Axisymmetric model using CAX8RH elements and a hyperelastic material.
pstc3ksyhic.inp: C3D20RH elements; requires two-dimensional input file pstca8yhhia.inp. Cornering and free spinning.
pstm34srhic.inp: M3D4R elements; requires two-dimensional input file pstma2srhia.inp. Cornering and free spinning.
pstm38srhic.inp: M3D8R elements; requires two-dimensional input file pstma3srhia.inp. Cornering only; no free spinning.
pstsf4shhic.inp: S4R elements; requires two-dimensional input file pstsa2shhia.inp. Cornering only; no free spinning.
psts68sheic.inp: S8R elements; requires two-dimensional input file pstsa3sheia.inp. Cornering and free spinning.

Straight line rolling tests:

pstca3shhia.inp: Axisymmetric model using CAX3H elements and a hyperelastic material.
pstc36shhis.inp: C3D6H elements; requires two-dimensional input file pstca3shhia.inp.
pstca4shhia.inp: Axisymmetric model using CAX4H elements and a hyperelastic material.
pstc38shhis.inp: C3D8H elements; requires two-dimensional input file pstca4shhia.inp.
pstca6shhia.inp: Axisymmetric model using CAX6H elements and a hyperelastic material.
pstc3fshhis.inp: C3D15H elements; requires two-dimensional input file pstca6shhia.inp.
pstca8shhia.inp: Axisymmetric model using CAX8H elements and a hyperelastic material.
pstc3kshhis.inp: C3D20H elements; requires two-dimensional input file pstca8shhia.inp.
pstma2srhia.inp: Axisymmetric model using CAX4H elements, MAX1 elements, and a hyperelastic material.
pstm34srhis.inp: M3D4R elements; requires two-dimensional input file pstma2srhia.inp.
pstma3srhia.inp: Axisymmetric model using CAX8H elements, MAX2 elements, and a hyperelastic material.
pstm38srhis.inp: M3D8R elements; requires two-dimensional input file pstma3srhia.inp.
pstsa2shhia.inp: Axisymmetric model using CAX4H elements, SAX1 elements, and a hyperelastic material.
pstsf4shhis.inp: S4R elements; requires two-dimensional input file pstsa2shhia.inp.
pstsf4shhis_po.inp: Postprocesssing analysis.
pstsa3sheia.inp: Axisymmetric model using CAX8RH elements, MAX2 elements, and an elastic material.
psts68sheis.inp: S8R elements; requires two-dimensional input file pstsa3sheia.inp.
pstcc9shhis.inp: CCL9H elements; requires two-dimensional input file pstca3shhia.inp.
pstcc12shhis.inp: CCL12H elements; requires two-dimensional input file pstca4shhia.inp.
pstcc18shhis.inp: CCL18H elements; requires two-dimensional input file pstca6shhia.inp.
pstcc24shhis.inp: CCL24RH elements; requires two-dimensional input file pstca8shhia.inp.

Cornering tests:

Material convection effects

Problem description

In this series of tests the effect of material convection with a viscoelastic material model is verified for different element types. The model consists of a ring with outer radius $R_{o}$ = 2.0 and inner radius $R_{i}$ = 1.0. The model is fully fixed on the inside, and plane strain boundary conditions are imposed along the axial direction.

An incompressible hyperelastic material is used, with long-term moduli $C_{1}^{\infty}$ = 80.0, $C_{2}^{\infty}$ = 20.0, shear relaxation coefficient of $g_{1}$ = 0.2, and relaxation time $τ$ = 0.1.

The first analysis step is a static step where the long-term response is requested and where the rigid surface is displaced a distance $Δ$ = 0.2 to establish a contact pressure. This step is followed by a steady-state transport analysis step where viscoelastic material effects are considered. No frictional stresses are transmitted so that the disk spins without translating along the foundation.

The problem is discretized with different types of three-dimensional elements. The models that are discretized with first-order elements use 30 element divisions along the circumference and 5 element divisions in the radial direction. The second-order and cylindrical element models use 20 elements along the circumference and 3 elements in the radial direction. All the models are discretized with one element in the axial direction.

Some other tests were performed to verify the effects of material convection when the material response is slightly compressible and allows for relaxation of the pressure stress; when viscoelastic effects take place in plane stress elements; when a hyperfoam material with relaxation is used; when the model contains viscoelastic rebars embedded in an elastic or viscoelastic material; and when an incompressible hyperelastic material with two terms defining the Prony series is used.

Results and discussion

The reaction force normal to the foundation and the torque around the axle are compared to results obtained from a transient Lagrangian analysis using the quasi-static analysis procedure that is run until steady-state conditions are achieved. A model with fine meshing (C3D8RH elements) along the entire circumference is used to obtain this reference solution.

Table 3 compares the solutions obtained using different element types with the reference solution.

Additional material convection tests

Additional tests were performed to verify the effects of material convection with an elastic-plastic or a viscoplastic material model. Both the Mises metal plasticity model with kinematic hardening and the two-layer viscoelastic-elastoplastic model, which is best suited for modeling the response of materials with significant time-dependent behavior as well as plasticity at elevated temperature, have been used. The model consists of a disc with outer radius of 4.0 mm, inner radius of 1.0 mm, and thickness of 3.0 mm. The disc is generated either by revolving the cross-section of an axisymmetric mesh about the symmetry axis or by revolving a single three-dimensional repetitive sector of the model about the symmetry axis. The bottom surface of the disc is fixed. The top surface is subjected either to a nonuniform distributed load or a nonuniform contact pressure and frictional stress due to a pad being applied to the top surface of the disc. For each of these tests the disc is assumed to rotate at an angular velocity of 87.2 rad/sec or 5 rad/sec, respectively.

Each model has been analyzed using both a quasi steady-state transport solution technique through the use of the steady-state transport pass-by-pass analysis technique and a directly sought steady-state solution technique. For each of the tests the circumferential stress and circumferential plastic strain are compared to results obtained from a transient Lagrangian analysis.

Input files

pstca3shhma.inp: Axisymmetric model using CAX3H elements and a hyperelastic material with viscosity.
pstc36shhvs.inp: C3D6H elements; requires two-dimensional input file pstca3shhma.inp.
pstca4syhma.inp: Axisymmetric model using CAX4RH elements and a hyperelastic material with viscosity.
pstc38syhvs.inp: C3D8RH elements; requires two-dimensional input file pstca4syhma.inp.
pstca4shhma.inp: Axisymmetric model using CAX4H elements and a hyperelastic material with viscosity.
pstc38shhvs.inp: C3D8H elements; requires two-dimensional input file pstca4shhma.inp.
pstca4sjhma.inp: Axisymmetric model using CAX4IH elements and a hyperelastic material with viscosity.
pstc38sjhvs.inp: C3D8IH elements; requires two-dimensional input file pstca4sjhma.inp.
pstca6shhma.inp: Axisymmetric model using CAX6H elements and a hyperelastic material with viscosity.
pstc3fshhvs.inp: C3D15H elements; requires two-dimensional input file pstca6shhma.inp.
pstca8srhma.inp: Axisymmetric model using CAX8RH elements and a hyperelastic material with viscosity.
pstc3ksrhvs.inp: C3D20RH elements; requires two-dimensional input file pstca8srhma.inp.
pstca8sfhma.inp: Axisymmetric model using CAX8H elements and a hyperelastic material with viscosity.
pstc3ksfhvs.inp: C3D20H elements; requires two-dimensional input file pstca8sfhma.inp.
pstcc9shhvs.inp: CCL9H elements; requires two-dimensional input file pstca3shhma.inp.
pstcc12shhvs.inp: CCL12H elements; requires two-dimensional input file pstca4shhma.inp.
pstcc18shhvs.inp: CCL18H elements; requires two-dimensional input file pstca6shhma.inp.
pstcc24shhvs.inp: CCL24H elements; requires two-dimensional input file pstca8sfhma.inp.
pstma2srhma.inp: Axisymmetric model using CAX4RH elements, MAX1 elements, and a hyperelastic material with viscosity.
pstm34srhvs.inp: M3D4R elements; requires two-dimensional input file pstma2srhma.inp.
pstma3srhma.inp: Axisymmetric model using CAX8RH elements, MAX2 elements, and a hyperelastic material with viscosity.
pstm38srhvs.inp: M3D8R elements; requires two-dimensional input file pstma3srhma.inp.
pstma2rbema.inp: Axisymmetric model using CAX4RH elements, MAX1 elements (with rebar layers), and a hyperelastic material with viscosity.
pstm34rbevs.inp: M3D4R elements with viscoelastic rebar; requires two-dimensional input file pstma2rbema.inp.
pstca8rbema.inp: Axisymmetric model using CAX8RH elements (with rebar layers) and a hyperelastic material, viscosity, and rebar.
pstc3krbevs.inp: C3D20RH elements with viscoelastic rebar; requires two-dimensional input file pstca8rbema.inp.
pstrebara.inp: Axisymmetric model using CAX4RH elements (with rebar layers) and a hyperelastic material, viscosity, and rebar.
pstrebar.inp: Viscoelastic continuum and viscoelastic rebar; requires two-dimensional input file pstrebara.inp.
pstpressa.inp: Axisymmetric model using RAX2 elements and CAX4RH elements with a hyperelastic material with viscosity.
pstpress.inp: Pressure stress relaxation; requires two-dimensional input file pstpressa.inp.
psthfoama.inp: Axisymmetric model using RAX2 elements and CAX4R elements with a hyperfoam material with viscosity.
psthfoam.inp: Hyperfoam material; requires two-dimensional input file psthfoama.inp.
pst2pronya.inp: Axisymmetric model using RAX2 elements and CAX4RH elements with a hyperelastic material with viscosity.
pst2prony.inp: Two-term Prony series; requires two-dimensional input file pst2pronya.inp.

Additional material convection tests:

sstransp_axi_pls_preload.inp: Axisymmetric model using CAX4 elements and an elastic-plastic material.
sstransp_axi_pls_dload_dir.f: User subroutine to apply distributed load.
sstransp_axi_pls_dload_dir.inp: Steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform distributed loads; requires two-dimensional input file sstransp_axi_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_axi_pls_dload_pbp.inp: Pass-by-pass steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform distributed loads; requires two-dimensional input file sstransp_axi_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_axi_pls_surf_dir.inp: Steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform contact pressure and frictional stress; requires two-dimensional input file sstransp_axi_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_axi_pls_surf_pbp.inp: Pass-by-pass steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform contact pressure and frictional stress; requires two-dimensional input file sstransp_axi_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_axi_visp_preload.inp: Axisymmetric model using CAX4 elements and a viscoplastic material.
sstransp_axi_visp_dload_dir.inp: Steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform distributed loads; requires two-dimensional input file sstransp_axi_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_axi_visp_dload_pbp.inp: Pass-by-pass steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform distributed loads; requires two-dimensional input file sstransp_axi_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_axi_visp_surf_dir.inp: Steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform contact pressure and frictional stress; requires two-dimensional input file sstransp_axi_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_axi_visp_surf_pbp.inp: Pass-by-pass steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform contact pressure and frictional stress; requires two-dimensional input file sstransp_axi_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_per_pls_preload.inp: 3D sector model using C3D8H elements and an elastic-plastic material. Preloaded with an internal pressure.
sstransp_per_pls_dload_dir.inp: Steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform distributed loads for a periodic disc; requires another three-dimensional input file sstransp_per_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_per_pls_dload_pbp.inp: Pass-by-pass steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform distributed loads for a periodic disc; requires another three-dimensional input file sstransp_per_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_per_pls_surf_dir.inp: Steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform contact pressure and frictional stress for a periodic disc; requires another three-dimensional input file sstransp_per_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_per_pls_surf_pbp.inp: Pass-by-pass steady-state transport analysis with linear kinematic hardening plasticity model subjected to nonuniform contact pressure and frictional stress for a periodic disc; requires another three-dimensional input file sstransp_per_pls_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_per_visp_preload.inp: 3D sector model using C3D8 elements and a viscoplastic material. Preloaded with an internal pressure.
sstransp_per_visp_dload_dir.inp: Steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform distributed loads for a periodic disc; requires another three-dimensional input file sstransp_per_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_per_visp_dload_pbp.inp: Pass-by-pass steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform distributed loads for a periodic disc; requires another three-dimensional input file sstransp_per_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_per_visp_surf_dir.inp: Steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform contact pressure and frictional stress for a periodic disc; requires another three-dimensional input file sstransp_per_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.
sstransp_per_visp_surf_pbp.inp: Pass-by-pass steady-state transport analysis with two-layer viscoplasticity model subjected to nonuniform contact pressure and frictional stress for a periodic disc; requires another three-dimensional input file sstransp_per_visp_preload.inp and user subroutine sstransp_axi_pls_dload_dir.f.

Tables

Table 3. Reaction forces and torques for analyses using different elements.
Element Type	Force	Torque
C3D6H	373.12	−9.55
C3D8RH	370.46	−9.49
C3D8H	373.98	−9.55
C3D8IH	376.92	−9.62
C3D15H	372.74	−9.66
C3D20RH	372.53	−9.91
C3D20H	373.43	−9.97
CCL9H	357.3	–10.02
CCL12H	375.5	–10.33
CCL18H	371.3	–10.04
CCL24H	373.3	–9.97
Reference solution	374.47	−9.77