4.10.8 3DNLG-8: Collapse of a straight pipe segment under pure bending

Product: ABAQUS/Standard

Elements tested

S3 S3R S4 S4R S4R5 S8R S8R5 S9R5

STRI3 STRI65

Problem description

Elastic and elastic-plastic cases of a beam under pure bending are tested. Nonlinear multipoint constraints and section ovalization are also tested.

Model:

A quarter model is used because of symmetry.

Material:

Young's modulus = 1.93 × 10⁵, Poisson's ratio = 0.26, plastic (isotropic hardening).

Equiv. Stress	Equiv. Plastic Strain
272	0.0
346	0.00473
379	0.01264
404	0.02836
424	0.0491

Boundary conditions:

Symmetry on plane z = 0 (; symmetry on plane x = 0 (; symmetry on rotated plane: i) Shell nodes must remain in rotated plane (Nonlinear MPC). ii) The z-rotation of shell nodes in the rotated plane must be the same as the master node (Linear MPC). iii) The component of rotation about the local -axis must be zero (Nonlinear MPC). The master node is constrained to move on the x-axis only.

Loading:

Z-rotation applied to pipe end ( = 0.004).

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test 3DNLG-8 from NAFEMS Publication R0024 “A Review of Benchmark Problems for Geometric Non-linear Behaviour of 3-D Beams and Shells (SUMMARY).”

The published results of this problem were obtained with ABAQUS. Thus, a comparison of ABAQUS and NAFEMS results is not an independent verification of ABAQUS. The NAFEMS study includes results from other sources for comparison that may provide a basis for verification of this problem.

Results and discussion

In the following table, the end moments for the elastic and elastic-plastic cases are reported at three rotation values of the master node. All the meshes have the same nodal spacing.

Element	= 0.0012		= 0.0024		= 0.004
	Moment × 10^–9		Moment × 10^–9		Moment × 10^–9
	Elastic	Plastic	Elastic	Plastic	Elastic	Plastic
S3/S3R	2.056	0.5146	3.488	0.5402	3.704	0.5066
S4	2.032	0.5088	3.452	0.5350	3.676	0.5042
S4R	2.020	0.5080	3.436	0.5332	3.664	0.5024
S4R5	2.018	0.5078	3.420	0.5334	3.578	0.5008
S8R	2.054	0.5116	3.448	0.5380	3.568	0.5024
S8R5	2.054	0.5118	3.452	0.5380	3.582	0.5030
S9R5	2.054	0.5118	3.452	0.5380	3.582	0.5030
STRI3	2.056	0.5144	3.474	0.5390	3.662	0.5036
STRI65	2.054	0.5122	3.452	0.5388	3.592	0.5046

Response predicted by ABAQUS

Essentially identical moment-rotation curves (at the master node) are obtained for all test cases. The response predicted using S8R5 elements is shown below.