2.3.7 Analysis of a twisted beam

Products: ABAQUS/Standard  ABAQUS/Explicit  

This problem examines 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).

Problem description

Results and discussion

Input files

References

Tables

Table 2.3.7–1 Tip displacements for 4-node shell meshes, thick case (b = 0.32 in).

LoadingIn-plane ( = 1.0 lb)Out-of-plane ( = 1.0 lb)
Reference solution5.424 × 10–3 (in)1.754 × 10–3 (in)
ElementMeshFE solution% errorFE solution% error
S42 × 125.440 × 10–30.291.730 × 10–3–1.37
4 × 245.428 × 10–30.071.747 × 10–3–0.40
8 × 485.427 × 10–30.051.753 × 10–3–0.06
S4R2 × 125.479 × 10–31.011.868 × 10–36.50
4 × 245.437 × 10–30.241.777 × 10–31.31
8 × 485.430 × 10–30.111.761 × 10–30.40
S4R52 × 125.443 × 10–30.351.879 × 10–37.10
4 × 245.418 × 10–3–0.101.768 × 10–30.78
8 × 485.416 × 10–3–0.151.755 × 10–30.05

Table 2.3.7–2 Tip displacements for 4-node shell meshes, thin case (b = 0.05 in).

LoadingIn-plane ( = 1.0 lb)Out-of-plane ( = 1.0 lb)
Reference solution1.390 (in)0.3431 (in)
ElementMeshFE solution% errorFE solution% error
S42 × 121.3910.070.3397–0.99
4 × 241.388–0.140.3421–0.29
8 × 481.388–0.140.3427–0.12
S4R2 × 121.3940.280.3403–0.81
4 × 241.389–0.070.3422–0.26
8 × 481.388–0.140.3428–0.09
S4R52 × 121.389–0.070.3388–1.25
4 × 241.387–0.220.3418–0.38
8 × 481.387–0.220.3426–0.15

Table 2.3.7–3 Tip displacements for continuum shell meshes, thick case (b = 0.32 in).

LoadingIn-plane ( = 1.0 lb)Out-of-plane ( = 1.0 lb)
Reference solution5.424 × 10–3 (in)1.754 × 10–3 (in)
ElementMeshFE solution% errorFE solution% error
SC8R2 × 12 × 17.819 × 10–344.22.428 × 10–338.4
2 × 12 × 25.254 × 10–3–3.131.887 × 10–37.59
2 × 12 × 45.352 × 10–3–1.331.874 × 10–36.82
2 × 12 × 85.410 × 10–3–0.271.873 × 10–36.79
4 × 24 × 17.696 × 10–341.92.388 × 10–336.2
4 × 24 × 25.229 × 10–3–3.591.798 × 10–32.53
4 × 24 × 45.349 × 10–3–1.381.777 × 10–31.28
4 × 24 × 45.395 × 10–3–0.531.775 × 10–31.17
8 × 48 × 17.635 × 10–340.82.380 × 10–335.7
8 × 48 × 25.220 × 10–3–3.761.781 × 10–31.54
8 × 48 × 45.331 × 10–3–1.721.781 × 10–30.3
8 × 48 × 85.393 × 10–3–0.571.757 × 10–30.19

Table 2.3.7–4 Tip displacements for continuum shell meshes, thin case (b = 0.05 in).

LoadingIn-plane ( = 1.0 lb)Out-of-plane ( = 1.0 lb)
Reference solution1.390 (in)0.3431 (in)
ElementMeshFE solution% errorFE solution% error
SC8R2 × 12 × 11.92738.60.582669.8
2 × 12 × 21.347–3.090.35744.17
2 × 12 × 41.366–1.730.3412–0.55
2 × 12 × 81.378–0.860.3384–1.37
4 × 24 × 11.90837.30.582869.9
4 × 24 × 21.346–3.170.36085.16
4 × 24 × 41.368–1.580.34510.58
4 × 24 × 41.381–0.650.3423–0.23
8 × 48 × 11.90336.90.582969.9
8 × 48 × 21.346–3.170.36175.42
8 × 48 × 41.368–1.580.34610.87
8 × 48 × 81.382–0.590.34330.06

Table 2.3.7–5 Tip displacements for 3-node shell meshes, thick case (b = 0.32 in).

LoadingIn-plane ( = 1.0 lb)Out-of-plane ( = 1.0 lb)
Reference solution5.424 × 10–3 (in)1.754 × 10–3 (in)
ElementMeshFE solution% errorFE solution% error
S3R4 × 65.262 × 10–3–2.991.400 × 10–3–20.18
8 × 125.361 × 10–3–1.161.581 × 10–3–9.86
16 × 245.405 × 10–3–0.351.696 × 10–3–3.31
STRI34 × 65.323 × 10–3–1.861.438 × 10–3–18.01
8 × 125.359 × 10–3–1.201.594 × 10–3–9.18
16 × 245.386 × 10–3–0.701.698 × 10–3–3.19

Table 2.3.7–6 Tip displacements for 3-node shell meshes, thin case (b = 0.05 in).

LoadingIn-plane ( = 1.0 lb)Out-of-plane ( = 1.0 lb)
Reference solution1.390 (in)0.3431 (in)
ElementMeshFE solution% errorFE solution% error
S3R4 × 61.352–2.730.3251–5.25
8 × 121.372–1.290.3381–1.46
16 × 241.383–0.500.3417–0.41
STRI34 × 61.383–0.500.3382–1.43
8 × 121.384–0.430.3413–0.52
16 × 241.386–0.290.3424–0.20

Table 2.3.7–7 Tip displacements for quadratic shell meshes, thick case (b = 0.32 in).

LoadingIn-plane ( = 1.0 lb)Out-of-plane ( = 1.0 lb)
Reference solution5.424 × 10–3 (in)1.754 × 10–3 (in)
ElementMeshFE solution% errorFE solution% error
STRI652 × 65.408 × 10–3–0.291.751 × 10–3–0.17
4 × 125.412 × 10–3–0.221.752 × 10–3–0.11
8 × 245.414 × 10–3–0.181.752 × 10–3–0.11
S8R1 × 65.376 × 10–3–0.881.745 × 10–3–0.51
2 × 125.411 × 10–3–0.241.752 × 10–3–0.11
4 × 245.415 × 10–3–0.171.752 × 10–3–0.11
S8R5 & S9R51 × 65.405 × 10–3–0.351.746 × 10–3–0.46
2 × 125.413 × 10–3–0.201.752 × 10–3–0.11
4 × 245.416 × 10–3–0.151.753 × 10–3–0.06

Table 2.3.7–8 Tip displacements for quadratic shell meshes, thin case (b = 0.05 in).

LoadingIn-plane ( = 1.0 lb)Out-of-plane ( = 1.0 lb)
Reference solution1.390 (in)0.3431 (in)
ElementMeshFE solution% errorFE solution% error
STRI652 × 61.384–0.430.3420–0.32
4 × 121.384–0.430.3429–0.06
8 × 241.386–0.290.3429–0.06
S8R1 × 61.214–12.660.3311–3.50
2 × 121.379–0.790.3427–0.11
4 × 241.387–0.220.3429–0.05
S8R5 & S9R51 × 61.386–0.290.3423–0.23
2 × 121.387–0.220.3429–0.05
4 × 241.387–0.210.3429–0.05

Table 2.3.7–9 Tip displacements for beam meshes, thick case (b = 0.32 in).

LoadingIn-plane ( = 1.0 lb)Out-of-plane ( = 1.0 lb)
Reference solution5.424 × 10–3 (in)1.754 × 10–3 (in)
ElementMeshFE solution% errorFE solution% error
B31125.422 × 10–3–0.041.753 × 10–3–0.06
245.428 × 10–30.071.750 × 10–3–0.23
485.429 × 10–30.091.750 × 10–3–0.23
B3265.429 × 10–30.091.750 × 10–3–0.23
125.429 × 10–30.091.750 × 10–3–0.23
245.429 × 10–30.091.750 × 10–3–0.23
B33125.430 × 10–30.111.743 × 10–3–0.63
245.429 × 10–30.091.743 × 10–3–0.63
485.428 × 10–30.071.743 × 10–3–0.63

Table 2.3.7–10 Tip displacements for beam meshes, thin case (b = 0.05 in).

LoadingIn-plane ( = 1.0 lb)Out-of-plane ( = 1.0 lb)
Reference solution1.390 (in)0.3431 (in)
ElementMeshFE solution% errorFE solution% error
B31121.3920.150.34380.26
241.3940.290.3430–0.03
481.3940.290.3428–0.03
B3261.3940.290.3427–0.03
121.3940.290.3427–0.03
241.3940.290.3427–0.03
B33121.3950.360.3417–0.32
241.3950.360.3418–0.32
481.3950.360.3421–0.32

Table 2.3.7–11 Tip displacements for S4RSW elements for the in-plane shear load case.

SolutionThick case (b = 0.32 in)Thin case (b = 0.05 in)
Reference solution0.0054241.3900
at node 1320.005422 (–0.036%)1.41703 (1.94%)

Table 2.3.7–12 Tip displacements for S4RSW elements for the out-of-plane shear load case.

Solution Thick case (b = 0.32 in)Thin case (b = 0.05 in)
Reference solution0.0017540.34310
at node 1320.001859 (5.98%)0.38134 (11.14%)


Figures

Figure 2.3.7–1 Twisted beam.

Figure 2.3.7–2 Variation of at node 132 with time, ABAQUS/Explicit analysis.

Figure 2.3.7–3 Energy variation with time, ABAQUS/Explicit analysis.