Non-local Models of Stiffness and Damping - Michael I Friswell

Figure 2 – A beam with a partial non-local foundation.
3. A NUMERICAL EXAMPLE
A simply supported uniform beam of length L = 6.096m rests on a uniform non-local elastic foundation is
considered, as shown in Figure 1 with x 0 = 0 and x2
= L (that is the foundation supports the whole beam). This
example was also analyzed by Lai et al. [8] and Thambiratnam and Zhuge [9]. The beam has Young’s modulus
-3 4
24.82GPa, second moment of area 1.439 × 10 m and mass per unit length 446.3kg/m . The stiffness of the
foundation is 16.55MNm -2 . Using the finite element method, the first four natural frequencies of free vibration
were obtained with 6, 8 and 10 elements, for different values of α for the exponential kernel. The results are
shown in Table 1 together with those from the analytical solution [10], given by
4
EI i π 4 H0
ωi
= + . (12)
ρA
4
L EI
Figure 3 shows the effect of varying α on these natural frequencies for the finite element model with 10
elements.
α ( m −1
)
2 2 2 5 10 50 ∞ ∞
N e
6 8 10 10 10 10 10 Analytical
Mode 1 32.137 32.137 32.137 32.758 32.862 32.897 32.898 32.898
Mode 2 55.310 55.287 55.281 56.495 56.728 56.808 56.812 56.808
Mode 3 110.89 110.62 110.54 111.61 111.86 111.95 111.95 111.90
Mode 4 194.85 193.36 192.92 193.74 193.98 194.07 194.08 193.76
Table 1 – Natural frequencies (Hz) of vibration of a simple beam on a non-local elastic foundation with an exponential
kernel. N e is the number of elements.
Figure 3 – The variation of the first four natural frequencies with α for the fully supported beam and an exponential kernel.
The natural frequencies have been normalized using the values at α = ∞ and 10 elements were used.