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

NON-LOCAL MODELS OF STIFFNESS AND DAMPING
Friswell, M.I, Adhikari, S.
Bristol University
Department of Aerospace Engineering, Queen’s Building, University Walk, Bristol BS8 1TR, UK
m.i.friswell@bristol.ac.uk
SUMMARY: Often stiffness and damping is modelled as local, where the force is proportional to the
instantaneous displacement and velocity at a given point. Often this model is not sufficient and spatial hysteresis
effects must be included in the dynamic analysis of structures. In non-local models, the stiffness and damping
force is obtained as a weighted average of the displacement and velocity field over the spatial domain,
determined by kernel functions. Approximate solutions for the complex eigenvalues and eigenvectors of beams
with non-local stiffness and damping may be obtained using the Galerkin or finite element methods. Numerical
examples will demonstrate the efficiency of these methods and highlight the effect on the dynamic response. The
source of the non-local effects will be highlighted using a detailed finite element model.
KEYWORDS: Non-local damping, kernel functions, finite element analysis, beams.
1. INTRODUCTION
Beam and beam-like structures resting on elastic or viscoelastic foundations have wide application in modern
engineering practices, such as, railway tracks, highway pavements, and continuously supported pipelines. As a
result, numerous studies have been performed to investigate the static deflection, the dynamic response, and the
dynamic stability of beams on elastic or viscoelastic foundations. However, in most of these studies, the
foundation was idealized as a one-parameter Winkler model, where the system is modelled as infinitely close
linear springs and the foundation pressure at any point is proportional to the deflection at that point. Several
more complicated models have been reported in the literature, such as the two-parameter Pasternak model, the
three-parameter Kerr model, the Vlasov model, or fractional derivative viscoelastic models. Friswell et al. [1]
discussed these models in detail.
The purpose of this paper is to develop and analyze general non-local models for elasticity and damping. In a
non-local model, the reaction force at any point is obtained as a weighted average of state variables over a spatial
domain via convolution integrals with spatial kernel functions that depend on a distance measure. The
development considers general kernel functions, and common kernel functions are highlighted and used in the
examples. In physical terms the non-local damping or elasticity property often arises when two dimensional
structures are modelled as one dimensional, and this interpretation is discussed in detail. Non-local damping and
stiffness are considered, and much of the analysis considers the free and forced vibration response. However
static problems of beams with non-local foundation stiffness may also be analyzed using this approach. Nonlocal
foundation models are considered in depth, but a similar analysis may be performed for non-local effects
within the beam [2-4].
2. NON-LOCAL FOUNDATION MODEL FOR BEAMS
Suppose that non-local foundation stiffness and damping exists for a beam between locations 1 x and 2 x , as
shown in Figure 1. The equation of motion for this beam may be expressed as the following integro-partialdifferential
equation,

( )
( )
2
2 2
∂ ⎛ ∂ wxt , ⎞ ∂ wxt ,
EI ( x)
A( x)
Q
2
⎜
2
⎟ +ρ +
2 NE x + QND
x = f x t
∂x ⎝ ∂x ⎠ ∂t
( ) ( ) ( , )
(1)
where EI ( x ) is the bending stiffness of the beam, ρ A( x)
is the mass per unit length of the beam, ( , )
transverse displacement, and f ( xt , ) is the distributed external force. Q ( x ) and Q ( )
wxt is the
NE
ND x are the forces due
to the foundation elasticity and damping, which are assumed to be non-local. These forces are given by
x
QNE
x t C x w t
x1
2
2
( , ) = ∫ E ( , ξ) ( ξ, ) dξ
and QND
( x t) CD
( x t )
x
x1
( ξ,
τ)
t
∂w
, = , ξ, −τ dτdξ
−∞
∂t
∫ ∫ . (2)
CE ( x , ξ ) and CD
( x, ξ, t−τ ) are the one-dimensional kernel functions for the elasticity and damping
respectively, defined over the spatial sub domain ( x1,
x 2)
. The appropriate boundary conditions must be
satisfied at x = 0 and x = L .
Figure 1 – A beam with a partial non-local foundation.
2.1. Typical Kernel Functions
The general kernel function allows for both time and spatial hysteresis, representing a non-local viscoelastic
damping model. If the kernel functions are assumed to be separable in space and time, then C ( x, ξ, t −τ ) and
CD ( x , ξ , t−τ ) are of the form
( , ξ ) = ( ) ( −ξ ) and C ( x, , t ) H ( x) c ( x ) g( t )
CE
x H x cE
x
D
ξ −τ = D −ξ −τ (3)
where H ( x ) indicates the presence of the foundation, c( x− ξ ) is the spatial kernel and g ( t ) is the time kernel.
The spatial kernel functions, c( x−ξ ) , where the subscript E or D is implicitly assumed, are usually normalised
to satisfy the condition
∞
−∞
∫ c( x)dx
= 1. (4)
Common choices for the spatial kernel function, c( x− ξ ) , are an exponential decay,
or an error function,
α −α x−ξ
c( x−ξ ) = e , (5)
2
E
( )
c x−ξ =
α
2π
e
α
2 −
2
( x−ξ) 2
, (6)
where α is the characteristic parameter of the non-local effect. Notice that, c( x−ξ) → 0 when α→∞ and
x ≠ ξ , so that c( x−ξ) →δ( x−ξ ) , which is the model for local elasticity or damping. Thus, 1/α is called the
influence distance.

Often the time kernel function, g () t , is assumed to be of the form
M
g()
t = g r
r μr
e −μ t
r=
1
∑ (7)
where μ r are the relaxation constants of the viscoelastic material, that may be determined by curve fitting of the
measured mechanical properties of the material. Other forms of g(t), such as the GHM model [5,6] may also be
analysed using the proposed method. The fractional derivative model [7] may also be used, but is better
described in the Laplace domain. These models are not considered further as the spatial hysteresis is the main
concern of this paper.
2.2. Finite Element Analysis of Beams with Non-local Foundations
For simple beam structures methods such as the transfer function method [3] or the Galerkin approach [4] may
be used. For the analysis of general beam structures, Friswell et al. [1] introduced the non-local effects into a
finite element analysis. Generally the approach adopted when developing models using finite element analysis is
to approximate the deformation within an element using nodal values of displacement and rotation. The kinetic
and strain energy for each element is then computed and the contributions of each element added together to
obtain a global model of the structure. The damping matrix is obtained is a similar way using the dissipation
function. One key aspect of finite element analysis is that the energy contribution from each element only
depends on the displacements and rotations at the nodes associated with that element. Clearly for non-local
elasticity this will not be the case, although the form of the exponential kernel function does lead to considerable
simplification [1].
A standard beam element is modelled using two nodes (at the ends of the beam element), and two degrees of
freedom per node (translation and rotation), as shown in Figure 2. The deformation within the e th element,
we
( η , t)
, is approximated using standard cubic shape functions for η ∈[ 0, l e ] , as
where
( η,
) ( η) ( )
w t = N q t
(8)
e e e
N ( η) =⎡ ( η) ( η) ( η) ( η)
⎤ and () t
e ⎣N N N N
1 2 3 4
The strain energy due to the foundation may then be written as
where
U
M
1 T
F i ij j
2
i, j=
1
⎦
() t
() t
() t
() t
⎧ we1
⎫
⎪ψ
e1
⎪
q e = ⎨ ⎬ . (9)
⎪we2
⎪
⎩
⎪ψ
⎪
e2
⎭
= ∑ q K q (10)
( ) ( ) ( )
l j li ˆ T
K ˆ ˆ
ij = H0 c ˆ ˆ ˆ
0 0
E xj + x−xi −ξ Ni ξ Nj
x dξdx, (11)
∫ ∫
ˆx and ˆξ are local co-ordinates, for example x = xj
+ xˆ
, x j is the location pf the j th node, and H ( x) = H0
.
Friswell et al. [1] gave further details, explicitly calculated the element matrices for the exponential decay and
error function kernels, and discussed the assembly process in more detail.

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.

Suppose now that the kernel consists of the error function. Figure 4 shows the first four natural frequencies for
the finite element method with 10 elements. Interestingly the amplitude of α required for the error function
kernel to approximate the Dirac delta function (local elasticity) is lower than that required for the exponential
kernel. This is because of the shape of the kernels close to the centre point. Note also that the ω 1 and ω 2 lines
cross in Figure 3 and do not cross in Figure 4. This change occurs because of differences in the relationships
between the rate of decay in the two spatial kernel functions and the mode shape deformation.
Figure 4 – The variation of the first four natural frequencies with α for the fully supported beam and an error function kernel.
The natural frequencies have been normalized using the values at α = ∞ and 10 elements were used.
4. THE ESTIMATION OF KERNEL FUNCTIONS
Thus far the spatial kernel function has been specified, and no indication given for its origin. There is little
information in the literature to help the specification of this kernel. The approach adopted here is to estimate the
spatial kernel functions based on a 2D finite element model of the foundation. Consider the external non-local
stiffness model; the damping model will be similar since often the stiffness and damping properties will be
distributed in a similar way. If the displacement in the beam was given by a Dirac delta function at the centre of
the beam, w( ξ) = δ( L/2)
and if the beam is relatively long (compared to the thickness and the non-local spatial
parameter) then from Equation (2),
( ) ( , /2)
≈ . (13)
QNE
x CE
x L
Assuming that CE
( x, L/2 ) ∝cE
( x− L/2)
then if QNE
( )
estimate of c ( x− ξ)
is immediately obtained.
E
Thus the key to the approach is to estimate QNE
( )
w( ξ) δ( L/2)
x is calculated from a finite element analysis, then an
x from a 2D finite element code, by approximating
= along the neutral axis of the beam, which is assumed to have negligible thickness. Including a
2D model of the beam would also be possible, although the effect of the beam stiffness would have to be
separated from the effect of the foundation stiffness. For a fine mesh the approximate displacement is obtained
by constraining the transverse displacement of the nodes on the neutral axis of the beam, except the one at the
centre which is free. All axial displacements on the neutral axis are zero since the beam is assumed to be axially
stiff. The nodes on the grounded boundary of the foundation are also constrained to have zero displacement. A
transverse unit force is then applied to the centre node, and the force QNE
( x ) is then obtained as the transverse
force at the constrained nodes along the beam neutral axis.
From a finite element analysis perspective, the symmetry of the beam and foundation problem means that only
half of the foundation needs to be modelled. From symmetry considerations the nodes on the line x = L /2 do
not displace in the x direction, and hence a boundary is introduced at x = L /2, with the nodes constrained to
have zero displacement in the x direction. Figure 5 shows a typical 2D mesh for the analysis, where, for clarity,
only a small number of elements are shown. Only the foundation material is considered as the beam thickness is
assumed to be negligible. The displacement constraints are indicated by triangles in the two directions. There are

two major problems with this approach to obtaining the kernel functions. First, a transverse force is applied to
the node at x = L /2, and the adjacent nodes are fixed. This means there is a significant deformation within the
element near this forced node that is unlikely to be captured accurately by the shape functions with the element.
In practice this means that the estimated kernel function is unlikely to be accurate close to x = L /2. Second,
normalization of the kernel function is difficult, because the local stiffness of the forced degree of freedom
changes with mesh density, the imposed displacement on the neutral axis is not a delta function, and also
because the inaccuracies of the deformation and constraining forces close to x = L /2.
Figure 5 – A typical 2D finite element mesh of the foundation. There are 20 elements along the beam axis and 4 across the
thickness of the foundation. All elements model the foundation.
Figure 6 shows the kernel function obtained for three different mesh densities and a foundation length of 1 m and
a foundation thickness of 0.2 m. The elements used were 4-noded rectangular elements (Dawe [11], p. 325). A
plane stress element was used, although the results are very similar for plane strain. The thickness and modulus
of the foundation material merely changes the normalization of the kernel. Poisson’s ratio was taken as 0.3. The
number of elements in each direction is chosen so that the elements are square. The results are shown on a semilog
plot, where the kernel is positive for x −ξ< 0.175 , and negative for x −ξ> 0.175 . Clearly the estimated
kernel functions are almost identical along most of the beam length. For x −ξ close to zero there are differences
in the estimated kernel for the reasons discussed above. Near the end of the foundation, x − ξ ≈ 1 , there are also
some differences, however the estimated kernel is unlikely to be accurate close to this location, as shown by the
increase in the kernel for all element mesh densities for x>0.95.
Figure 6 – The estimated kernel function for three different mesh densities.
Figure 6 shows two regions where the plot is approximately linear, namely 0.05 < x −ξ < 0.15 and
0.35 < x −ξ < 0.85 , showing that the kernel may be approximated using the sum of two exponential functions.
Neglecting the estimated kernel for x − ξ < 0.05 and x −ξ> 0.92 for the 8000 element example, and curve

fitting the sum of two exponential functions using a simplex method in MATLAB, gives the estimated kernel
function as
( − ) = 0.112exp( −29.0 − ) −0.00511exp( −9.81
− )
ξ ξ ξ . (14)
cE
x x x
This fitted kernel function is compared to the estimated kernel function in Figure 7.
Figure 7 – The estimated and fitted kernel functions.
5. CONCLUSIONS
In this paper, a method to analyze beams with a non-local foundation was demonstrated based on a onedimensional
finite element method. In a non-local foundation model the reaction force of the foundation on the
beam at a given point depends on the beam displacement within a spatial domain. Examples were given to
illustrate the effect the parameters of the non-local model can have on the natural frequencies of the system. A
method to estimate the spatial kernel functions of beams supported on foundations based on a more detailed
finite element model was developed. This suggested a kernel function consisting of the sum of two exponential
functions should be used, although further work is needed to model a range of foundation dimensions, and also
to model the beam itself.
ACKNOWLEDGEMENTS
Michael Friswell gratefully acknowledges the support of the Royal Society through a Royal Society-Wolfson
Research Merit Award. Sondipon Adhikari gratefully acknowledges the support of the Engineering and Physical
Sciences Research Council through the award of an Advanced Research Fellowship.
4. REFERENCES
[1] Friswell, M.I., Adhikari, S., Lei, Y., “Vibration Analysis of Beams with Nonlocal Foundations using the
Finite Element Method”, International Journal for Numerical Methods in Engineering, in press.
[2] Friswell, M.I., Adhikari, S., Lei, Y., “Non-local Finite Element Analysis of Damped Beams”, International
Journal of Solids and Structures, in review.
[3] Adhikari, S., Lei, Y., Friswell, M.I., “Modal Analysis of Non-viscously Damped Beams”, Journal of
Applied Mechanics, in press.
[4] Lei, Y., Friswell, M.I., Adhikari, S., Lei, Y., “A Galerkin Method for Distributed Systems with Non-local
Damping”, International Journal of Solids and Structures, 43(11-12), 2006, pp. 3381-3400.

[5] Kim, T.-W., Kim, J.-H., “Eigensensitivity Based Optimal Distribution of a Viscoelastic Damping Layer for
a Flexible Beam”, Journal of Sound and Vibration, 273(1-2), 2004, pp. 201-218.
[6] Friswell, M.I, Inman, D.J., Lam, M.J., “On the Realisation of GHM Models in Viscoelasticity”,. Journal of
Intelligent Material Systems and Structures, 8(11), 1997, pp. 986-993.
[7] Bagley, R.L., Torvik, P.J., “Fractional Calculus – A Different Approach to the Analysis of Viscoelastically
Damped Structures”, AIAA Journal, 21(5), 1983, pp. 741–748.
[8] Lai, Y.C., Ting, B.Y., Lee, W.S., Becker, W.R., “Dynamic Response of Beams on Elastic Foundation”,
Journal of Structural Engineering, 1992, 118, pp. 853–858.
[9] Thambiratnam, D., Zhuge, Y., “Free Vibration Analysis of Beams on Elastic Foundation”, Computers and
Structures, 1996, 60, pp. 971–980.
[10] Timoshenko, S., Young, D.H., Weaver, W., “Vibration Problems in Engineering”, fourth edition, John
Wiley and Sons, 1974.
[11] Dawe, D.J., “Matrix and finite displacement analysis of structures”, Oxford Engineering Science Series,
1984.