Tip in-plane shear load on a cantilevered hook

The Raasch Challenge problem is used to demonstrate how Abaqus models in-plane shear loading in a curved strip using shell elements. Transverse shear flexibility and proper treatment of twisting deformations of the shell elements are important factors in determining the bending behavior. The results are compared to the reference solution by Knight (1997).

This page discusses:

Problem description
Results and discussion
Input files
References
Tables
Figures

ProductsAbaqus/Standard

Problem description

The geometry consists of a “hook” in the form of a curved strip rigidly clamped at one end and loaded with a unit in-plane shear along the width at the other end. It has two circular segments that are connected at the tangent point. The smaller segment has a mean radius of 0.3556 m (14 inches) and spans 60° from the clamped end to the tangent point. The larger segment spans 150° from the tangent point to the free end and has a mean radius of 1.1684 m (46 inches). The hook is 0.0508 m (2 inches) thick and 0.508 m (20 inches) wide, modeled as linear elastic with an elastic modulus of 22.77 MPa (3300 psi) and a Poisson's ratio of 0.35. In most tests the shear force is applied through the use of a distributing coupling constraint. The coupling constraint provides coupling between a reference node on which the load is prescribed and the nodes located on the free end. The distributed nodal loads on the free end are equivalent to a uniformly distributed load of 8.7563 N/m (0.05 lb/in). In two of the tests an equivalent shear force is applied as a distributed shear traction instead.

The problem is modeled using fully integrated S4 shell elements with five different meshes: 1 × 9, 3 × 18, 5 × 36, 10 × 72, and 20 × 144. For comparison the problem is also analyzed with S4R shell elements and SC8R continuum shell elements that use reduced integration. The reference solution is obtained with a refined mesh using C3D20R continuum elements with reduced integration.

Results and discussion

The solution reported is the in-plane displacement along the centerline of the loaded end. A comparison of the tip displacement normalized with the reference solution is shown in Table 1. The reduced-integration elements, S4R and SC8R, show excessively large displacement for the coarse mesh (1 × 9) because of the elements' poor treatment of in-plane bending deformations with coarse meshes. The coarse mesh for S4 elements gives a solution that is only about 3.5% stiffer than the reference solution. The refined meshes for both types of shell elements give comparable solutions, which are 0.2% more flexible than the reference solution.

The solution for the continuum shell mesh with a single element in the thickness direction converges to an excessively large displacement. This may be due to the element's poor treatment of drill stiffness. Stacking two or more elements yields the exact solution even for a coarse mesh (3 × 18 × 2).

The solutions using a distributed shear traction to apply the load agree exactly with the solutions using a coupling constraint.

Input files

C3D20R elements:

raasch_c3d20r_20x144x2.inp: 20 × 144 × 2 mesh.

S4 elements:

raasch_s4_1x9.inp: 1 × 9 mesh.
raasch_s4_1x9_edld.inp: 1 × 9 mesh loaded with a distributed edge traction.
raasch_s4_3x18.inp: 3 × 18 mesh.
raasch_s4_5x36.inp: 5 × 36 mesh.
raasch_s4_10x72.inp: 10 × 72 mesh.
raasch_s4_20x144.inp: 20 × 144 mesh.

S4R elements:

raasch_s4r_1x9.inp: 1 × 9 mesh.
raasch_s4r_3x18.inp: 3 × 18 mesh.
raasch_s4r_5x36.inp: 5 × 36 mesh.
raasch_s4r_10x72.inp: 10 × 72 mesh.
raasch_s4r_20x144.inp: 20 × 144 mesh.

SC8R elements:

raasch_sc8r_1x9x1.inp: 1 × 9 × 1 mesh.
raasch_sc8r_3x18x1.inp: 3 × 18 × 1 mesh.
raasch_sc8r_5x36x1.inp: 5 × 36 × 1 mesh.
raasch_sc8r_10x72x1.inp: 10 × 72 × 1 mesh.
raasch_sc8r_20x144x1.inp: 20 × 144 × 1 mesh.
raasch_sc8r_1x9x2.inp: 1 × 9 × 2 mesh.
raasch_sc8r_3x18x2.inp: 3 × 18 × 2 mesh.
raasch_sc8r_3x18x2_trvec.inp: 3 × 18 × 2 mesh loaded with a distributed general traction.
raasch_sc8r_5x36x2.inp: 5 × 36 × 2 mesh.
raasch_sc8r_10x72x2.inp: 10 × 72 × 2 mesh.
raasch_sc8r_20x144x2.inp: 20 × 144 × 2 mesh.

References

Knight, N. F., Jr., “The Raasch Challenge for Shell Elements,” AIAA Journal, vol. 35, no. 2, pp. 375–381, 1997.

Tables

Table 1. Comparison of tip deflections (normalized with continuum solution) in the direction of load. (Continuum solution is 5.020 for 20 × 144 × 2 mesh of C3D20R elements.)
Mesh	S4	S4R	SC8R
Mesh	S4	S4R	single element	two elements stacked
1 × 9	0.967	2.951	2.622	1.693
3 × 18	0.972	0.979	1.534	1.019
5 × 36	0.987	0.989	1.522	1.007
10 × 72	0.998	0.999	1.530	1.011
20 × 144	1.003	1.003	1.535	1.015

Figures