Love-Kirchhoff beams and shells

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/Standard

Elements tested

B23

B23H

B33

B33H

STRI3

STRI65

Problem description

A three-dimensional problem is shown here. It may 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, $M_{x} = M_{y} = M_{z} = $ 100.0 at end B. Only $F_{x}$ , $F_{z}$ , and $M_{y}$ are applied for shell models.

General:

Gauss integration is used for the shell cross-section for element STRI3.

Reference solution

Displacements in beam elements

$u_{x}$ = 1.667 × 10⁻⁵, $u_{y}$ = −1.333 × 10⁻³, $u_{z}$ = 0.01467 at node A,

$ϕ_{x}$ = 4.92 × 10⁻³, $ϕ_{y}$ = 5.2 × 10⁻³, $ϕ_{z}$ = 1.2 × 10⁻³ at node $B$ .

Stress resultants in beam elements

$N_{x} = $ −25.0, $M_{11} = $ 25(1 − x), $M_{22} = $ 100 + 25(5 − $x$ ), $M_{x} = $ 100.0.

Displacements in shell elements

$u_{x} = $ 1.667 × 10⁻⁵, $u_{z} = $ 0.01467 at node A, $ϕ_{y} = $ 5.2 × 10⁻³ at node B.

Results and discussion

Beam elements yield exact solutions. 3-node shell elements yield exact solutions for $u_{x}$ and $ϕ_{y}$ but yield a value of 0.01412 for $u_{z}$ . 6-node shell elements yield exact solutions for $u_{x}$ and $ϕ_{y}$ but yield a value of 0.01464 for $u_{z}$ .

Input files

eb2arxs4.inp: B23 elements.
eb2jrxs4.inp: B23H elements.
eb3arxs4.inp: B33 elements.
eb3jrxs4.inp: B33H elements.
es63sxs4.inp: STRI3 elements.
es56sxs4.inp: STRI65 elements.