Analysis of a twisted beam

This problem illustrates the accuracy of shell and beam finite element solutions for bending of warped structures. The responses of both a thick and thin twisted cantilever beam subjected to either an in-plane or out-of-plane shear load are obtained. The test was proposed by MacNeal and Harder (1985), who provided the analytical solution for the thick twisted beam. The reference solution for the thin twisted beam was provided by Simo et al. (1989).

This page discusses:

Problem description
Results and discussion
Input files
References
Tables
Figures

ProductsAbaqus/StandardAbaqus/Explicit

Problem description

The structure is a cantilever beam, 12.0 in long and 1.1 in wide, that twists 90° from end to end, as shown in Figure 1. The beam is aligned with the x-axis. Its thickness, b, is 0.32 in for the thick case and 0.05 in for the thin case.

The beam is modeled in Abaqus/Standard with 4-node shell elements (S4, S4R, and S4R5), 3-node shell elements (S3R and STRI3), quadratic shell elements (STRI65, S8R, S8R5, and S9R5), continuum shell elements (SC8R), and beam elements (B31, B32, and B33). Three mesh densities are considered for each element type. The coarsest mesh of 4-node shell elements (2 × 12 with a warp angle of 7.5° per element length) is illustrated in Figure 1. The 3-node shell mesh has the same number of elements as the equivalent 4-node shell mesh. The quadratic shell mesh has half as many elements in each direction (in general, the same number of degrees of freedom) as the corresponding linear shell mesh. The coarsest mesh of beam elements uses 12 linear elements.

The beam is modeled in Abaqus/Explicit with a 2 × 12 mesh of S4R, S4RS, or S4RSW elements.

The material is steel with a Young's modulus of 29.0 Msi and a Poisson's ratio of 0.22. A point load of 1.0 lb is applied at the center of the free end in the y- and z-directions, respectively.

Results and discussion

The results are listed in Table 1 to Table 12. The tip displacements in the load directions are compared with the analytical solution.

Abaqus/Standard results

The shell element models all converge to the analytical solution for both load cases and thicknesses. Even for the coarsest meshes, where for the 4-node shells the warp angle is 7.5° per element, the results are in good agreement.

The 4-node quadrilateral results are listed in Table 1 and Table 2, the 3-node triangular results are listed in Table 5 and Table 6, and the second-order shell results are listed in Table 7 and Table 8. For the coarsest meshes the first-order shell results are somewhat better for the in-plane than for the out-of-plane loading case. The out-of-plane loading case causes in-plane bending deformation at the built-in end, where the maximum bending moments occur (refer to Figure 1). First-order triangular and reduced-integration quadrilateral elements require mesh refinement to model this in-plane bending accurately. The first-order fully integrated shell, S4, and the second-order reduced-integration elements capture the correct in-plane bending behavior.

The continuum shell results are listed in Table 3 and Table 4 for both load cases and thicknesses. Results are compared for cases in which 1, 2, 4, and 8 elements are stacked in the thickness direction. For the case with a single element stacked in the thickness direction, the results show excessively large displacements. This may be due to the element's poor treatment of drill stiffness. The results show good agreement for cases with multiple elements (even two elements) stacked in the thickness direction.

The beam element models accurately reproduce the analytical result for both load cases and thicknesses; see Table 9 and Table 10. Since the coarsest mesh is sufficiently refined to capture the analytical solution, the results do not improve with mesh refinement.

Abaqus/Explicit results

Figure 2 shows the time history of the tip displacement and various energies for the in-plane shear load case when the beam has a thickness of 0.32 in. The tip displacement values indicated in the tabulated results are the displacement values at node 132 in the direction of the applied tip load. Table 11 and Table 12 compare the solutions obtained with elements S4R, S4RS, and S4RSW. The response of S4RS is quite similar to that of S4R.

Input files

B31 elements:

twistedbeam_b31_12_thick.inp: 12-element, thick beam.
twistedbeam_b31_12_thin.inp: 12-element, thin beam.
twistedbeam_b31_24_thick.inp: 24-element, thick beam.
twistedbeam_b31_24_thin.inp: 24-element, thin beam.
twistedbeam_b31_48_thick.inp: 48-element, thick beam.
twistedbeam_b31_48_thin.inp: 48-element, thin beam.

B32 elements:

twistedbeam_b32_6_thick.inp: 6-element, thick beam.
twistedbeam_b32_6_thin.inp: 6-element, thin beam.
twistedbeam_b32_12_thick.inp: 12-element, thick beam.
twistedbeam_b32_12_thin.inp: 12-element, thin beam.
twistedbeam_b32_24_thick.inp: 24-element, thick beam.
twistedbeam_b32_24_thin.inp: 24-element, thin beam.

B33 elements:

twistedbeam_b33_12_thick.inp: 12-element, thick beam.
twistedbeam_b33_12_thin.inp: 12-element, thin beam.
twistedbeam_b33_24_thick.inp: 24-element, thick beam.
twistedbeam_b33_24_thin.inp: 24-element, thin beam.
twistedbeam_b33_48_thick.inp: 48-element, thick beam.
twistedbeam_b33_48_thin.inp: 48-element, thin beam.

S3R elements:

twistedbeam_s3r_2x12_thick.inp: 2 × 12 mesh, thick beam.
twistedbeam_s3r_2x12_thin.inp: 2 × 12 mesh, thin beam.
twistedbeam_s3r_4x24_thick.inp: 4 × 24 mesh, thick beam.
twistedbeam_s3r_4x24_thin.inp: 4 × 24 mesh, thin beam.
twistedbeam_s3r_8x48_thick.inp: 8 × 48 mesh, thick beam.
twistedbeam_s3r_8x48_thin.inp: 8 × 48 mesh, thin beam.

S4 elements:

twistedbeam_s4_2x12_thick.inp: 2 × 12 mesh, thick beam.
twistedbeam_s4_2x12_thin.inp: 2 × 12 mesh, thin beam.
twistedbeam_s4_4x24_thick.inp: 4 × 24 mesh, thick beam.
twistedbeam_s4_4x24_thin.inp: 4 × 24 mesh, thin beam.
twistedbeam_s4_8x48_thick.inp: 8 × 48 mesh, thick beam.
twistedbeam_s4_8x48_thin.inp: 8 × 48 mesh, thin beam.

S4R elements:

twistedbeam_s4r_2x12_thick.inp: 2 × 12 mesh, thick beam.
twistedbeam_s4r_2x12_thin.inp: 2 × 12 mesh, thin beam.
twistedbeam_s4r_4x24_thick.inp: 4 × 24 mesh, thick beam.
twistedbeam_s4r_4x24_thin.inp: 4 × 24 mesh, thin beam.
twistedbeam_s4r_8x48_thick.inp: 8 × 48 mesh, thick beam.
twistedbeam_s4r_8x48_thin.inp: 8 × 48 mesh, thin beam.

S4R5 elements:

twistedbeam_s4r5_2x12_thick.inp: 2 × 12 mesh, thick beam.
twistedbeam_s4r5_2x12_thin.inp: 2 × 12 mesh, thin beam.
twistedbeam_s4r5_4x24_thick.inp: 4 × 24 mesh, thick beam.
twistedbeam_s4r5_4x24_thin.inp: 4 × 24 mesh, thin beam.
twistedbeam_s4r5_8x48_thick.inp: 8 × 48 mesh, thick beam.
twistedbeam_s4r5_8x48_thin.inp: 8 × 48 mesh, thin beam.

2 × 12 mesh of S4RSW elements in Abaqus/Explicit:

twistedbeam_thick_fy.inp: Thick beam, tip load in y-direction.
twistedbeam_thick_fz.inp: Thick beam, tip load in z-direction.
twistedbeam_thin_fy.inp: Thin beam, tip load in y-direction.
twistedbeam_thin_fz.inp: Thin beam, tip load in z-direction.

2 × 12 mesh of S4RS elements in Abaqus/Explicit:

twistedbeam_s4rs_thick_fy.inp: Thick beam, tip load in y-direction.
twistedbeam_s4rs_thick_fz.inp: Thick beam, tip load in z-direction.
twistedbeam_s4rs_thin_fy.inp: Thin beam, tip load in y-direction.
twistedbeam_s4rs_thin_fz.inp: Thin beam, tip load in z-direction.

2 × 12 mesh of S4R elements in Abaqus/Explicit:

twistedbeam_s4r_thick_fy.inp: Thick beam, tip load in y-direction.
twistedbeam_s4r_thick_fz.inp: Thick beam, tip load in z-direction.
twistedbeam_s4r_thin_fy.inp: Thin beam, tip load in y-direction.
twistedbeam_s4r_thin_fz.inp: Thin beam, tip load in z-direction.

S8R elements:

twistedbeam_s8r_2x12_thick.inp: 2 × 12 mesh, thick beam.
twistedbeam_s8r_2x12_thin_s8r.inp: 2 × 12 mesh, thin beam.
twistedbeam_s8r_4x24_thick.inp: 4 × 24 mesh, thick beam.
twistedbeam_s8r_4x24_thin_s8r.inp: 4 × 24 mesh, thin beam.
twistedbeam_s8r_8x48_thick.inp: 8 × 48 mesh, thick beam.
twistedbeam_s8r_8x48_thin_s8r.inp: 8 × 48 mesh, thin beam.

S8R5 elements:

twistedbeam_s8r5_2x12_thick.inp: 2 × 12 mesh, thick beam.
twistedbeam_s8r5_2x12_thin.inp: 2 × 12 mesh, thin beam.
twistedbeam_s8r5_4x24_thick.inp: 4 × 24 mesh, thick beam.
twistedbeam_s8r5_4x24_thin.inp: 4 × 24 mesh, thin beam.
twistedbeam_s8r5_8x48_thick.inp: 8 × 48 mesh, thick beam.
twistedbeam_s8r5_8x48_thin.inp: 8 × 48 mesh, thin beam.

S9R5 elements:

twistedbeam_s9r5_2x12_thick.inp: 2 × 12 mesh, thick beam.
twistedbeam_s9r5_2x12_thin.inp: 2 × 12 mesh, thin beam.
twistedbeam_s9r5_4x24_thick.inp: 4 × 24 mesh, thick beam.
twistedbeam_s9r5_4x24_thick.inp: 4 × 24 mesh, thin beam.
twistedbeam_s9r5_8x48_thick.inp: 8 × 48 mesh, thick beam.
twistedbeam_s9r5_8x48_thin.inp: 8 × 48 mesh, thin beam.

STRI3 elements:

twistedbeam_stri3_2x12_thick.inp: 2 × 12 mesh, thick beam.
twistedbeam_stri3_2x12_thin.inp: 2 × 12 mesh, thin beam.
twistedbeam_stri3_4x24_thick.inp: 4 × 24 mesh, thick beam.
twistedbeam_stri3_4x24_thin.inp: 4 × 24 mesh, thin beam.
twistedbeam_stri3_8x48_thick.inp: 8 × 48 mesh, thick beam.
twistedbeam_stri3_8x48_thin.inp: 8 × 48 mesh, thin beam.

STRI65 elements:

twistedbeam_stri65_2x12_thick.inp: 2 × 12 mesh, thick beam.
twistedbeam_stri65_2x12_thin.inp: 2 × 12 mesh, thin beam.
twistedbeam_stri65_4x24_thick.inp: 4 × 24 mesh, thick beam.
twistedbeam_stri65_4x24_thin.inp: 4 × 24 mesh, thin beam.
twistedbeam_stri65_8x48_thick.inp: 8 × 48 mesh, thick beam.
twistedbeam_stri65_8x48_thin.inp: 8 × 48 mesh, thin beam.

SC8R elements:

twistedbeam_sc8r_2x12x1_thick.inp: 2 × 12 × 1 mesh, thick beam.
twistedbeam_sc8r_2x12x2_thick.inp: 2 × 12 × 2 mesh, thick beam.
twistedbeam_sc8r_2x12x4_thick.inp: 2 × 12 × 4 mesh, thick beam.
twistedbeam_sc8r_2x12x8_thick.inp: 2 × 12 × 8 mesh, thick beam.
twistedbeam_sc8r_4x24x1_thick.inp: 4 × 24 × 1 mesh, thick beam.
twistedbeam_sc8r_4x24x2_thick.inp: 4 × 24 × 2 mesh, thick beam.
twistedbeam_sc8r_4x24x4_thick.inp: 4 × 24 × 4 mesh, thick beam.
twistedbeam_sc8r_4x24x8_thick.inp: 4 × 24 × 8 mesh, thick beam.
twistedbeam_sc8r_8x48x1_thick.inp: 8 × 48 × 1 mesh, thick beam.
twistedbeam_sc8r_8x48x2_thick.inp: 8 × 48 × 2 mesh, thick beam.
twistedbeam_sc8r_8x48x4_thick.inp: 8 × 48 × 4 mesh, thick beam.
twistedbeam_sc8r_8x48x8_thick.inp: 8 × 48 × 8 mesh, thick beam.
twistedbeam_sc8r_2x12x1_thin.inp: 2 × 12 × 1 mesh, thin beam.
twistedbeam_sc8r_2x12x2_thin.inp: 2 × 12 × 2 mesh, thin beam.
twistedbeam_sc8r_2x12x4_thin.inp: 2 × 12 × 4 mesh, thin beam.
twistedbeam_sc8r_2x12x8_thin.inp: 2 × 12 × 8 mesh, thin beam.
twistedbeam_sc8r_4x24x1_thin.inp: 4 × 24 × 1 mesh, thin beam.
twistedbeam_sc8r_4x24x2_thin.inp: 4 × 24 × 2 mesh, thin beam.
twistedbeam_sc8r_4x24x4_thin.inp: 4 × 24 × 4 mesh, thin beam.
twistedbeam_sc8r_4x24x8_thin.inp: 4 × 24 × 8 mesh, thin beam.
twistedbeam_sc8r_8x48x1_thin.inp: 8 × 48 × 1 mesh, thin beam.
twistedbeam_sc8r_8x48x2_thin.inp: 8 × 48 × 2 mesh, thin beam.
twistedbeam_sc8r_8x48x4_thin.inp: 8 × 48 × 4 mesh, thin beam.
twistedbeam_sc8r_8x48x8_thin.inp: 8 × 48 × 8 mesh, thin beam.

References

MacNeal, R. H., and R. L. Harder, “A Proposed Standard Set of Problems to Test Finite Element Accuracy,” Finite Elements in Analysis Design, vol. 11, pp. 3–20, 1985.
Simo, J. C., D. D. Fox, and M. S. Rifai, “On a Stress Resultant Geometrically Exact Shell Model. Part II: The Linear Theory; Computational Aspects,” Computational Methods in Applied Mechanical Engineering, vol. 73, pp. 53–92, 1989.

Tables

Table 1. Tip displacements for 4-node shell meshes, thick case (b = 0.32 in).
Loading		In-plane ( $F_{y}$ = 1.0 lb)		Out-of-plane ( $F_{z}$ = 1.0 lb)
Reference solution		5.424 × 10⁻³ (in)		1.754 × 10⁻³ (in)
Element	Mesh	FE solution	% error	FE solution	% error
S4	2 × 12	5.440 × 10⁻³	0.29	1.730 × 10⁻³	−1.37
	4 × 24	5.428 × 10⁻³	0.07	1.747 × 10⁻³	−0.40
	8 × 48	5.427 × 10⁻³	0.05	1.753 × 10⁻³	−0.06
S4R	2 × 12	5.479 × 10⁻³	1.01	1.868 × 10⁻³	6.50
	4 × 24	5.437 × 10⁻³	0.24	1.777 × 10⁻³	1.31
	8 × 48	5.430 × 10⁻³	0.11	1.761 × 10⁻³	0.40
S4R5	2 × 12	5.443 × 10⁻³	0.35	1.879 × 10⁻³	7.10
	4 × 24	5.418 × 10⁻³	−0.10	1.768 × 10⁻³	0.78
	8 × 48	5.416 × 10⁻³	−0.15	1.755 × 10⁻³	0.05

Table 2. Tip displacements for 4-node shell meshes, thin case (b = 0.05 in).
Loading		In-plane ( $F_{y}$ = 1.0 lb)		Out-of-plane ( $F_{z}$ = 1.0 lb)
Reference solution		1.390 (in)		0.3431 (in)
Element	Mesh	FE solution	% error	FE solution	% error
S4	2 × 12	1.391	0.07	0.3397	−0.99
	4 × 24	1.388	−0.14	0.3421	−0.29
	8 × 48	1.388	−0.14	0.3427	−0.12
S4R	2 × 12	1.394	0.28	0.3403	−0.81
	4 × 24	1.389	−0.07	0.3422	−0.26
	8 × 48	1.388	−0.14	0.3428	−0.09
S4R5	2 × 12	1.389	−0.07	0.3388	−1.25
	4 × 24	1.387	−0.22	0.3418	−0.38
	8 × 48	1.387	−0.22	0.3426	−0.15

Table 3. Tip displacements for continuum shell meshes, thick case (b = 0.32 in).
Loading		In-plane ( $F_{y}$ = 1.0 lb)		Out-of-plane ( $F_{z}$ = 1.0 lb)
Reference solution		5.424 × 10⁻³ (in)		1.754 × 10⁻³ (in)
Element	Mesh	FE solution	% error	FE solution	% error
SC8R	2 × 12 × 1	7.819 × 10⁻³	44.2	2.428 × 10⁻³	38.4
	2 × 12 × 2	5.254 × 10⁻³	–3.13	1.887 × 10⁻³	7.59
	2 × 12 × 4	5.352 × 10⁻³	–1.33	1.874 × 10⁻³	6.82
	2 × 12 × 8	5.410 × 10⁻³	–0.27	1.873 × 10⁻³	6.79
	4 × 24 × 1	7.696 × 10⁻³	41.9	2.388 × 10⁻³	36.2
	4 × 24 × 2	5.229 × 10⁻³	–3.59	1.798 × 10⁻³	2.53
	4 × 24 × 4	5.349 × 10⁻³	–1.38	1.777 × 10⁻³	1.28
	4 × 24 × 4	5.395 × 10⁻³	–0.53	1.775 × 10⁻³	1.17
	8 × 48 × 1	7.635 × 10⁻³	40.8	2.380 × 10⁻³	35.7
	8 × 48 × 2	5.220 × 10⁻³	–3.76	1.781 × 10⁻³	1.54
	8 × 48 × 4	5.331 × 10⁻³	–1.72	1.781 × 10⁻³	0.3
	8 × 48 × 8	5.393 × 10⁻³	–0.57	1.757 × 10⁻³	0.19

Table 4. Tip displacements for continuum shell meshes, thin case (b = 0.05 in).
Loading		In-plane ( $F_{y}$ = 1.0 lb)		Out-of-plane ( $F_{z}$ = 1.0 lb)
Reference solution		1.390 (in)		0.3431 (in)
Element	Mesh	FE solution	% error	FE solution	% error
SC8R	2 × 12 × 1	1.927	38.6	0.5826	69.8
	2 × 12 × 2	1.347	–3.09	0.3574	4.17
	2 × 12 × 4	1.366	–1.73	0.3412	–0.55
	2 × 12 × 8	1.378	–0.86	0.3384	–1.37
	4 × 24 × 1	1.908	37.3	0.5828	69.9
	4 × 24 × 2	1.346	–3.17	0.3608	5.16
	4 × 24 × 4	1.368	–1.58	0.3451	0.58
	4 × 24 × 4	1.381	–0.65	0.3423	–0.23
	8 × 48 × 1	1.903	36.9	0.5829	69.9
	8 × 48 × 2	1.346	–3.17	0.3617	5.42
	8 × 48 × 4	1.368	–1.58	0.3461	0.87
	8 × 48 × 8	1.382	–0.59	0.3433	0.06

Table 5. Tip displacements for 3-node shell meshes, thick case (b = 0.32 in).
Loading		In-plane ( $F_{y}$ = 1.0 lb)		Out-of-plane ( $F_{z}$ = 1.0 lb)
Reference solution		5.424 × 10⁻³ (in)		1.754 × 10⁻³ (in)
Element	Mesh	FE solution	% error	FE solution	% error
S3R	4 × 6	5.262 × 10⁻³	−2.99	1.400 × 10⁻³	−20.18
	8 × 12	5.361 × 10⁻³	−1.16	1.581 × 10⁻³	−9.86
	16 × 24	5.405 × 10⁻³	−0.35	1.696 × 10⁻³	−3.31
STRI3	4 × 6	5.323 × 10⁻³	−1.86	1.438 × 10⁻³	−18.01
	8 × 12	5.359 × 10⁻³	−1.20	1.594 × 10⁻³	−9.18
	16 × 24	5.386 × 10⁻³	−0.70	1.698 × 10⁻³	−3.19

Table 6. Tip displacements for 3-node shell meshes, thin case (b = 0.05 in).
Loading		In-plane ( $F_{y}$ = 1.0 lb)		Out-of-plane ( $F_{z}$ = 1.0 lb)
Reference solution		1.390 (in)		0.3431 (in)
Element	Mesh	FE solution	% error	FE solution	% error
S3R	4 × 6	1.352	−2.73	0.3251	−5.25
	8 × 12	1.372	−1.29	0.3381	−1.46
	16 × 24	1.383	−0.50	0.3417	−0.41
STRI3	4 × 6	1.383	−0.50	0.3382	−1.43
	8 × 12	1.384	−0.43	0.3413	−0.52
	16 × 24	1.386	−0.29	0.3424	−0.20

Table 7. Tip displacements for quadratic shell meshes, thick case (b = 0.32 in).
Loading		In-plane ( $F_{y}$ = 1.0 lb)		Out-of-plane ( $F_{z}$ = 1.0 lb)
Reference solution		5.424 × 10⁻³ (in)		1.754 × 10⁻³ (in)
Element	Mesh	FE solution	% error	FE solution	% error
STRI65	2 × 6	5.408 × 10⁻³	−0.29	1.751 × 10⁻³	−0.17
	4 × 12	5.412 × 10⁻³	−0.22	1.752 × 10⁻³	−0.11
	8 × 24	5.414 × 10⁻³	−0.18	1.752 × 10⁻³	−0.11
S8R	1 × 6	5.376 × 10⁻³	−0.88	1.745 × 10⁻³	−0.51
	2 × 12	5.411 × 10⁻³	−0.24	1.752 × 10⁻³	−0.11
	4 × 24	5.415 × 10⁻³	−0.17	1.752 × 10⁻³	−0.11
S8R5 & S9R5	1 × 6	5.405 × 10⁻³	−0.35	1.746 × 10⁻³	−0.46
	2 × 12	5.413 × 10⁻³	−0.20	1.752 × 10⁻³	−0.11
	4 × 24	5.416 × 10⁻³	−0.15	1.753 × 10⁻³	−0.06

Table 8. Tip displacements for quadratic shell meshes, thin case (b = 0.05 in).
Loading		In-plane ( $F_{y}$ = 1.0 lb)		Out-of-plane ( $F_{z}$ = 1.0 lb)
Reference solution		1.390 (in)		0.3431 (in)
Element	Mesh	FE solution	% error	FE solution	% error
STRI65	2 × 6	1.384	−0.43	0.3420	−0.32
	4 × 12	1.384	−0.43	0.3429	−0.06
	8 × 24	1.386	−0.29	0.3429	−0.06
S8R	1 × 6	1.214	−12.66	0.3311	−3.50
	2 × 12	1.379	−0.79	0.3427	−0.11
	4 × 24	1.387	−0.22	0.3429	−0.05
S8R5 & S9R5	1 × 6	1.386	−0.29	0.3423	−0.23
	2 × 12	1.387	−0.22	0.3429	−0.05
	4 × 24	1.387	−0.21	0.3429	−0.05

Table 9. Tip displacements for beam meshes, thick case (b = 0.32 in).
Loading		In-plane ( $F_{y}$ = 1.0 lb)		Out-of-plane ( $F_{z}$ = 1.0 lb)
Reference solution		5.424 × 10⁻³ (in)		1.754 × 10⁻³ (in)
Element	Mesh	FE solution	% error	FE solution	% error
B31	12	5.422 × 10⁻³	−0.04	1.753 × 10⁻³	−0.06
	24	5.428 × 10⁻³	0.07	1.750 × 10⁻³	−0.23
	48	5.429 × 10⁻³	0.09	1.750 × 10⁻³	−0.23
B32	6	5.429 × 10⁻³	0.09	1.750 × 10⁻³	−0.23
	12	5.429 × 10⁻³	0.09	1.750 × 10⁻³	−0.23
	24	5.429 × 10⁻³	0.09	1.750 × 10⁻³	−0.23
B33	12	5.430 × 10⁻³	0.11	1.743 × 10⁻³	−0.63
	24	5.429 × 10⁻³	0.09	1.743 × 10⁻³	−0.63
	48	5.428 × 10⁻³	0.07	1.743 × 10⁻³	−0.63

Table 10. Tip displacements for beam meshes, thin case (b = 0.05 in).
Loading		In-plane ( $F_{y}$ = 1.0 lb)		Out-of-plane ( $F_{z}$ = 1.0 lb)
Reference solution		1.390 (in)		0.3431 (in)
Element	Mesh	FE solution	% error	FE solution	% error
B31	12	1.392	0.15	0.3438	0.26
	24	1.394	0.29	0.3430	−0.03
	48	1.394	0.29	0.3428	−0.03
B32	6	1.394	0.29	0.3427	−0.03
	12	1.394	0.29	0.3427	−0.03
	24	1.394	0.29	0.3427	−0.03
B33	12	1.395	0.36	0.3417	−0.32
	24	1.395	0.36	0.3418	−0.32
	48	1.395	0.36	0.3421	−0.32

Table 11. Tip displacements for 4-node shell 2 x 12 mesh in Abaqus/Explicit, thick case (b = 0.32 in).
Loading		In-plane ( $F_{y}$ = 1.0 lb)		Out-of-plane ( $F_{z}$ = 1.0 lb)
Reference solution		5.424 × 10⁻³ (in)		1.754 × 10⁻³ (in)
Element	Mesh	FE solution	% error	FE solution	% error
S4R	2 × 12	5.542 x 10⁻³	2.18	1.800 × 10⁻³	2.62
S4RS	2 × 12	5.438 × 10⁻³	2.57	1.802 × 10⁻³	2.74
S4RSW	2 × 12	5.435 × 10⁻³	0.20	1.869 × 10⁻³	6.56

Table 12. Tip displacements for 4-node shell 2 x 12 mesh in Abaqus/Explicit, thin case (b = 0.05 in).
Loading		In-plane ( $F_{y}$ = 1.0 lb)		Out-of-plane ( $F_{z}$ = 1.0 lb)
Reference solution		1.390 (in)		0.3431 (in)
Element	Mesh	FE solution	% error	FE solution	% error
S4R	2 × 12	1.366	-1.73	0.3443	0.35
S4RS	2 × 12	1.376	-1.01	0.3390	−1.19
S4RSW	2 × 12	1.424	2.45	0.3821	11.37

Figures