clockwork theoretical manual

Machine Foundations Analysis

THEORETICAL MANUAL REV. 12.2

CLOCKWORK THEORETICAL MANUAL

I.C.:TM-12.2 Page: 2

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CONTENT

I. INTRODUCTION .................................................................................................................................... 4

1. MACHINE FOUNDATION DESIGN AND CLOCKWORK ..................................................................... 6

2. THE IMPEDANCE METHOD ................................................................................................................. 7

2.1 COORDINATE SYSTEM ...........................................................................................................7

2.2. IMPEDANCES......................................................................................................................8

3. LIBRARY OF FOUNDATION IMPEDANCES ..................................................................................... 10

4.1. INDIVIDUAL PILE................................................................................................................................ 15

4. OTHER DYNAMIC CAPABILITIES ..................................................................................................... 18

REFERENCES ............................................................................................................................................... 19



I. INTRODUCTION

The fundamental problem of Machine Foundation analysis can be posed as illustrated in Fig. I.1

Fig I.1. a) Machine foundation harmonically excited over two layers and half-space medium. b)

Machine foundation embedded over half-space upon an arbitrary excitation. c) Piled Machine

foundation harmonically excited over half-space medium.

A rigid foundation bears on an elastic soft soil which extends to infinity. Time varying forces,

produced from the operation of a machine, acts on different points of the foundation.

If we isolate the foundation (free body diagram) for dynamic equilibrium we will need reaction

forces at the foundation soil interface.

The dynamic response of the foundation interacting with the soil shall be determined and compared

with threshold values that are acceptable for a proper operation of the machine. If the response

results higher than the threshold limits, the foundation dimensions shall be revised. It is an iterative

procedure that is typical in the engineering design process.

As a first approach for modeling the problem we can assume that the soil is elastic. In this case seems

natural to model the soil by linear springs. Such a model, masses supported on elastic springs,

exhibits natural modes of vibration. It is well known that these systems when excited at certain

frequencies vibrates with very large displacements .In fact, with infinite displacements according to

the mathematical model (Resonance phenomena)

This phenomenon is not observed in the practice because the soil extents are infinite and we are in

presence of an open system, instead of a closed one. In open systems the kinetic energy of the



particles in the vicinity of the foundation escapes in the form of travelling waves propagating in the

soil.

In a closed system the energy cannot escape as waves, because these are reflected in the boundaries

of the closed system. The kinetic interaction of the out-coming waves with the reflected (or

refracted) waves determines the so called vibration modes. We can say that vibration modes are

trapped waves inside the limits of the closed model.

The fundamental problem, for the Machine Foundation Problem, is to define simple models that can

represent the dissipation of energy from the travelling waves.

Being a dissipation of energy process it is natural to associate this behavior to damping mechanisms.

Radiation (or Geometric) Damping thus stands for the dissipation of elastic energy in the forms of

waves.

To model the Radiation Damping coming from an unbounded soil has been the object of rigorous

studies of continuum mechanics. The history of all these efforts is well summarized for example in

[15].

Now it is known that simplified procedures are available to solve these problems accepting

engineering simplifications.

These solutions are based in representing the soil by means of equivalent lumped parameter models.

The equivalent models are constituted of discrete springs, dashpots and eventually masses. The

parameters that defines the models are, in general, dependent of the exciting frequency,

Note that for static loads the equivalent model can consist uniquely of springs. For dynamic loads the

existence of travelling waves that drains the energy of the system needs special modeling.



1. MACHINE FOUNDATION DESIGN AND CLOCKWORK

The design procedure of a typical machine foundation requires to follows the following steps. In each

of them CW will help the task of the designer:

- Estimate-the magnitude and characteristics of the dynamical loads. Typically the equipment

provider will help in this task. CW presents many options to evaluate the dynamical loads

when detailed information is missing. These are taken from well-known literature such as

[19]

- Establish the soil profile. Following parameter are significant:

o Elastic constants (Shear modulus, Poisson ratio).

o Mass density

o Material Damping ratio

CW facilitates the inputting of such information and provides typical parameters for soil

types.

- Define the Design Criteria. In cooperation with the vendor, and/or the client and considering

relevant codes. CW provides charts with well-known Acceptance Criteria for different

machines and condition.

- Define the trial dimensions and type of yours foundation. With CW this is easy using the

advanced graphics interface of the software.

- Calculate the dynamic response of the machine foundation system; CW accepts to define, for

some parameters, ranges of values allowing to evaluate the variation of results.

- Check if the response is acceptable. Otherwise change foundations dimensions and

recalculate response.



2. THE IMPEDANCE METHOD

2.1 COORDINATE SYSTEM

Rigid body motion has six degrees of freedom: 3 translations and 3 rotations. If a point of the body is

selected as a reference, once known the displacements velocity and acceleration of the 6 DOF of this

node, the total kinematics of the body is recoverable.

In CW the reference point (or master node) is the following:

- For bearing foundations the geometric center of gravity of the Soil Contact Area (SCA).

- For piled foundation the geometric center of gravity of piles section areas.

Using the reference point as a coordinate origin X and Y axis are traced as indicated in the User

Graphic Interface. The vertical Z axis results directed upward. See Fig. 2.1

Fig. 2.1 – Right-hand convention



2.2. IMPEDANCES

The interaction between the foundation and the supporting soil can be investigated using either a

steady state analysis in the frequency domain or a direct time domain approach. In CW the frequency

domain analysis is utilized.

CW is an implementation of the Impedance Method

The dynamic impedance is defined “as the ratio between the steady-state force (or moment) and the

resulting displacement (or rotation) at the base of the massless foundation”. For each DOF (vertical,

horizontal, rocking, torsional) a discrete model that in most cases are independent of the others, and

acting in the corresponding direction is introduced.

The evaluation of the complex valued impedances functions at the interface points of the soil-

foundation system becomes relevant in this method.

Collecting the six DOF the Foundation Impedance Matrix is a complex matrix S ,where the real and

imaginary parts represent respectively the stiffness and radiation damping of the soil foundation

system. Generally the matrix terms are frequency dependent.

Normal for shallow foundations, and if the DOF are related to a baricentric coordinate system, S is a

diagonal matrix without coupling of the different directions.

Impedance terms are conventionally given in relation to a dimensionless frequency factor

where:

- is the excitation frequency.

- is a characteristic dimension of the foundation (r= radius for circular foundation)

- is the shear wave velocity of the soil.

Typically each term of the complex S matrix is in the following form:

( ) ( ( ) ( ))( )

Where is the static stiffness of the relevant DOF.



Note that the Soil Internal Damping is incorporated using the damping ratio and that the Radiation

Damping is represented by ( ).

The real parts of the impedance term represents force components that are in phase of the

displacements, and in this sense can be considered as a standard stiffness parameter. The imaginary

term are force components in phase with the velocities and are interpreted as energy dissipation by

radiation (Radiation Damping).

For a harmonically varying load {P}, statically transported to the origin (0, 0, 0) with frequency w

applied to the massless foundation.

{ } { }

The response, after the transient phase at the beginning will be:

{ } { }

Where { } { } are complex amplitudes of the applied loads and displacements.

For dynamic equilibrium of a mass less foundation the equation of motion is:

{ } [ ]{ } ([ ] [ ])( ){ }

With [ ] the massless foundation impedance, and:

{ } [ ] { }

To incorporate the foundation and machine mass, a mass matrix { } is generated and concentrated

at the coordinate origin. This matrix is usually sparse because the origin does not coincide with the

center of gravity of the foundation-machine system.

Incorporating this contribution to the equations of motion we obtain:

{ } [ ] { } {[ ] [ ]}{ }

Where is the total system impedance matrix.


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Once obtained { },, that represents the displacements of the origin of the coordinate system, it is

easy to calculate the displacement in any other point of the rigid foundation. These points, where

results are needed, are user defined and called Control Points in CW.

3. LIBRARY OF FOUNDATION IMPEDANCES

Clockwork 12.0 modeling capabilities are shown in Table I. Follows a brief description of each type

of model; for complete formulations see the indicated references.

Standard Fundamental Cones Piles

Soil Type

Half-Space Soil

Multiple Layers over Half-Space - - -

Layers over Rock - - -

Foundation Shape

Disk

Rectangle - -

Arbitrary - -

Embedded Foundation

In One Layer (Half-Space) - -

In Multiple Layers - - -

Pile Theory

Winkler - - -

Gazetas - - - -

Natural Frequency

Coupled Modes -

Uncoupled Modes -

Dynamic Loads

Harmonic

Periodic

Transient - - - - Table 3.1- Clockwork capabilities summary table.


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a) Standard Model

The static-stiffness coefficients of the half-space, for each of the different DOF, are taken as

the direct springs K. These are given in Table 3.1 and Table 3.2.

The model consists of spring, dashpot and masses. These last shall be added to the

Foundation-machine masses. See Table 3.4. See Figure 3.1

Fig 3.1 – Standard Lumped-Parameter for Translational and Rotational motions

Table 3.2 - Static stiffness on Homogeneous Half-Space for disk and rectangle shapes.


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Table 3.3 - Static stiffness on Homogeneous Half-Space for an arbitrary foundation shape.

Table 3.4 – Dimensionless Coefficients of standard lumped parameter model with mass on

homogeneous half-space.


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b) Fundamental Model

The same Tables 3.2 and 3.3 are used for static values K.

The model can be visualized as shown in the Figure 3.2. See Table 3.5. For complete

formulation see [13].

Fig. 3.2 – Spring-Dashpot model and Monkey Tail Model

Table 3.5 – Dimensionless Coefficients for Dashpots and Masses used in Fundamental Model.


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c) Cone Models

Theory developed by Wolf and Deek [10], [11], [12] allows to numerically simulate the

reflection, refraction of waves phenomena of a foundation above a stratified soil. Also

embedding can be simulated. See fig jjj for typical results of impedances parameters b k(a0)

and c(a0) of disk on one layer above a half space.

Table 3.6 – Cone and spring-dashpot-mass model for foundation on surface of homogeneous

half-space

Fig 3.3 – Vertical dynamic-stiffness coefficient of disk on layer fixed at base numerically

obtained.


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4. PILED FOUNDATIONS

4.1. INDIVIDUAL PILE

The impedance of an individual pile is determined based on a Winkler beam model (Beam in an

elastic foundation).

In this approach the soil is modeled by means of Sub grade Coefficients for the displacement DOFs.

For vertical and horizontal displacements these parameters are to be entered by the user or

alternatively selected from values proposed by CW. These last selected from the available technical

literature [6], [17].

The impedance is determined for two conditions that shall be selected: floating pile and pile

supported on rock at its tip.

The imaginary part of the impedance is calculated assuming a viscous damping Subgrade Coefficient

associated with the type of wave that is expected to be radiated by the pile motions and from the

elastic curve compatible with the selected condition of the pile. See [4].These parameters were

assumed frequency independents (k(a0) y c(a0) =1)

Table 4.1 – Stiffness and Damping for Vertical DOF


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Table 4.2 – Stiffness and Damping for Horizontal and Rocking DOF


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4.2 PILED FOUNDATION

The individual pile impedances are assembled in the standard way using 6 DOF rigid body

kinematics and the global impedance matrix of the mass less foundation S is obtained.

Before using, this matrix shall be corrected to account for pile group effects. These corrections are

done following the procedure indicated in [2] and [13]. Note that this procedure is consistent with

the wave approach used for generation of the individual pile impedances.

The group effect is frequency dependent and can result in reduction or amplification of final dynamic

response of the foundation.

Fig 4.1 - From [14]


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4. OTHER DYNAMIC CAPABILITIES

4.1. VIBRATION NATURAL FREQUENCIES AND DAMPING RATIO

CW allows to determine the six undamped natural frequencies of the foundation-machine system,

solving the classic eigenvalue problem. The static values of the stiffness are used.

Sometimes the engineering practice refers to critical damping; for example some codes [19] and

[20]), recommend to limit damping ratios for vibration controlled design,. Damping ratios are always

associated to uncoupled one DOF systems. For this purpose CW also calculates approximate natural

frequencies and damping ratios for each of the 6 DOF using only the diagonal terms of the Stiffness,

Damping and Mass matrices. With the calculated value the designer can take design decisions. For

this purpose CW allows, to escalate the damping ratio, using the “Effective Damping” parameter.

4.2 FOURIER ANALYSIS FOR PERIODICAL LOADS

The response of the foundation-machine system when excited by periodical loads can be determined

in CW decomposing the excitation in Fourier series. The forces can be in any direction but the forces

components (Fx, Fy, Fz) shall have the same period.

Transient load analysis may be approximated with this option using a larger period that contains the

excitation and trailing zeros.

At this date Fourier analysis is not available for piled foundations.


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REFERENCES

- [1] J.P. Wolf and J.W. Deeks, “Foundation Vibration Analysis a Strength of Materials Approach”,

Elsevier, 2007

- [2] R. Dobry and G. Gazetas, Simple Method for Dynamic Stiffness and Damping of Floating Pile

Groups, Géotechnique, 38 (1988): 557-554.

- [3] G. Gazetas, Foundation Vibrations, in Foundation Engineering Handbook, 2nd Edition, edited by H.-

Y. Fang, Chapter 15, pp. 553-593 (New York: Van Nostrand Reinhold, 1991)

- [4] R. Flores and Manuel Silva, “Engineering Approach to modeling of Piled Systems”, Seventh World

Conference on Earthquake Engineering, 1980

- [5] H.G. Poulos and E.H. Davis, Pile Foundation Analysis and Design. New York: Wiley, 1980.

- [6] F.E. Richart, R.D. Woods and J.R. Hall, Vibrations of Soils and Foundations. Englewood Cliffs, NJ:

Prentice-Hall, 1970.

- [7] J.P. Wolf, Soil-Structure-Interaction Analysis in Time Domain. Englewood Cliffs, NJ: Prentice-Hall,

1988.

- [8] J.P. Wolf, Consistent Lumped-Parameter Models for Unbounded Soil: Physical Representation,

earthquake Engineering and Structural Dynamics, 20 (1991): 11-32.

- [9] J.P. Wolf, Consistent Lumped- Parameter Models for Unbounded Soil: Frequency-Independent

Stiffness, Damping and Mass Matrices, Earthquake Engineering and Structural Dynamics, 20 (1991):

33-41.

- [10] J.P Wolf and J.W. Meek, Cone Models for a Soil Layer on Flexible Rock Half-space, Earthquake

Engineering and Structural Dynamics, 22 (1993): 185-193.

- [11] J.P. Wolf and J.W. Meek, Rotational Cone Models for a Soil Layer on Flexible Rock Half-space,

Earthquake Engineering and Structural Dynamics, 23 (1994), in press.

- [12] J.P. Wolf and J.W. Meek, Dynamic Stiffness of Foundation on or Embedded on Layered Soil

Halfspace, Earthquake Engineering and Structural Dynamics Using Cone Frustrums, 23 (1994), in

press.


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- [13] J.P. Wolf, “Foundation Vibration Analysis Using Simple Physical Models”, PTR Prentice-Hall,

1994.

- [14] R. Flores and F. Costa Reis, Interação Solo-Estrutura para fundacoes estaqueadas V Seminario

Nacional de Produção Energia Electrica, Recife 1979

- [15] G. Gazetas, “Analysis of Machine Foundations: State of the Art,” Soil Dynamics and Earthquake

Engineering, 2 (1983), 2-42.

- [16] J. Lysmer and R.L. Kuhlemeyer, “Finite Dynamic Model for Infinite Media,” Journal of the

Engineering Mechanics Division, ASCE, 95 (1969), 859-877.

- [17] R. F. Scott, Foundations Analysis, Prentice Hall, 1981

- [18] J.P. Wolf, Dynamic Soil-Structure Interaction (Englewood Cliffs: NJ: Prentice Hall, 1985).

- [19] ACI 351.3R-04, Foundations for Dynamic Equipment

- [20] DIN 4024, Machine Foundations

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