Three-dimensional solid elements

This problem contains basic test cases for one or more Abaqus elements and features.

This page discusses:

Elements tested
Problem description
Results and discussion
Input files

ProductsAbaqus/Standard

Elements tested

C3D4

C3D4H

C3D5

C3D5H

C3D6

C3D6H

C3D8

C3D8H

C3D8I

C3D8IH

C3D8R

C3D8RH

C3D10

C3D10H

C3D10HS

C3D10M

C3D10MH

C3D15

C3D15H

C3D15V

C3D15VH

C3D20

C3D20H

C3D20R

C3D20RH

C3D27

C3D27H

C3D27R

C3D27RH

CSS8

Problem description

Material:

Linear elastic, Young's modulus = 30 × 10⁶, Poisson's ratio = 0.3.

Boundary conditions:

$u_{x}^{A}$ = $u_{y}^{A}$ = $u_{z}^{A}$ = 0, $u_{y}^{B}$ = 0, $u_{z}^{D}$ = 0, $u_{x}^{E}$ = 0.

Step 1

A distributed pressure of 1000/area is applied on each face, and equivalent concentrated forces for shear loading, defined such that all three shear stresses are of magnitude −1000.

Response:
Stresses: $σ_{x x} = σ_{y y} = σ_{z z} = σ_{x y} = σ_{x z} = σ_{y z} = $ −1000 at every integration point.
Strains: $ε_{x x} = ε_{y y} = ε_{z z} = $ −1.3333 × 10⁻⁵, $γ_{x y} = γ_{y z} = γ_{x z} = $ −8.6667 × 10⁻⁵.
Displacements: $u_{x} = x ε_{x x} + y γ_{x y}, u_{y} = y ε_{y y} + z γ_{y z}, u_{z} = z ε_{z z} + x γ_{x z}$ .

For lower-order elements the test description is complete. For higher-order elements another step definition is included.

Step 2

Hydrostatic pressure loading is applied to the four vertical faces, varying from 0 at top to 1000/area at bottom, in addition to the Step 1 loads.

Response:

Stresses

$σ_{x x} = σ_{y y} = $ −1000(2 − z), $σ_{z z} = $ −1000, $σ_{x y} = σ_{x z} = σ_{y z} = $ −1000.

Strains

$ε_{x x} = ε_{y y} = $ 3.333 × 10⁻⁵(0.7z − 1.1), $ε_{z z} = $ 3.333 × 10⁻⁵(0.2 − 0.6z),

$γ_{x y} = γ_{x z} = γ_{y z} = $ −8.66667 × 10⁻⁵.

Results and discussion

Elements using reduced integration may have additional boundary conditions to those specified above.

Elements C3D27R and C3D27RH employ 21 nodes in this test to produce the exact solutions. The lack of midface nodes is consistent with the elements' intended use, since no contact elements are present.

All elements that do not use the modified formulation, except C3D20RH, yield exact solutions. The stresses calculated for element C3D20RH are correct.

The modified tetrahedral element formulation cannot exactly capture a linearly varying gradient field due to the piecewise linear interpolation used for the unknown field. However, the numerical solution will converge to the exact solution as the mesh is refined.

Section output requests to the results (.fil) file and to the data (.dat) file are used in some of the input files to output accumulated quantities on the face in the y-z plane. The area of the face is 2.0 in both steps. The accumulated force is reported in a coordinate system that is local to the section. In Step 1 the force is 2000 in each local direction. In Step 2 the total force component in the local 1-direction (normal to the face) changes to 3000.

Input files

ec34sfs2.inp: C3D4 elements.
ec34shs2.inp: C3D4H elements.
ec35sfs2.inp: C3D5 elements.
ec35shs2.inp: C3D5H elements.
ec36sfs2.inp: C3D6 elements.
ec36shs2.inp: C3D6H elements.
ec38sfs2.inp: C3D8 elements.
ec38shs2.inp: C3D8H elements.
ec38sis2.inp: C3D8I elements.
ec38sjs2.inp: C3D8IH elements.
ec38srs2.inp: C3D8R elements.
ec38sys2.inp: C3D8RH elements.
ec3asfs2.inp: C3D10 elements.
ec3ashs2.inp: C3D10H elements.
ec3asis2.inp: C3D10HS elements.
ec3asks2.inp: C3D10M elements.
ec3asls2.inp: C3D10MH elements.
ec3fsfs2.inp: C3D15 elements.
ec3fshs2.inp: C3D15H elements.
ec3isfs2.inp: C3D15V elements.
ec3ishs2.inp: C3D15VH elements.
ec3ksfs2.inp: C3D20 elements.
ec3kshs2.inp: C3D20H elements.
ec3ksrs2.inp: C3D20R elements.
ec3ksys2.inp: C3D20RH elements.
ec3rsfs2.inp: C3D27 elements.
ec3rshs2.inp: C3D27H elements.
ec3rsrs2.inp: C3D27R elements.
ec3rsys2.inp: C3D27RH elements.
ecss8sis2.inp: CSS8 elements.