2.3.4 Skew sensitivity of shell elements

Products: ABAQUS/Standard  ABAQUS/Explicit  

This example is intended to provide an evaluation of the sensitivity of the shell elements in ABAQUS to skew distortion when they are used as thin plates. An analytical series solution to the boundary value problem is available in Morley (1963), and an identical evaluation of elements in numerous other commercial codes is presented by Robinson (1985).

Problem description

Results and discussion

Parametric study using a parametric study script

The skew sensitivity investigation discussed in this example can be performed conveniently as a parametric study using the Python scripting capabilities offered in ABAQUS. As an example we perform a parametric study in ABAQUS/Standard in which 15 analyses are automatically executed; these analyses correspond to combinations of five different values of the skew angle (: 90°, 80°, 60°, 40°, and 30°) for three different element types (S8R, S4R, and S4). We also perform a parametric study in ABAQUS/Explicit in which 12 analyses are executed automatically; these analyses correspond to combinations of three different values of the skew angle (: 90°, 60°, and 30°), two different element types (S4R and S4RS), and two mesh discretizations (4 × 4 and 8 × 8 elements).

skewshell_parametric.inp shows the parametrized template input data used to generate the parametric variations of the ABAQUS/Standard parametric study. The parametric study script file (skewshell_parametric.psf) is used to perform the parametric study. The vertical displacement in the center of the plate is reported in the following table for each of the analyses of the parametric study:

   ________________________________________________

        Parametric study: skewshell_parametric
   ________________________________________________

         elemType,         delta,      N405_U.3,
   ________________________________________________

              s8r,            90,   -0.00150858,
              s4r,            90,   -0.00149891,
               s4,            90,   -0.00144697,
              s8r,            80,   -0.00141673,
              s4r,            80,   -0.00143168,
               s4,            80,   -0.00137446,
              s8r,            60,  -0.000845317,
              s4r,            60,  -0.000969093,
               s4,            60,  -0.000885679,
              s8r,            40,  -0.000258699,
              s4r,            40,  -0.000371343,
               s4,            40,  -0.000315966,
              s8r,            30,   -9.5434e-05,
              s4r,            30,  -0.000153366,
               s4,            30,  -0.000130785,
   ________________________________________________
These results match the corresponding results found in Table 2.3.4–5 to Table 2.3.4–8.

skew_discr.inp shows the parametrized template input data used to generate the parametric variations for the ABAQUS/Explicit parametric study. The parametric study script file (skew_discr.psf) is used to perform the parametric study. The vertical displacement at the center of the plate is reported in the following table for each analysis of the parametric study:

                      Parametric study: skewXpl
   ________________________________________________________________

            level,      elemType,         delta,      N405_U.3,
   ________________________________________________________________

                1,           s4r,            90,   -0.00144092,
                2,           s4r,            90,   -0.00144511,
                1,          s4rs,            90,   -0.00155302,
                2,          s4rs,            90,   -0.00147813,
                1,           s4r,            60,  -0.000954238,
                2,           s4r,            60,  -0.000925741,
                1,          s4rs,            60,   -0.00102325,
                2,          s4rs,            60,  -0.000963277,
                1,           s4r,            30,  -0.000151794,
                2,           s4r,            30,  -0.000148982,
                1,          s4rs,            30,  -0.000161744,
                2,          s4rs,            30,  -0.000162303,
   ________________________________________________________________
The results match the corresponding results found in Table 2.3.4–11 to Table 2.3.4–13.

Input files

References

Tables

Table 2.3.4–1 Skewed plate results: S3R, ABAQUS/Standard analysis.

Skew angleMesh
(mm)Error(× 10–2 N-m/m)Error(× 10–2 N-m/m)Error
90°Series      
 solution1.478 4.79 4.79 
 4 × 41.214–17.9%4.03–15.9%3.97–17.1%
 8 × 81.425–3.6%4.861.5%4.841.0%
 14 × 141.462–1.1%4.810.4%4.800.2%
80°Series      
 solution1.409 4.86 4.48 
 4 × 41.148–18.5%4.09–15.8%3.60–19.6%
 8 × 81.343–4.7%4.911.0%4.44–0.9%
 14 × 141.391–1.3%4.870.2%4.510.7%
60°Series      
 solution0.932 4.25 3.33 
 4 × 40.615–34.0%2.98–29.9%1.93–42.0%
 8 × 80.812–12.9%3.82–10.1%2.82–15.3%
 14 × 140.913–2.0%4.19–1.4%3.31–0.6%
40°Series      
 solution0.349 2.81 1.80 
 4 × 40.213–39.0%1.86–33.8%0.88–51.1%
 8 × 80.292–16.3%2.42–13.9%1.39–22.8%
 14 × 140.346–0.8%2.810.0%1.821.1%
30°Series      
 solution0.148 1.91 1.08 
4 × 40.080–45.9%1.14–40.3%0.46–57.4%
 8 × 80.125–15.5%1.60–16.2%0.80–25.9%
 14 × 140.1480.0%1.89–1.0%1.080.0%

Table 2.3.4–2 Skewed plate results: STRI3, ABAQUS/Standard analysis.

Skew angleMesh
(mm)Error(× 10–2 N-m/m)Error(× 10–2 N-m/m)Error
90°Series      
 solution1.478 4.79 4.79 
 4 × 41.4880.7%5.228.9%5.228.9%
 8 × 81.4810.2%4.892.0%4.892.0%
 14 × 141.4800.1%4.820.6%4.820.6%
80°Series      
 solution1.409 4.86 4.48 
 4 × 41.4190.7%5.3710%4.837.8%
 8 × 81.4100.1%4.982.4%4.572.0%
 14 × 141.4090.0%4.890.7%4.510.7%
60°Series      
 solution0.932 4.25 3.33 
 4 × 40.9653.5%4.8614%3.628.8%
 8 × 80.9400.8%4.434.2%3.412.4%
 14 × 140.9350.3%4.311.4%3.360.9%
40°Series      
 solution0.349 2.81 1.80 
 4 × 40.39012%3.4021%2.1519%
 8 × 80.3634.2%3.058.5%1.937.4%
 14 × 140.3572.4%2.913.4%1.874.1%
30°Series      
 solution0.148 1.91 1.08 
4 × 40.17316%2.3523%1.3525%
 8 × 80.1586.6%2.1211%1.2213%
 14 × 140.1543.8%2.015.3%1.167.5%

Table 2.3.4–3 Skewed plate results: STRI65, ABAQUS/Standard analysis.

Skew angleMesh
(mm)Error(× 10–2 N-m/m)Error(× 10–2 N-m/m)Error
90°Series      
 solution1.478 4.79 4.79 
 4 × 41.481–0.2%5.116.7%4.994.2%
 8 × 81.4860.5%4.912.5%4.871.7%
 14 × 141.4840.4%4.830.8%4.810.4%
80°Series      
 solution1.409 4.86 4.48 
 4 × 41.377–2.3%4.890.6%4.653.8%
 8 × 81.4130.3%4.931.4%4.612.9%
 14 × 141.4140.3%4.890.6%4.541.3%
60°Series      
 solution0.932 4.25 3.33 
 4 × 40.825–3.5%3.95–7%3.06–8.2%
 8 × 80.919–0.8%4.270.5%3.360.9%
 14 × 140.9340.3%4.270.5%3.360.9%
40°Series      
 solution0.349 2.81 1.80 
 4 × 40.273–22%2.45–13%1.41–21%
 8 × 80.333–4.8%2.76–1.8%1.73–3.8%
 14 × 140.3500.6%2.810.0%1.821.1%
30°Series      
 solution0.148 1.91 1.08 
 4 × 40.114–23%1.6423%0.80–25%
 8 × 80.143–3.4%1.8711%1.03–5%
 14 × 140.1522.7%1.925.3%1.112.7%

Table 2.3.4–4 Skewed plate results: S4R5, ABAQUS/Standard analysis.

Skew angleMesh
(mm)Error(× 10–2 N-m/m)Error(× 10–2 N-m/m)Error
90°Series      
 solution1.478 4.79 4.79 
 4 × 41.5021.6%4.23–12%4.23–12%
 8 × 81.4850.5%4.65–2.8%4.65–2.8%
 14 × 141.4820.3%4.75–0.9%4.75–0.9%
80°Series      
 solution1.409 4.86 4.48 
 4 × 41.4361.9%4.29–11%3.96–12%
 8 × 81.4150.5%4.71–3.0%4.36–2.6%
 14 × 141.4120.2%4.81–1.0%4.45–0.7%
60°Series      
 solution0.932 4.25 3.33 
 4 × 40.9815.3%3.78–11%2.88–14%
 8 × 80.9431.2%4.12–3.1%3.25–2.3%
 14 × 140.9370.5%4.21–0.9%3.31–0.7%
40°Series      
 solution0.349 2.81 1.80 
 4 × 40.38410%2.58–8.1%1.45–19%
 8 × 80.3654.8%2.74–2.6%1.80–0.0%
 14 × 140.3572.3%2.79–0.8%1.83–1.7%
30°Series      
 solution0.148 1.91 1.08 
 4 × 40.1607.7%1.74–9.0%0.80–26%
 8 × 80.1607.9%1.89–1.3%1.080.0%
 14 × 140.1554.6%1.91–0.2%1.134.3%

Table 2.3.4–5 Skewed plate results: S4R, ABAQUS/Standard analysis.

Skew angleMesh
(mm)Error(× 10–2 N-m/m)Error(× 10–2 N-m/m)Error
90°Series      
 solution1.478 4.79 4.79 
 4 × 41.4981.4%4.22–12%4.22–12%
 8 × 81.4850.5%4.65–2.9%4.65–2.9%
 14 × 141.4830.3%4.75–0.9%4.75–0.9%
80°Series      
 solution1.409 4.86 4.48 
 4 × 41.4311.6%4.28–12%3.94–12%
 8 × 81.4150.4%4.71–3.0%4.36–2.7%
 14 × 141.4140.4%4.81–0.9%4.45–0.7%
60°Series      
 solution0.932 4.25 3.33 
 4 × 40.9694.0%3.76–12%2.84–15%
 8 × 80.9370.5%4.11–3.4%3.23–3.1%
 14 × 140.9360.4%4.21–0.9%3.30–1.0%
40°Series      
 solution0.349 2.81 1.80 
 4 × 40.3716.3%2.52–10%1.41–22%
 8 × 80.3531.1%2.68–4.5%1.72–4.4%
 14 × 140.3510.4%2.76–1.9%1.77–1.4%
30°Series      
 solution0.148 1.91 1.08 
 4 × 40.1533.4%1.70–11%0.78–27%
 8 × 80.1512.0%1.82–4.8%1.00–6.9%
 14 × 140.1490.7%1.86–2.7%1.05–2.5%
 4 × 4*0.1565.4%1.7211%0.79–27%
 8 × 8*0.1554.7%1.862.6%1.05–2.7%
 14 × 14*0.1501.3%1.88–1.5%1.10–1.8%
*ABAQUS/Standard finite-strain element with enhanced hourglass control.

Table 2.3.4–6 Skewed plate results: S4, ABAQUS/Standard analysis.

Skew angleMesh
(mm)Error(× 10–2 N-m/m)Error(× 10–2 N-m/m)Error
90°Series      
 solution1.478 4.79 4.79 
 4 × 41.447–2.1%4.78–0.2%4.78–0.2%
 8 × 81.474–0.3%4.800.2%4.800.2%
 14 × 141.4810.2%4.800.2%4.800.2%
80°Series      
 solution1.409 4.86 4.48 
 4 × 41.375–2.4%4.84–0.4%4.520.9%
 8 × 81.402–0.5%4.860.0%4.500.4%
 14 × 141.4100.1%4.860.0%4.500.4%
60°Series      
 solution0.932 4.86 3.33 
 4 × 40.886–2.4%4.84–0.4%3.381.5%
 8 × 80.910–0.5%4.860.0%3.31–0.6%
 14 × 140.9250.1%4.860.0%3.32–0.3%
40°Series      
 solution0.349 2.81 1.80 
 4 × 40.316–4.9%2.63–6.4%1.70–5.6%
 8 × 80.323–2.4%2.71–3.6%1.74–3.3%
 14 × 140.327–0.8%2.76–1.9%1.77–1.4%
30°Series      
 solution0.148 1.91 1.08 
 4 × 40.131–11.0%1.67–12.6%0.92–14.8%
 8 × 80.133–10.1%1.80 –5.8%1.02–5.6%
 14 × 140.141 –4.7%1.85 –3.1%1.06–1.9%

Table 2.3.4–7 Skewed plate results: S8R5, S9R5; ABAQUS/Standard analysis.

Skew angleMesh
(mm)Error(× 10–2 N-m/m)Error(× 10–2 N-m/m)Error
90°Series      
 solution1.478 4.79 4.79 
 4 × 41.4830.3%5.167.8%5.167.8%
 8 × 81.4810.2%4.881.9%4.881.9%
 14 × 141.4830.3%4.820.7%4.820.7%
80°Series      
 solution1.409 4.86 4.48 
 4 × 41.4130.3%5.186.5%4.919.6%
 8 × 81.4110.2%4.941.6%4.592.4%
 14 × 141.4130.3%4.890.6%4.531.0%
60°Series      
 solution0.932 4.25 3.33 
 4 × 40.9451.4%4.434.3%3.8415%
 8 × 80.9370.6%4.311.4%3.453.5%
 14 × 140.9380.7%4.280.8%3.381.5%
40°Series      
 solution0.349 2.81 1.80 
 4 × 40.3706.0%2.924.0%2.3531%
 8 × 80.3572.5%2.851.3%1.979.3%
 14 × 140.3572.5%2.851.4%1.884.7%
30°Series      
 solution0.148 1.91 1.08 
 4 × 40.16410%2.057.4%1.5140%
 8 × 80.1564.9%1.941.7%1.2616%
 14 × 140.1554.5%1.952.2%1.178.2%

Table 2.3.4–8 Skewed plate results: S8R, ABAQUS/Standard analysis.

Skew angleMesh
(mm)Error(× 10–2 N-m/m)Error(× 10–2 N-m/m)Error
90°Series      
 solution1.478 4.79 4.79 
 4 × 41.5092.1%5.229.1%5.229.1%
 8 × 81.4941.1%4.912.5%4.912.5%
 14 × 141.4921.0%4.851.2%4.851.2%
80°Series      
 solution1.409 4.86 4.48 
 4 × 41.4170.6%5.258.0%4.899.1%
 8 × 81.4210.9%4.972.3%4.602.8%
 14 × 141.4210.9%4.911.1%4.551.5%
60°Series      
 solution0.932 4.25 3.33 
 4 × 40.845–9.3%4.464.8%3.391.8%
 8 × 80.915–1.8%4.301.2%3.32–0.1%
 14 × 140.9330.2%4.290.8%3.350.7%
40°Series      
 solution0.349 2.81 1.80 
 4 × 40.259–26%2.73–3.0%1.50–17%
 8 × 80.308–12%2.68–4.7%1.57–13%
 14 × 140.332–4.7%2.75–2.2%1.70–5.5%
30°Series      
 solution0.148 1.91 1.08 
 4 × 40.095–36%1.72–10%0.78–28%
 8 × 80.119–20%1.70–11%0.82–24%
 14 × 140.1490.5%1.910.2%1.090.8%

Table 2.3.4–9 Skewed plate results: SC6R, ABAQUS/Standard analysis.

Skew angleMesh
(mm)Error(× 10–2 N-m/m)Error(× 10–2 N-m/m)Error
90°Series      
 solution1.478 4.79 4.79 
 4 × 41.214–17.8%4.00–6.5%4.00–16.5%
 8 × 81.425–3.6%4.851.3%4.851.3%
 14 × 141.462–1.1%4.800.3%4.800.3%
80°Series      
 solution1.409 4.86 4.48 
 4 × 41.147–18.5%4.07–16.3%3.54–20.9%
 8 × 81.343–4.7%4.921.1%4.43–1.1%
 14 × 141.391–1.3%4.870.2%4.500.5%
60°Series      
 solution0.932 4.25 3.33 
 4 × 40.615–34%3.02–28.9%1.94–41.8%
 8 × 80.812–12.9%3.83–9.8%2.81–15.5%
 14 × 140.913–2.1%4.19–1.5%3.31-0.7%
40°Series      
 solution0.349 2.81 1.80 
 4 × 40.213–38.9%1.89–32.9%0.89–50.7%
 8 × 80.292–16.3%2.43–13.6%1.39–22.8%
 14 × 140.346–0.8%2.810%1.82–1.1%
30°Series      
 solution0.148 1.91 1.08 
 4 × 40.080–46.2%1.16–39.4%0.46–56.9%
 8 × 80.125–15.8%1.61–15.9%0.80–26.1%
 14 × 140.1480.1%1.911.9%1.070.5%

Table 2.3.4–10 Skewed plate results: SC8R, ABAQUS/Standard analysis.

Skew angleMesh
(mm)Error(× 10–2 N-m/m)Error(× 10–2 N-m/m)Error
90°Series      
 solution1.478 4.79 4.79 
 4 × 41.5031.7%4.23–11.6%4.23–11.6%
 8 × 81.4870.6%4.66–2.8%4.66–2.8%
 14 × 141.485–0.5%4.75–0.8%4.75–0.8%
80°Series      
 solution1.409 4.86 4.48 
 4 × 41.4361.9%4.29–11.8%3.96—11.6%
 8 × 81.4170.6%4.72–2.9%4.43–2.5%
 14 × 141.414–0.4%4.82–0.8%4.46–0.5%
60°Series      
 solution0.932 4.25 3.33 
 4 × 40.981–5.2%3.78–11.0%2.87–13.7%
 8 × 80.9431.2%4.12–3.1%3.25–2.3%
 14 × 140.9380.6%4.22–0.7%3.31–0.5%
40°Series      
 solution0.349 2.81 1.80 
 4 × 40.3839.6%2.58–8.2%1.45–19.5%
 8 × 80.3634.1%2.73–2.8%1.79–0.5%
 14 × 140.3551.7%2.78–0.9%1.82–1.3%
30°Series      
 solution0.148 1.91 1.08 
 4 × 40.1586.8%1.74–9.1%0.80–25.9%
 8 × 80.1586.8%1.88–1.6%1.07–0.8%
 14 × 140.533.4%1.90–0.7%1.113.1%

Table 2.3.4–11 Skewed plate results: S3R, ABAQUS/Explicit analysis.

Skew AngleMesh
(mm)Error(×10–2 N-m/m)Error(×10–2N-m/m)Error
90°Series      
solution1.478 4.79 4.79 
2 × 2 × 40.949–36.2%2.69–44.0%2.69–44.0%
4 × 4 × 41.325–10.4%4.30–10.2%4.30–10.2%
8 × 8 × 41.413–4.4%4.56–4.8%4.56–4.8%
80°Series      
solution1.409 4.86 4.48 
2 × 2 × 40.941–33.0%2.71–44.1%2.65–40.8%
4 × 4 × 41.257–10.7%4.26–12.3%4.18–6.6%
8 × 8 × 41.347–4.4%4.654.0%4.26–4.9%
60°Series      
solution0.932 4.25 3.33
2 × 2 × 40.783–15.9%2.60–38.1%2.19–34.3%
4 × 4 × 40.822–11.8%3.99–6.1%2.98–10.5%
8 × 8 × 40.897–3.7%4.14–2.5%3.21–3.6%
40°Series      
solution0.349 2.81 1.80
2 × 2 × 40.332–4.9%1.78–36.6%1.10–38.8%
4 × 4 × 40.326–6.6%2.63–6.4%1.65–8.3%
8 × 8 × 40.3480.2%2.841.0%1.800.0%
30°Series      
solution0.148 1.91 1.08
2 × 2 × 40.145–2.0%1.16–39.2%0.60–44.4%
4 × 4 × 40.148–0.0%1.78–6.8%0.98–9.3%
8 × 8 × 40.1522.7%1.862.6%1.101.9%

Table 2.3.4–12 Skewed plate results: S4R, ABAQUS/Explicit analysis.

Skew AngleMesh
(mm)Error(×10–2 N-m/m) Error(×10–2N-m/m)Error
90° Series      
 solution1.478 4.79 4.79 
 4 × 41.444–2.3%3.95–17%3.95–17%
 8 × 81.450–1.9%4.49–6.0%4.49–6.0%
 14 × 141.445–2.2%4.60–3.9%4.60–3.9%
80°Series      
 solution1.409 4.86 4.48 
 4 × 41.356–4.0%4.08–16%3.71–17%
 8 × 81.372–2.6%4.57–6.0%4.20–6.2%
 14 × 141.378–2.2%4.68–2.5%4.32–3.5%
60°Series      
 solution0.932 4.25 3.33 
 4 × 40.9572.7%3.53–16%2.54–23%
 8 × 80.930–0.2%4.02–5.4%3.09–7.2%
 14 × 140.922–1.0%4.15–2.3%3.23–3.0%
40°Series      
 solution0.349 2.81 1.80
 4 × 40.305–12%2.21–21%1.17–35%
 8 × 80.328–6.0%2.59–7.9%1.57–13%
 14 × 140.344–1.5%2.74–2.5%1.73–4.0%
30°Series      
 solution0.148 1.91 1.08 
 4 × 40.1522.7%1.45–24%0.63–42%
 8 × 80.1512.0%1.71–10%0.88–26%
 14 × 140.144–2.7%1.84–3.7%1.01–6.4%

Table 2.3.4–13 Skewed plate results: S4RS, ABAQUS/Explicit analysis.

Skew AngleMesh
(mm)Error(×10–2 N-m/m)Error(×10–2N-m/m)Error
90°Series      
solution1.478 4.79 4.79 
4 × 41.553+5.1%4.29–10%4.29–10%
8 × 81.477–0.1%4.60–4.0%4.60–4.0%
14 × 141.458–1.4%4.63–3.3%4.63–3.3%
80°Series      
solution1.409 4.86 4.48 
4 × 41.516+7.6%4.46–8.2%4.08–8.9%
8 × 81.4090.0%4.72–2.9%4.26–4.9%
14 × 141.391–1.3%4.73–2.7%4.32–3.6%
60°Series      
solution0.932 4.25 3.33 
4 × 41.026+10%3.91–8.0%2.96–11%
8 × 80.954+2.1%4.20–1.2%3.21–3.6%
14 × 140.942+1.0%4.19–1.4%3.330.0%
40°Series      
solution0.349 2.81 1.80 
4 × 40.397+14%2.65–5.7%1.49–17%
8 × 80.371+6.3%2.78–1.1%1.82+1.1%
14 × 140.361+3.4%2.80–0.5%1.87+4.0%
30°Series      
solution0.148 1.91 1.08 
4 × 40.162+9.5%1.76–7.9%0.82–24%
8 × 80.162+9.5%1.910.0%1.09+1.0%
14 × 140.156+5.4%1.92+0.5%1.14+5.6%

Table 2.3.4–14 Skewed plate results: SC8R, ABAQUS/Explicit analysis.

Skew AngleMesh
(mm)Error(×10–2 N-m/m) Error(×10–2N-m/m)Error
90° Series      
 solution1.478 4.79 4.79 
 4 × 41.466–0.8%4.15–13%4.15–13%
 8 × 81.446–2.1%4.53–5.4%4.53–5.4%
 14 × 141.443–2.4%4.61–3.8%4.61–3.8%
80°Series      
 solution1.409 4.86 4.48 
 4 × 41.403–0.4%4.20–14%3.87–14%
 8 × 81.381–2.0%4.59–5.6%4.25–5.1%
 14 × 141.378–2.2%4.68–3.7%4.33–3.3%
60°Series      
 solution0.932 4.25 3.33 
 4 × 40.9571.6%3.72–13%2.80–16%
 8 × 80.926–0.6%4.06–4.5%3.18–4.5%
 14 × 140.924–0.9%4.16–2.1%3.26–2.1%
40°Series      
 solution0.349 2.81 1.80 
 4 × 40.3675.2%2.50–11%1.39–23%
 8 × 80.3500.3%2.67–5.0%1.71–5%
 14 × 140.3490.0%2.76–1.8%1.78–1.1%
30°Series      
 solution0.148 1.91 1.08 
 4 × 40.1522.0%1.68–12%0.77–29%
 8 × 80.1490.7%1.81–5.2%0.99–8.3%
 14 × 140.1490.7%1.87–2.1%1.06–1.9%


Figures

Figure 2.3.4–1 Simply supported skew plate with uniform distributed load.

Figure 2.3.4–2 4 × 4 mesh for the complete plate of quadrilateral elements.

Figure 2.3.4–3 2 × 2 × 4 mesh for the complete plate of triangular elements.

Figure 2.3.4–4 Energy balance for 14 × 14 mesh at 40° (ABAQUS/Explicit analysis).