COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Simplified Doubly Asymptotic Approximation Boundary for
Foundations Dynamic Analysis
Wenjun Lei, Demin Wei*
College of Architecture and Civil Engineering, South China Univ. of Tech., Guangzhou, 510640 China
Email: [email protected]
Abstract The paper presents a simplified doubly asymptotic approximation boundary (spring-viscous
boundary) for infinite medium used in finite element method, which can be applied to both static and
dynamic foundation problems. The boundary can be realized by using special boundary elements. The
spring part of this boundary can be determined by Mindlin Equations, and frequency dependent viscous
material (resistance proportional to velocity) is introduced into the boundary elements to simulate dashpots
of the viscous boundary. When choosing appropriate parameters of Young’s modulus, Poisson’s ratio and
material damping ratio, these boundary elements can simulate not only the elasticity recovery capacity of
the far field media, but also the radiation damping. Two case studies justify the validity and practicability
of this simplified boundary, which proves that it is applicable to the dynamic soil-structure interaction
analysis.
Key words：soil dynamics, dynamic interaction, finite element, artificial boundary, Mindlin equations
INTRODUCTION
When finite element (FE) method is used in foundation analysis, infinite extend soil must be correctly
modeled. For static problems, simply fixed boundary conditions are generally set at the edges of the finite
element domain, which is relatively far from the foundations. For dynamic problems, it is necessary to
consider the radiation of waves into the far-field by imposing appropriate conditions in the boundary of the
finite element domain. By now, several local artificial boundary conditions have been presented, such as
the viscous damper boundary [1], the consistent boundary [2], the paraxial boundary [3], the extrapolation
algorithm [4], etc. These local artificial boundary conditions are all derived from the plane wave hypothesis,
accompanied with low frequency stability problems. Underwood et al [5] present doubly asymptotic
approximation boundary on the basis of the viscous damper boundary, which is asymptotically valid at
both high and low frequencies. Based on the viscous damper boundary, an additional stiffness matrix is
added to the global stiffness matrix, which represents the far-field static stiffness and can be constructed by
boundary-element method. The doubly asymptotic approximation boundary can overcome the low
frequency stability problem of the viscous damper boundary. However, since the additional stiffness matrix
is constructed by the boundary-element method, it is difficult to be used in commercial FE programs.
In this paper, based on the Mindlin equations [6] of elastic half-space, single parameter elastic boundary
(springs) of foundation problems is deduced. The single parameter elastic boundary is a kind of
simplification of far-field stiffness, when combined with viscous damper, it can replace doubly asymptotic
approximation boundary in dynamic foundation analysis, and it is easy to realize in commercial FE program.
⎯ 1120 ⎯
LOCATING THE ARTIFICIAL BOUNDARY AND ASCERTAINING THE SINGLE PARAMETER
SPRING STIFFNESS
The position of the single parameter elastic boundary varies with different problems, this choice ensure
that there is sufficient accuracy but no too much computing effort. The position is determined by the
following principle: firstly, a big finite element model with fixed boundary should be handled, and the
displacement of the soils in the big model can be solved, then the elastic boundary can be set in the
position where the soil displacement is equal to about one tenth of the foundation displacement. In this
paper, several engineering problems are studied, including laterally and vertically loaded footing, vertically
and laterally loaded single pile, etc.
The spring parameter of the elastic boundary is determined by Mindlin equations. For shallow foundations,
reaction of the foundation on the base is assumed as evenly distributed; for pile foundations, the reaction
on the base is calculated by beam-on-Winkler-foundation model. Discretizing these reactions into
concentrated loads and taking them into Mindlin equations, one can get the stresses and strains on the
boundary position. The spring parameters that reflect the far-field stiffness can be got by the ratio of the
stress to the strain. The spring parameters derived from this method are uncoupled at the boundary. In order
to get a single spring parameter, weighted average technical is used, and the weight function is the square
of displacements that was the loaded foundation induced. There are three degrees of freedom (DOF) for
each nodes of the FE model, and we can get the spring parameters of all the three DOFs in the boundary
position by the method mentioned above. If a layer of narrow boundary element is used to simulate those
springs, there are only two elastic constants (Young’s modulus, E and Poisson’s ratio, v) that can be chosen,
springs stiffness of the three directions can’t be satisfied simultaneously. Hence, only spring stiffness of the
main direction will be calculated. For example, for the boundary surface perpendicular to the loading
direction, only the spring stiffness perpendicular to the surface is considered and for this reason, the narrow
boundary elements are added according to this spring stiffness. The narrow boundary elements can also
provides shear stiffness for the nodes at the boundary position, though the shear stiffness are different from
the calculated springs, it can be acceptable since those springs are approximation for far-field stiffness, and
the shear stiffness of the surface perpendicular to the loading direction is relatively unimportant. When the
spring parameter has been specified, we can use a very simple boundary to replace the far-field, which can
be realized by appending a layer of narrow element on the boundary of FE model. By this means,
commercial FE program can be used in the dynamic foundations analysis.
BOUNDARY POSITION AND SPRING STIFFNESS
1. Laterally and vertically loaded footings The basal area of square footing is 2b×2b, and the soil FE
model truncated from the half-space is a cuboid with the size of 2B×2B×Z, in which Z is the height of the
model. According to the results of big model analysis, the B and Z are taken as B≥9b，Z≥12b, respectively.
For laterally loaded footing, Z can be taken a smaller value as Z≥6b. The vertical spring stiffness, kxp, on the
side surface of the FE model, and the vertical spring stiffness, kzp, on the bottom surface of FE model is
calculated respectively,
k xp = 0.65Es / B （N/m3）
(1a)
k zp = 0.8 Es / Z （N/m3）
(1b)
where Es is soil Young’s modulus, all the springs’ stiffness are stiffness of per unit area. The Eq. (1) is for
square footing, but in practical problem, rectangular and circular foundations are usually adopted. According
to the principle of Saint Venant, Eq. (1) can also be applied to rectangular and circular foundations to get the
equivalent square foundation, which has the same basal area as the rectangular or circular foundation.
⎯ 1121 ⎯
2. Laterally loaded single piles The diameter of the pile is d, its length is L, and the soil FE model truncated
from the half-space is a cuboid with the size of 2B×2B×L. When the slenderness ratio of the pile, i.e. L/d, is
relatively small, the height of the FE model should be slightly taller than the length of the pile. The result of
the big model analysis shows that the value of B is primarily determined by the stiffness ratio of the pile to
the soil, i.e. Ep/Es, the value of B can be approximately calculated as follows:
B = 5d log( E p / Es )
(2)
In this paper, only lateral loaded long piles (L/d>4.0/λ) are considered. As for short piles, the boundary
spring stiffness should be taken between the circumstance of long piles and shallow foundations. Like
shallow foundation, only spring stiffness in the primary direction is considered. The vertical spring
stiffness on the side surface of the FE model, kxp, and the vertical spring stiffness on the bottom surface of
the FE model, kzp, is calculated respectively as follows:
k xp = 0.64 Es / B （N/m3）
(3a)
k zp = 1.2 Es / L （N/m3）
(3c)
3. Vertically loaded single piles The diameter of the pile is d, and its length is L. The FE model for this
problem is a cylinder with the dimension of R×Z, in which R is the radius of the cylinder. According to the
principle mentioned above, the values of R and Z are related to the stiffness ratio of the pile to the soil, Ep/Es,
and slenderness ratio of the pile, L/D. R and Z can be approximately determined as follows:
R = 2.7d × log( E p / Es ) × [( L / d ) 0.4 − 0.5)]
(4a)
Z = 2d × [log( E p / Es )]1.8 × ( L / d ) 0.45 − 20d × [log( E p / Es ) − 2.5)]
(4b)
The vertical spring stiffness, kzp, on the bottom of the FE model and the shear spring stiffness, kt, on the
side surface of the cylinder is calculated as follows:
kt = 0.123Es / R （N/m3）
(5a)
k zp = 1.05Es / Z （N/m3）
(5b)
DETERMINATION OF THE BOUNDARY ELEMENT MATERIAL PARAMETERS
A layer of narrow boundary elements were set on the FE model surfaces to simulate not only the elasticity
recovery capacity of the far field media, but also act as the viscous damper boundary to absorb the elastic
wave. The Poisson’s ratio of the boundary element, v, is calculated by
2(1 − vs )
2(1 − v)
=
1 − 2v
1 − 2v s
(6)
where vs is the Poisson’s ratio of the soil, when vs<1/3, its value can be taken as 1/3. The Young’s modulus
of the boundary element, Ed, is calculated by
Ed = k xp (k zp )
(1 + v)(1 - 2v)
×l
1- v
Ed = 2kt × l × (1 + ν )
（vertical springs）
（shear springs）
(7a)
(7b)
where l is the thickness of the boundary elements. The material of the boundary elements has viscous kind
of damping. In the FE equation, viscous damping matrix, [C], satisfies the equation, [C ] = β[K ] , the viscous
⎯ 1122 ⎯
damping ratio of the boundary elements can be calculated by
β=
β=
ρV p
k zp (k xp )
ρVs
kt
（when the boundary elements dominated by vertical spring）
（when the boundary elements dominated by shear spring）
(8a)
(8b)
where ρ is the mass density of the soils, Vp and Vs are velocities of dilatational and shear waves in the
soils, respectively.
CASE STUDIES
Two case studies are presented to validate this boundary. The first case is an axially loaded semi-infinite
circular rod, which has a diameter of 1 meter. The material of the rod is homogenous elastic, The Yang’s
modulus E=3×108N/m2, Poisson’s ratio ν=0.15, mass density ρ = 2500 kg/m3. The rod with the length of 30
meters is modeled which is truncated by spring-viscous boundary. The meshed element has the length of 1
meter along the rod. Since the static stiffness of semi-infinite rod is zero, a very small Yang’s modulus is
adopted for the boundary element (spring-viscous boundary). The FE model is shown in Fig. 1. The
purpose of this case study is to check the validity of this boundary and study the utmost permissible
element length. In dynamic FE analysis, for low-order elements, the dimension should be less than
(1/8~1/6) the possible shortest wavelength [7, 8]. In this case, 20-node equivalent parameter elements are
adopted to mesh the rod, and the consistent mass matrix is used in the dynamic analysis.
Figure 1: The FE model of semi-infinite circular rod
When the harmonic load, P = e iωt kN, acts on the rod top, the closed-form solution of the displacement of
rod top is
Uz =
P
iωρVA
（9）
where Uz is the displacement of the rod top, ω is circle frequency, V = E / ρ = 346.4m / s and is the
elastic wave velocity, A = πD 2 / 4 = 0.785 and is the area of rod. The comparison of the proposed FE model
results with closed-form solution is shown in Fig. 2, in which the closed form solution is shown as solid
lines and FE model results as dash lines. It is evident, that when the vibration frequency is less than 150Hz,
the two results are almost consistent. For this frequency, the elastic wavelength λ = 346.4 / 150 ≈ 2.3m , and it
is almost 2.3 times of the element length; when the vibration frequency is less than 120Hz, the two results
coincide very well. For this frequency, the elastic wavelength λ = 346.4 / 120 ≈ 2.9m , and it is almost 2.9
times of the element length. Through this case, it can be concluded: firstly, spring-viscous boundary can
absorb elastic wave very well; secondly, the largest meshed element size should satisfy ΔL ≤ λ / 3 when
20-node equivalent parameter elements are adopted in the dynamic analysis.
The second case is a rigid circular disk supported at the surface of a viscoelastic half space and subjected
vertical harmonic oscillations. The analysis results of the steady-state response obtained by FE model put
forward in this paper will be compared with the Veletsos and Verbic’s solutions [9].
The Young’s modulus (Es) of the foundation soils is taken as 3×107N/m2, hysteresis damping ratio (β) is
⎯ 1123 ⎯
Displacement (m)
Displacement (m)
taken as 0.05, Poisson’s ratio (v) is taken as 1/3, and the diameter of the rigid circular disk is 1 meter.
Axisymmetrical 4-node plane element is used in meshing, and the results of meshing and boundary of
model are shown in Fig.3. Since the same mesh will be used in the analysis of different frequency
harmonic oscillations, a fine mesh is used in Fig. 3 in order to satisfy the largest element size requirement
for the high frequency analysis. The action of the rigid circular disk is simulated by the constraint
equations; it will ensure all the nodes of the disk have the same vertical displacements, and the horizontal
displacement will not be restricted.
f (Hz)
f (Hz)
a) The real part
b) The imaginary part
Figure 2: Displacement of the top of the semi-infinite circular rod under the harmonic load
The static vertical stiffness of the rigid disk rests on the surface of the elastic half-space, obtained by F.E.
analysis model presented in this paper is K v = 3.409 × 107 , which is coincident with the theoretical solution
of the elastic half-space problems, i.e. K v =
4Gr
= 3.375 × 107 . When the disk is subjected to vertical
1− v
harmonic oscillations, the vertical dynamic load-displacement relationship given by Veletsos and Verbic is
expressed as:
weiωt = P
1− v
( F1 + iF2 )eiωt
4Gr
(10)
Table 1 Comparison of FE model results and the solutions of Veletsos & Vebic
a0
Veletsos &
Vebic
FE model
0
0.5
1.0
1.5
2.0
F1
0.99
0.84
0.52
0.27
0.15
F2
−0.10
−0.44
−0.60
−0.55
−0.45
F1
0.99
0.84
0.52
0.27
0.13
F2
−0.10
−0.45
−0.58
−0.56
−0.46
where w is vertical displacement of disk; P is amplitude of the harmonic load; G is the shear modulus of
soil, r is the radius of the disk; F1 and F2 are dimensionless flexibility function. The comparison of the FE
model results and the solutions of Veletsos and Verbic are presented in Fig.4, and several frequencies
results list in table 1. As shown in the figure, the two results tally with each other very well, including at
the low frequency. The comparison proved that the low frequency stability of the foundation dynamic
analysis is ensured by combining the viscous boundary with an elastic boundary. At the low frequencies,
there is little departure between the two results, this probably stems from the fact that the viscous boundary
⎯ 1124 ⎯
can’t perfectly absorb incident Rayleigh waves. The low frequencies results can be improved with no
increase in the computing time by extending FE model domain in horizontal direction, and increasing the
element size at the same time. It should be pointed out that there is slight departure in flexibility function
F1 between the two results at high frequencies, the cause is that Veletsos and Verbic’s get a some little high
value at these frequencies.
Boundary element
600cm
600cm
50cm
Figure 3: The FE model mesh
Figure 4: Flexibility function F1 and F2: comparison of the proposed FE model results with Veletsos & Verbic’s results
CONCLUSIONS
A simplified doubly asymptotic approximation boundary (spring-viscous boundary) for infinite medium
applied in finite element method analysis is put forward. The spring part of this boundary is calculated by
Mindlin equations. Through combining the spring with the viscous damper boundary, a spring-viscous
boundary is constructed. The boundary can not only simulate the elasticity recovery capacity of the far
field media, but also absorb the elastic wave. Hence, it can be applied to static or dynamic foundation-base
interaction problems; when it is applied to the dynamic foundation analysis, the low frequency stability
problem in FE method with the viscous damper boundary will be overcome. Since this boundary is very
simple, it can be easily realized in commercial FE programs by setting a layer of boundary elements with
suitable material constants. It will be promising in the engineering application.
REFERENCE
1. Lysmer J, Kuhlemeyer RL. Finite dynamic model for infinite media. Journal of the Engineering
Mechanics, ASCE, 1969; 95(4): 859-877.
⎯ 1125 ⎯
2. Lysmer J, Waas G. Shear waves in plane infinite structures. Journal of the Engineering Mechanics,
ASCE, 1972; 98(1): 85-105.
3. Engquist B, Majda A. Absorbing boundary conditions for the numerical simulation of wave. Math.
Comput., 1977; 31: 629-651.
4. Liao ZP, Huang KL, Yang BP, Yuan YF. Transmitting boundary of transient waves. Science in China
Series A, 1984; 27(6): 556-564.
5. Underwood P, Geers TL. Doubly asymptotic, boundary-element analysis of Dynamic soil-structure
interaction. Journal of solid structure, 1981; 17(8): 687-697.
6. Mindlin RD. Force at a point in the interior of a semi-infinite solid. Physics, 1936; 7: 195-202.
7. Kuhlemeyer RL, Lysmer J. Finite element method accuracy for wave propagation problems. Journal
of the Soil Mechanics and Foundations, ASCE, 1973; 99(5): 421-427.
8. Kausel E, Roesset JM. Dynamic stiffness of circular foundations. Journal of Engineering Mechanics,
ASCE, 1975; 101(6): 771-785.
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Dynamics, 1973; 2(1): 87-102.
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