The finite element method

EXERCISE No. 1

1.1 For the gravity dam with the cross section shown in the following figure, determine the displacements, the state of stress at the bottom of downstream slope and the reaction forces at the contact with the rock layer (along the embedment).

Problem's data:

a). Geometry of the dam:

- height H = 27+15,0 m= 42 m

- width of embedment B = 27+10,0 m= 37 m

b). Material properties:

- elasticity module E = 200.000 Dan/ cm2= 20.000.000 kN/m2

- Poisson's coefficient ? = 0,16

c). Assumptions:

- water level at the crest of the dam;

- plane strain state;

- only water pressure effect is taken into account.

Theoretical considerations:

o First stage for every analysis is the domain meshing.

The problem will be solved using the F.E.M., by decomposing (meshing) the structure into 4 identical, triangular, linear finite elements.

o Second stage consists in computing the elemental stiffness matrix [K]e for the triangular element.

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o Boundary conditions:

The boundary conditions are expressed by displacement contraints applied

for nodes 1, 2 and 3 - all degrees of freedom u1 , v1 , u2 , v2, u3 , v3 are suppressed. The applied forces are calculated by assesing the volume of the triangular prism diagram of water preasure, as shown in the next figure.

Project: The finite element method