Objective
To determine the stress-strain state of a thick circular slab rigidly fixed along the contour and subjected to a uniformly distributed load applied to the upper surface, in accordance with the three-dimensional theory of elasticity.
Reference
B.M. Lisitsyn, Projection and projection-grid methods, Kyiv, Vyshcha Shkola, 1991
Problem statement
To determine the vertical displacements Z, the normal stress σz, and the radial stress σr at the centre of the slab, as well as the radial stress σr* at the fixed edge for values of 1; 0,5; 0; −0,5 and −1 at γ = a/h = 2 and 5.
Design model
A thick slab rigidly fixed along the contour, circular in plan, made of an isotropic linear-elastic material and subjected to one-sided uniformly distributed pressure.
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Variant 1
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Variant 2
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Geometry
Slab diameter 2a = 20 m
Slab thickness 2h = 10 m and 4 m
Material properties
Modulus of elasticity Е = 1 * 107 kPa
Poisson's ratio ν = 0,25.
Boundary conditions
Restraints in all degrees of freedom (DOF) are applied along the contour of the slab.
Loads
Uniformly distributed load applied over the area of the upper surface of the slab: q = 10 kN/m2.
Note
The problem is solved in a 3D formulation (model type 5).
The model is generated with FE type 36 – arbitrary 3D (8-node) isoparametric solid and FE type 34 – arbitrary 3D (6-node) isoparametric solid.
The local axes of solids for the results are aligned in such a way that each local X1-axis is directed away from the central axis of the plate (the Z1-axis is directed upward).
The finite element mesh of each slab is divided on plan:
in the radial direction — with a step of 0,5 m,
in the tangential direction — with a step of 5º and
through the thickness — with a step of 0,5 m.
Variant 1: Slab with thickness 10 m (γ = a/h = 2).
Nodes: 30261. Elements: 28800.
Variant 2: Slab with thickness 4 m (γ = a/h = 5).
Nodes: 12969. Elements: 11520.
Output data
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Variant 1
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Variant 2
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Variant 2
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Variant 1
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Variant 2
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Comparison of calculation results
Without additional side nodes:
| The unknown | Point z/h | Analytical solution | LIRA-FEM | Error, % | |
| σr / q | Var. 1 | 1 | -1,036 | -0,945 | 8,7838 |
| 0,5 | -0,395 | -0,408 | 3,1863 | ||
| 0 | -0,131 | -0,12 | 8,3969 | ||
| -0,5 | 0,133 | 0,125 | 6,015 | ||
| -1 | 0,531 | 0,59 | 10 | ||
| Var. 2 | 1 | -3,634 | -3,463 | 4,7056 | |
| 0,5 | -1,499 | -1,635 | 8,318 | ||
| 0 | -0,151 | -0,155 | 2,5806 | ||
| -0,5 | 1,202 | 1,323 | 9,1459 | ||
| -1 | 3,209 | 3,151 | 1,8074 | ||
| σr* / q | Var. 1 | 1 | 2,432 | 2,634 | 7,6689 |
| 0,5 | -0,282 | -0,145 | 48,5816 | ||
| 0 | -0,164 | -0,19 | 13,6842 | ||
| -0,5 | -0,241 | -0,209 | 13,278 | ||
| -1 | -1,403 | -1,296 | 7,6265 | ||
| Var. 2 | 1 | 5,939 | 2,186 | 63,1925 | |
| 0,5 | 1,544 | 0,447 | 71,0492 | ||
| 0 | -0,268 | -0,064 | 76,1194 | ||
| -0,5 | -1,803 | -0,558 | 69,0516 | ||
| -1 | -5,866 | -2 | 65,9052 | ||
| σz / q | Var. 1 | 1 | -1,113 | -1,037 | 6,8284 |
| 0,5 | -0,839 | -0,86 | 2,4419 | ||
| 0 | -0,534 | -0,531 | 0,5618 | ||
| -0,5 | -0,18 | -0,177 | 1,6667 | ||
| -1 | 0,096 | 0,032 | 66,6667 | ||
| Var. 2 | 1 | -1,107 | -1,252 | 11,5815 | |
| 0,5 | -0,935 | -0,828 | 11,4439 | ||
| 0 | -0,488 | -0,499 | 2,2044 | ||
| -0,5 | -0,116 | -0,171 | 32,1637 | ||
| -1 | 0,108 | 0,108 | 57,1429 | ||
| w / qa | Var. 1 | 1 | -1,156 | -1,148 | 0,692 |
| 0,5 | -0,99 | -0,99 | 0 | ||
| 0 | -0,848 | -0,845 | 0,3538 | ||
| -0,5 | -0,76 | -0,759 | 0,1316 | ||
| -1 | -0,709 | -0,703 | 0,8463 | ||
| Var. 2 | 1 | -4,558 | -4,482 | 1,6674 | |
| 0,5 | -4,575 | -4,512 | 1,377 | ||
| 0 | -4,543 | -4,489 | 1,1886 | ||
| -0,5 | -4,489 | -4,428 | 1,3589 | ||
| -1 | -4,382 | -4,314 | 1,5518 | ||
With additional side nodes:
| The unknown | Point z/h | Analytical solution | LIRA-FEM | Error, % | |
| σr / q | Var. 1 | 1 | -1,036 | -0,863 | 16,6988 |
| 0,5 | -0,395 | -0,412 | 4,1262 | ||
| 0 | -0,131 | -0,12 | 8,3969 | ||
| -0,5 | 0,133 | 0,127 | 4,5113 | ||
| -1 | 0,531 | 0,513 | 3,3898 | ||
| Var. 2 | 1 | -3,634 | -2,897 | 20,2807 | |
| 0,5 | -1,499 | -1,656 | 9,4807 | ||
| 0 | -0,151 | -0,156 | 3,2051 | ||
| -0,5 | 1,202 | 1,343 | 10,4989 | ||
| -1 | 3,209 | 2,584 | 19,4765 | ||
| σr* / q | Var. 1 | 1 | 2,432 | 1,182 | 51,398 |
| 0,5 | -0,282 | -0,159 | 43,617 | ||
| 0 | -0,164 | -0,201 | 18,408 | ||
| -0,5 | -0,241 | -0,223 | 7,4689 | ||
| -1 | -1,403 | -0,8514 | 39,3158 | ||
| Var. 2 | 1 | 5,939 | 8,544 | 30,4892 | |
| 0,5 | 1,544 | 1,404 | 9,0674 | ||
| 0 | -0,268 | -0,216 | 19,403 | ||
| -0,5 | -1,803 | -1,785 | 0,9983 | ||
| -1 | -5,866 | -7,36 | 20,2989 | ||
| σz / q | Var. 1 | 1 | -1,113 | -0,997 | 10,4223 |
| 0,5 | -0,839 | -0,861 | 2,5552 | ||
| 0 | -0,534 | -0,531 | 0,5618 | ||
| -0,5 | -0,18 | -0,175 | 2,7778 | ||
| -1 | 0,096 | -0,003 | - | ||
| Var. 2 | 1 | -1,107 | -1,015 | 8,3107 | |
| 0,5 | -0,935 | -0,833 | 10,9091 | ||
| 0 | -0,488 | -0,499 | 2,2044 | ||
| -0,5 | -0,116 | -0,165 | 29,697 | ||
| -1 | 0,108 | 0,013 | 87,963 | ||
| w / qa | Var. 1 | 1 | -1,156 | -1,15 | 0,519 |
| 0,5 | -0,99 | -0,993 | 0,3021 | ||
| 0 | -0,848 | -0,848 | 0 | ||
| -0,5 | -0,76 | -0,761 | 0,1314 | ||
| -1 | -0,709 | -0,706 | 0,4231 | ||
| Var. 2 | 1 | -4,558 | -4,525 | 0,724 | |
| 0,5 | -4,575 | -4,555 | 0,4372 | ||
| 0 | -4,543 | -4,532 | 0,2421 | ||
| -0,5 | -4,489 | -4,471 | 0,401 | ||
| -1 | -4,382 | -4,357 | 0,5705 | ||
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