1.3.9 Shear flexible beams and shells: I

Product: ABAQUS/Standard

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:

at end A, at end B.

Loading:

25.0 at end A. Only and are applied for shell models.

Section properties:

0.25, 1 × 10⁶, 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.

Reference solution

Displacements in beam elements

at node A.

Regular and open section elements

1.667 × 10^–5, 4.333 × 10^–5.

Pipe elements

2.792 × 10^–5, 2.194 × 10^–3.

Stress resultants in beam and pipe elements

–25.0, 25(5 – x), 25(5 – x),

Transverse shear: 25.0, –25.0.

Displacements in shell elements

1.667 × 10^–5, 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 1.3.9–1.

Table 1.3.9–1 Pipe element solutions.


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