HzPhase of uhx
5 Soil – Structure Interaction
5.4 Solution Of The Equation Of Motion
S S F
IS S + II I + II I − I =
K u K u K u u 0 (5.13)
Whence the equation of motion becomes
S S
In the case, when the free-field is known, the equation of motion is:
5.4 Solution Of The Equation Of Motion
From equation (5.10), (5.11), (5.12), (5.14) and (5.15) the displacements for the DOFs can be obtained for the given loads.
For each degree of freedom, i, we have:
(5.16)
The time-harmonic displacements are:
( )
Literature
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Vibration in Semi-infinite Solids Due to Periodic Surface Loading Harvard University, Sc.D. Thesis
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The Elastic Theory of Soil Dynamics
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Vertical Motions of Rigid Footings
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Design Procedures for Dynamically Loaded Foundations
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Vibrations of Soils and Foundations
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Lateral and Rocking Vibrations of Footings
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Appendix A – Energy considerations
Given a displacement vector u (n,1) and its corresponding force vector P (n,1) of an elastic system. They are related by stiffness K
=
P K u (A.1)
Assume a kinematic constraint equation
= 0
u au (A.2)
where the vector u0 has dimension (m,1), m<n. Its corresponding force vector is P0. They are related by
0 = 0 0
P K u (A.3)
The energy of the deformation may be expressed in term of both pairs of variables:
0 0
T T
E=P u=P u (A.4)
or using (A.2)
0 0 0
T T
E=P au =P u (A.5)
from there
0
= T
P a P (A.6)
Connecting (A.4) and (A.2) yields
0 0 0
T = T
P au P u (A.7)
From (A.4), (A.1), (A.3) and then (A.2) one obtains
0 0 0
0 0 0 0 0
T T
T T T
=
= u K u u K u
u a Kau u K u (A.8)
From there
0
= T
K a K a (A.9)
Appendix B – Dimension-less equation of motion in frequency domain The equation of motion with dimensions is
K K 2
Definitions of dimension-less quantities:
0 2 3
or reciprocal
2 3
u′ ′ϕ – dimension-less displacement (-) and rotation (-), respectively ,M
h r
P′ ′ - dimension-less force (-) and moment (-), respectively
2
Finally the equation of motion in dimension-less notation:
2
Appendix C – Impedance functions of square foundation
Impedance function in this appendix are given for the four ensuing cases of embedment depth t:
Impedance functions have been calculated using Boundary Element Method in frequency domain. Rigid, mass-less square foundation lays on elastic half-space with Poisson’ s ratio = 0.4.
Presented are in dimension-less notation separately as - real part KR and
Partitioning of the impedance matrix:
,