Results.
| Example | Reference | Records | Largest deviation |
|---|---|---|---|
| Comparison with ETABS | Independent program | 23 | 0.255% |
| Comparison with the closed-form solution | Closed form | 12 | 0.949% |
Problem. An open-top reinforced concrete cylindrical water tank with a fixed base is solved separately with MindFEM and with ETABS, and the results are also compared with the closed-form solution of the cylindrical shell. The water pressure grows linearly with depth and the water surface passes through a row of elements. The hoop force, the vertical bending moment, the base shear and the displacement are compared along the height of the wall.
Reference document. The first reference is the same tank modelled in ETABS 20.1.0 with its own objects, with the water pressure given as a non-uniform area load. The second is the exact solution of the axisymmetric cylindrical shell equations including shear deformation.
The radial displacement w of the wall, its rotation φ, the vertical bending moment M and the shear force Q are linked by these four equations. D is the bending stiffness and κGt the shear stiffness (κ = 5/6). Below the water surface the pressure is p(z) = γ(d − z), above it zero, with d the water depth and γ the unit weight of water.
The hoop force follows from the radial displacement.
Numerical model. The mid-surface radius is 6 m, the wall 6 m high and 0.30 m thick, C30/37 concrete, with a fixed base and a free top edge, and the water is 5.3 m deep. The wall is divided into 72 flat panels around and into quadrilateral elements 0.125 m high.
Comparison with ETABS
Independent calculation. ETABS 20.1.0, the same structure modelled with ETABS's own objects
23-row table · largest deviation 0.255%
| Quantity | Reference | Computed | Deviation |
|---|---|---|---|
| G: total vertical reaction [kN] | 1,663.13 | 1,663.13 | < 0.001% |
| W: hoop force at z = 1 m [kN/m] | 116.473 | 116.386 | 0.0742% |
| W: hoop force at z = 2 m [kN/m] | 181.564 | 181.766 | 0.111% |
| W: hoop force at z = 3 m [kN/m] | 150.02 | 150.089 | 0.0464% |
| W: hoop force at z = 4 m [kN/m] | 85.7423 | 85.72 | 0.0261% |
| W: hoop force at z = 5 m [kN/m] | 26.001 | 25.9463 | 0.210% |
| W: vertical bending moment at z = 0 m [kNm/m] | -21.6275 | -21.6342 | 0.0311% |
| W: vertical bending moment at z = 1 m [kNm/m] | 3.88901 | 3.89173 | 0.0701% |
| W: vertical bending moment at z = 1.5 m [kNm/m] | 5.68376 | 5.68926 | 0.0968% |
| W: vertical bending moment at z = 2 m [kNm/m] | 4.7122 | 4.71721 | 0.106% |
| W: base shear [kN/m] | 47.7251 | 47.847 | 0.255% |
| W: radial displacement at z = 2 m [mm] | 0.113467 | 0.113298 | 0.149% |
| W: radial displacement at z = 3 m [mm] | 0.0937593 | 0.0937135 | 0.0489% |
| G + W: hoop force at z = 1 m [kN/m] | 112.489 | 112.396 | 0.0832% |
| G + W: hoop force at z = 2 m [kN/m] | 181.099 | 181.299 | 0.111% |
| G + W: hoop force at z = 3 m [kN/m] | 150.384 | 150.439 | 0.0363% |
| G + W: hoop force at z = 4 m [kN/m] | 85.9697 | 85.9304 | 0.0457% |
| G + W: hoop force at z = 5 m [kN/m] | 26.0398 | 26.0531 | 0.0509% |
| G + W: vertical bending moment at z = 0 m [kNm/m] | -22.2455 | -22.2519 | 0.0290% |
| G + W: vertical bending moment at z = 1 m [kNm/m] | 3.99284 | 3.99561 | 0.0693% |
| G + W: vertical bending moment at z = 1.5 m [kNm/m] | 5.84051 | 5.84648 | 0.102% |
| G + W: vertical bending moment at z = 2 m [kNm/m] | 4.84342 | 4.84902 | 0.116% |
| G + W: base shear [kN/m] | 49.0666 | 49.1657 | 0.202% |
Note. G is the self weight of the wall, W the 5.3 m of water inside the tank. The values are read on the line of nodes at θ = 0. The section forces are the mean of the values of the elements sharing the node. The hoop force acts in the horizontal direction of the element, the vertical bending moment in its vertical direction. The ETABS moment sign convention is the opposite of MindFEM's, and the table uses MindFEM's sign. The base shear is the horizontal reaction of the base node divided by that node's share of the circumference.
Comparison with the closed-form solution
Closed form. The axisymmetric closed-form solution of the cylindrical shell, with shear deformation
| Quantity | Reference | Computed | Deviation |
|---|---|---|---|
| W: hoop force at z = 1 m [kN/m] | 116.383 | 116.386 | 0.0024% |
| W: hoop force at z = 2 m [kN/m] | 181.479 | 181.766 | 0.158% |
| W: hoop force at z = 3 m [kN/m] | 150.046 | 150.089 | 0.0291% |
| W: hoop force at z = 4 m [kN/m] | 85.8083 | 85.72 | 0.103% |
| W: hoop force at z = 5 m [kN/m] | 25.9963 | 25.9463 | 0.192% |
| W: vertical bending moment at z = 0 m [kNm/m] | -21.6672 | -21.6342 | 0.152% |
| W: vertical bending moment at z = 1 m [kNm/m] | 3.85515 | 3.89173 | 0.949% |
| W: vertical bending moment at z = 1.5 m [kNm/m] | 5.66647 | 5.68926 | 0.402% |
| W: vertical bending moment at z = 2 m [kNm/m] | 4.70771 | 4.71721 | 0.202% |
| W: base shear [kN/m] | 47.84 | 47.847 | 0.0145% |
| W: radial displacement at z = 2 m [mm] | 0.113424 | 0.113298 | 0.111% |
| W: radial displacement at z = 3 m [mm] | 0.0937785 | 0.0937135 | 0.0694% |
Note. In the closed-form solution the displacement and rotation are zero at the base, the moment and shear force zero at the top edge. Below and above the water surface are two separate regions, and the displacement, rotation, moment and shear force are continuous at the surface. The thin-shell solution, which ignores shear deformation, gives a base moment 4.5 per cent larger, because the base moment dies out over about 1 m, and the 0.30 m wall thickness is not small next to that length. The moment at z = 1 m changes quickly near the point where it passes through zero. Here the difference is 3.9 per cent with elements 0.25 m high and 0.95 per cent with elements 0.125 m high.
Model files
The ETABS model that produced the results in the table can be downloaded. It opens in ETABS 20
and later with File > Import > ETABS .e2k File.