Computational Methods - Civil Engineering : Example problems for test 2 1
Problem 1. For the following problem of heat flow, discretized using FEM, determine the right-hand side vector of the obtained set of algebraic equations.
1 2 3
T =15◦C 4
qn= 0
qn= 4 J/m2s
qn= 6 J/m2s
1.5m 2m
1m 1m
x y
Problem 2. Calculate the heat flux density vector q and temperature at point A(1.0,1.5) of the configu- ration discretized using 1 finite element. The input data contain: dof vector θ, conductivity matrix k and shape functions.
3 m
2 m A
1 3 2
x y
θ=
T1
T2
T3
=
3.5
2 4
◦C
k=
2 0 0 2
J/◦Cms
N1(x, y) = −1 2y+ 1 N2(x, y) =1
3x N3(x, y) = −1
3x+1 2y
Problem 3. The following panel is discretized with one three-noded FE. Determine the right-hand side vector used in FEM computations.
5 kN/m 8 kN/m
X Y
1 2
3
3 m
4 m N1(x, y) = −41y+ 1 N2(x, y) = −31x+14y N3(x, y) = 13x
Computational Methods - Civil Engineering : Example problems for test 2 2
Problem 4. For the panel (plane stress problem) discretized with 4 finite elements determine the global loading vector.
1 2 3
4 5 6
7 8 9
1m 1m
2m 2m
5kN
Problem 5. Derive the shape functions for the triangular finite element.
3
2 1
x y
1 m 2 m
2 m
Problem 6. The following plane stress structure has been computed using FEM. For the plotted element the solution vector is given. Compute the strain and the stress vectors at point A with coordinates (1,1).
de=0 − 12 · 10−4 0 0 0 0 12 · 10−4 0
1
4 3
2 A(1, 1)
X Y
3 m
4 m E = 50 GPa
ν = 0 h = 0.2 m
D= E 1 − ν2
1 ν 0
ν 1 0 0 0 1−ν2
N1(x, y) = −13x+ 1
−
1 4y+ 1 N2(x, y) = 13x
−
1 4y+ 1 N3(x, y) = 13x 1
4y N4(x, y) = −13x+ 1 1
4y
