Objective
To determine the stress state of a triangular dam of unit thickness in Cartesian coordinates under the self‑weight and hydrostatic pressure.
Reference
V.I.Samul, Fundamentals of elasticity and plasticity theory. - Moscow: Vysshaya Shkola, 1982.
Problem statement
To determine the components of the stress tensor in Cartesian coordinates σx, σy, τxy in the horizontal cross‑section of the dam located at a depth of y0 = 5.0 m from the top of the dam.
Design model
Horizontal load linearly distributed with unit weight γ is applied to the vertical face of a triangular dam of unit thickness in its plane
The dam is also subjected to its self-weight.
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Initial geometry of analytical model
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Initial geometry of FE model
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Geometry
Dam thickness h = 1,0 m
Apex angle of the triangular dam β = 30º
Dam height H = 15,0 m
Material properties
Modulus of elasticity Е = 3,0 * 107 Pa
Poisson's ratio of the dam material μ = 0,2
Unit weight of the fluid γ = 10,0 kN/m3
Unit weight of the dam material γ1 = 20,0 kN/m3
Note
Design model – 2D frame; elements of the plate – 452 eight‑node elements of type 30 and 23 six‑node elements of type 25.
The FE mesh is divided with a step of 0.25 m in the horizontal X and vertical Y-directions.
Internal forces are to be determined along the X and Y-axes of the global coordinate system. Since at the fixed end the dam does not experience forces distributed according to the law of the analytical solution, in order to obtain an accurate solution at a depth of y0 = 5.0 m from the top of the dam under the self‑weight and hydrostatic pressure, the height of the dam up to the fixed end is taken as H = 15.0 m.
Nodes: 1506.
Output data
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Stress σx (kNm/m2)
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Stress σy (kNm/m2)
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Stress τxy (kN/m2)
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Analytical solution
For the analytical solution, the stress components σx, σy, τxy in the dam body can be calculated using the following formulas (V. I. Samul, Fundamentals of elasticity and plasticity theory, Vysshaya shkola, 1982, p. 77):
Comparison of calculation results
| Parameters |
On the inclined face of the dam (x = y0 · tgβ = 2,8868 m) |
On the vertical face of the dam (x = 0,00 m) |
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| Analytical solution | LIRA-FEM | Error, % | Analytical solution | LIRA-FEM | Error, % | |
| σx(kN/m2) | -50,00 | -49,2 | 1,6 | -50,00 | -47,5 | 5 |
| σy(kN/m2) | -150,00 | -148,00 | 1,33 | 50,00 | 46,6 | 6,8 |
| τxy(kN/m2) | -86,6 | -85,1 | 1,73 | 0,00 | 0,0323 | - |
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