For the beam element compute nodal forces on the basis of the global dof vector (mind the global node numbers) and draw diagrams of section forces

Computational Methods - Civil Engineering : Example problems for test 1 1

Problem 1. Solve the beam using FEM (compute the vectors of dofs and reactions, draw cross-section force diagrams). Compute the deflection at point A.

10kN

3m

∆=5cm

EA= 3 · 10⁵kN

A

EI= 250 kNm²

Problem 2.

Knowing Hermite interpolation functions determine substitute nodal force z¹ for the two-noded beam element and the parabolic loading.

12 kN

4 m x

y

1 2

d1

d2

d³ d4

Problem 3. Use the FEM algorithm to solve the beam shown, i.e. calculate the vectors of nodal di- splacements and reactions, and make diagrams of cross-section force distributions.

4 m 6 m

2EI EI

1 2 3

e= 1 e= 2 x EI=18000 kNm²

y 13 kN

Problem 4. For the beam element compute nodal forces on the basis of the global dof vector (mind the global node numbers) and draw diagrams of section forces.

l=2m

x y

py=6kN/m

E= 20 · 10⁶kPa I= 3 · 10⁻⁴m⁴ d= {0 0 0 2 0 -2 0 2 -1 2 0 -3} · 10⁻³m

3 2

Problem 5. Solve the truss using FEM.

3 3

4

2

1 X

Y

30 kN/m a

a=5 cm – horizontal support translation EA=10⁴ kN

a) Compute nodal displacements b) Compute forces in elements c) Compute constraint reactions d) Check equilibrium at node 3

Computational Methods - Civil Engineering : Example problems for test 1 2

Problem 6. Find the second component of the substitute nodal force vector for the truss element loaded as shown in the figure. The values of distributed load intensity are given at element ends and at its centre.

5 kN/m

12 kN/m 9 kN/m

1 4 m 2

d1 d2

Problem 7. Solve the truss using FEM. Compute the vectors of nodal forces for each element using the FEM procedure. Draw the cross-section force diagram. Adopt 1 dof at each node.

0.005 m

EA=20 MN EA=45 MN X

2 m 3 m

K¯^e= E^eA^e l^e

 1 −1

−1 1



Problem 8. Compute the nodal forces for elements no 1 and 4, i.e. perform the ”return to element”

procedure. Draw diagrams of force N^efor these elements.

3 m 3 m 3 m

4 m

EA=25·10³ kN

8 kN/m 20 kNm

1

2

3

4

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89:;1

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89:;2

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89:;3

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89:;4

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89:;5 x

y d=

























0 0 0.7500 -0.2500 0 0 1.1100 1.4575









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

· 10⁻²m

Problem 9. Determine the right-hand side vector in FEM equation set for the frame shown.

5m

4m x

y

1

2 3

12 kN/m 14 kN

Problem 10. Present in a graphical manner the assembly of stiffness matrices for elements 2 and 3 of the frame shown below. Determine the global right-hand side vector f in FEM equation set.

y x

1 1

2 2

3

3

4

4 12 kN

14 kNm

15 kN