Shell elements – benchmark studies

A series of benchmark tests are presented for shell elements used in IDEA StatiCa Steel. The article shows mesh convergence to analytical solution with varying plate thickness and mesh distortion.

Bending of Quad 1

 Theoretical results:

  • Deformation at point C: \(w_c=0.096/T+13333.33333/T^3\)
  • Bending moment at A: \(m_x=800\)

Distortion coefficient s of mesh is defined by moving all internal nodes by 

\[x'=x+r_c s \Delta x\]

\[y'=y+r_c s \Delta y\]

where \(r_c\) is a random number in interval \(<-1, 1>\), \(\Delta x\) or \(\Delta y\) are typical element lengths.

Mesh examples:

Comparison of numerical and theoretical results:

Bending of Tria 1

 Theoretical results:

  • deformation at point C: \(w_c = 0.096/T+13333.33333/T^3\)
  • moment at A: \(m_x=800\)

Distortion coefficient s of mesh is defined as moving all internal nodes by

\[x'=x+r_c s \Delta x\]

\[y'=y+r_c s \Delta y\]

where \(r_c\) is a random number in interval \(<-1, 1>\), \(\Delta x\) or \(\Delta y\) are typical element lengths.

Mesh examples:

Comparison of numerical and theoretical results:

Bending of Quad 2

Simple supported square plate

\[L=1000;\,E=2.1e5;\,μ=0.3;\,p=1e-3\]

Theoretical solution (by sinus series expansion):

  • Deflection at C: \(w_t = 211.2423380/T^3+0.1094543906e-2/T\)
  • Bending moment \(m_x\) at C: \(m_{x,C} = 47.88629590\)

Distortion is defined as in the first benchmark.

Mesh examples:

Comparison of numerical and theoretical results:

Bending of Tria 2

Simple supported square plate

\[L=1000;\,E=2.1e5;\,μ=0.3;\,p=1e-3\]

Theoretical solution (by sinus series expansion):

  • Deflection at C: \(wt = 211.2423380/T^3+0.1094543906e-2/T\)
  • Moment \(m_x\) at C: \(m_{x,C} = 47.88629590\)

Distortion is defined as in the first benchmark.

Mesh examples:

Comparison of numerical and theoretical results:

Conclusion

Generally, the convergence to analytical solution for plate bending is very fast. Quads do not have any issues even with a high distortion coefficient. Extreme distortion may be an issue for Trias; it is recommended to watch out for Tria elements with very bad aspect ratio. IDEA StatiCa Steel greatly prefers Quads in automatic meshing.

The results do not vary significantly for different thicknesses.


Related articles

Plate model and mesh convergence