Shear flexible beams and shells: I

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

This page discusses:

Elements tested
Problem description
Reference solution
Results and discussion
Input files

ProductsAbaqus/StandardAbaqus/Explicit

Elements tested

B21
B21H
B22
B22H
B31
B31H
B31OS
B31OSH
B32
B32H
B32OS
B32OSH

PIPE21
PIPE21H
PIPE22
PIPE22H
PIPE31
PIPE31H
PIPE32
PIPE32H

S4
S4R
S4R5
S8R
S8R5
S9R5

Problem description

A three-dimensional problem is shown here, which can be particularized for two-dimensional beam elements.

Material:

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

Boundary conditions:

$ϕ_{x} = ϕ_{y} = ϕ_{z} = 0$ at end A, $u_{x} = u_{y} = u_{z} = 0$ at end B.

Loading:

$F_{x} = F_{y} = F_{z} = $ 25.0 at end A. Only $F_{x}$ and $F_{z}$ are applied for shell models.

Section properties

$A = $ 0.25, $I_{11} = I_{22} = $ 1 × 10⁶, $J = $ 0.0104167. The bending inertias have intentionally been chosen as very large values in order to test the shear-only modes.

For pipe elements a circular cross-section of outer radius 0.5 and wall thickness 0.05 is used. For this case a different analytical solution based upon Timoshenko theory is used for comparison.

Analogous problems are modeled in Abaqus/Explicit using linear beam and pipe elements. Unit density is prescribed for the material, and the solution is computed for unit time. Loads are applied smoothly for a quasi-static solution, similar to that from static analysis. The results using pipe elements are consistent to that using beam elements, both of which match the static analysis.

Reference solution

Displacements in beam elements

$u_{x} = \frac{P L}{E A}, u_{y} = u_{z} = \frac{P L^{3}}{3 E I} (1 + \frac{3 E I}{k G A L^{2}})$ at node A.

Regular and open section elements

$u_{x} = $ 1.667 × 10⁻⁵, $u_{y} = u_{z} = $ 4.333 × 10⁻⁵.

Pipe elements

$u_{x} = $ 2.792 × 10⁻⁵, $u_{y} = u_{z} = $ 2.194 × 10⁻³.

Stress resultants in beam and pipe elements

$S F 1 = $ −25.0, $S M 1 = $ 25(5 − x), $S M 2 = $ 25(5 − x),

Transverse shear: $S F 3 = $ 25.0, $S F 2 = $ −25.0.

Displacements in shell elements

$u_{x} = $ 1.667 × 10⁻⁵, $u_{z} = $ 4.333 × 10⁻⁵ at node A.

Results and discussion

All beam and shell elements yield exact solutions. Pipe element solutions are given in Table 1.

Table 1. Pipe element solutions.
	$u_{x}$	$u_{y} = u_{z}$
Analytical solution	2.792 × 10⁻⁵	2.194 × 10⁻³
Linear pipe elements	2.792 × 10⁻⁵	2.093 × 10⁻³
Quadratic pipe elements	2.792 × 10⁻⁵	2.106 × 10⁻³

Input files

eb22gxs5.inp: B21 elements.
eb2hgxs5.inp: B21H elements.
eb23gxs5.inp: B22 elements.
eb2igxs5.inp: B22H elements.
eb32gxs5.inp: B31 elements.
eb3hgxs5.inp: B31H elements.
ebo2gxs5.inp: B31OS elements.
ebohgxs5.inp: B31OSH elements.
eb33gxs5.inp: B32 elements.
eb3igxs5.inp: B32H elements.
ebo3gxs5.inp: B32OS elements.
eboigxs5.inp: B32OSH elements.
ep22pxs5.inp: PIPE21 elements.
ep2hpxs5.inp: PIPE21H elements.
ep23pxs5.inp: PIPE22 elements.
ep2ipxs5.inp: PIPE22H elements.
ep32pxs5.inp: PIPE31 elements.
ep3hpxs5.inp: PIPE31H elements.
ep33pxs5.inp: PIPE32 elements.
ep3ipxs5.inp: PIPE32H elements.
ese4sgs5.inp: S4 elements.
esf4sgs5.inp: S4R elements.
es54sgs5.inp: S4R5 elements.
es68sgs5.inp: S8R elements.
es58sgs5.inp: S8R5 elements.
es59sgs5.inp: S9R5 elements.
es56sgs5.inp: STRI65 elements.
force_shearflex_beam2d_xpl.inp: B21 elements in Abaqus/Explicit.
force_shearflex_beam3d_xpl.inp: B31 elements in Abaqus/Explicit.
force_shearflex_pipe2d_xpl.inp: PIPE21 elements in Abaqus/Explicit.
force_shearflex_pipe3d_xpl.inp: PIPE31 elements in Abaqus/Explicit.