4.10.1 3DNLG-1: Elastic large deflection response of a Z-shaped cantilever under an end load

Product: ABAQUS/Standard  

Elements tested

B31    B31H    B32    B32H    B33    B33H   

S4    S4R   

SC6R    SC8R   

Problem description

Model:

Uniform thickness (t = 1.7).

Material:

Linear elastic, Young's modulus = 2.05 × 105, Poisson's ratio = 0.3.

Boundary conditions:

All degrees of freedom restrained at built-in end.

Loading:

Concentrated end load (P = 4000).

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test 3DNLG-1 from NAFEMS Publication R0024 “A Review of Benchmark Problems for Geometric Non-linear Behaviour of 3-D Beams and Shells (SUMMARY).”

The published results for 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

Displacements converge faster than stresses. Even though the displacements seem to have converged, the lower-order elements need more refined meshes (compared to the higher-order elements) before the stresses are observed to converge. Stresses are most accurate at the integration points within the element. When stress values are extrapolated from the integration points to the nodes and then averaged, the stress values calculated may not capture the peak values if a stress gradient is present. Since the higher-order elements (B32, B32H, B33, and B33H) use linear extrapolation within an element, a stress gradient in an element may be captured adequately when extrapolating stresses to the nodes. However, constant extrapolation is used for linear elements (B31, B31H, S4, S4R, and SC8R), which results in slow convergence of nodal stress values. Higher mesh refinement near stress gradients is needed for such elements.

Tip Displacement
Element TypeNumber of ElementsApplied Load
104.51263.04000.0
B317280.42 133.1143.5
B31H7280.42133.1 143.5
B32980.42133.1143.4
B32H980.42133.1143.4
B33980.42133.1143.4
B33H980.42133.1143.4
S41 × 72 80.42133.1143.5
S4R1 × 72 80.42133.1143.5
SC6R2 × 72 × 180.56133.1143.5
SC8R1 × 72 × 179.28133.1143.5

Moment at A
Element TypeNumber of ElementsApplied Load
104.51263.04000.0
B3172–8333–55109921
B31H72–8333–55099922
B329–8308–496310742
B32H9–8308–496210743
B339–8316–498210659
B33H9–8317–498310661
S41 × 72–8334–55109934
S4R1 × 72–8334–55109934
SC6R2 × 72 × 1–8315–54819831
SC8R1 × 72 × 1–8333–55079939

Response predicted by ABAQUS (element B32)

Input files

n3g1x33x_b31.inp

B31 elements.

n3g1x33x_b31h.inp

B31H elements.

n3g1x33x_b32.inp

B32 elements.

n3g1x33x_b32h.inp

B32H elements.

n3g1x33x_b33.inp

B33 elements.

n3g1x33x_b33h.inp

B33H elements.

n3g1x33x_s4.inp

S4 elements.

n3g1x33x_s4r.inp

S4R elements.

nlg1_std_sc6r.inp

SC6R elements.

nlg1_std_sc8r.inp

SC8R elements.