Machine Foundation Design - An Introduction

Machine Foundation Design | Lawrence Galvez

MACHINE FOUNDATION DESIGN

Lawrence O. Galvez, LM.PICE, M.ASEP

ABSTRACT: This lecture paper serves as an introduction in the complex analysis and design of machine foundation which involves familiarity in areas of structural dynamics, structural engineering and geotechnical engineering. This paper is aimed for graduate or practicing civil or structural design engineers who are new to the subject but is familiar with the areas of civil engineering knowledge cited above; although this paper can also be a handy reference for those already doing machine foundation design. It discusses a brief review of structural and soil dynamics which both play important roles in response of a machine-foundation system due to vibratory loadings induced by the operation of the machine. Current state-of-the-art approach in creating mathematical models using finite element method is discussed together with a recommended step-by-step procedure in doing the analysis and design to satisfy strength and serviceability requirements. An example design of a block foundation is shown to help the reader familiarize in the design process and use rational assumptions in getting a good preliminary geometry of foundation before starting with a detailed analytical model. Main references for this paper were taken from the course syllabus of Advance Foundation Engineering & Design in Graduate School of Engineering of University of the City of Manila but other salient references were gathered from internationally recognized codes, standards and practices.

KEYWORDS: machine foundation; resonance; rocking vibration; soil dynamics, block foundation

1. INTRODUCTION

Foundations supporting vibratory loadings are often present in the industries of oil & gas, petrochemicals, heavy manufacturing and power generations. These vibratory loadings are caused by the unbalanced machine forces as well as the static weight of the machine. If these vibrations are excessive they may damage the machine or cause it not to function properly. It should be noted that the initial cost of construction of a machine foundation is generally a small fraction of the total cost of the machine, accessories and installation, but the failure of the foundation as a result of poor design or construction can translate to heavy dollar losses (Prakash and Puri, 1988). For example, a turbo-generator unit in a power plant may cost around $20 to $ 50 million and an estimated loss of $250,000 per day if it malfunctioned; it is therefore crucial for a foundation engineer to give special attention to the foundation design of vibrating machines.

Historically, analysis and design of machine foundations were done by approximate rule-of-thumb procedure which consist of sizing the foundation based on the weight and type of vibrating machines being supported or by strengthening the soil beneath the foundation by using piles. These procedures generally works; however, this often results to considerable overdesign (Bowles, 1997).

With the advent of improved manufacturing technology, machines now are of higher ratings which give rise to considerably higher dynamic forces and thereby higher stresses which demands improved performance and safety (Bhatia, 2008).

The analysis and design of foundation and structures subjected to vibratory loads is considered a very complex problem because of the interaction of structural engineering, geotechnical engineering and the theory of vibrations (Arya, 1979); added to these, from author’s experience, is that a designer should be keen in the details of machine’s (equipment’s) loadings, configurations or general arrangements and design operations which are already a basic part of mechanical engineering. However, due to practical availability of commercial finite element analysis software nowadays, a designer can create a good mathematical model of machine-foundation system with reasonable time to

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have a reliable prediction of the dynamic response of the system and create necessary changes in the model to mitigate unwanted amplitudes of vibrations i.e. to control the vibrations in the system.

2. STRUCTURAL DYNAMICS

2.1 Static vs Dynamic Analysis of Foundations

Design of concrete foundations supporting static or pseudo-static loadings usually involves computation of internal forces (e.g. axial, shear, moments, etc.) by satisfying the equations of equilibrium (FH = 0; FV = 0; FM = 0) and then check for strength and serviceability requirements. Static loadings are dead loads or permanent loads while the pseudo-static are occupancy live loads, wind and earthquake loads. (Note that earthquake loads mentioned denotes the use of approximate equivalent static lateral force procedure per building codes). For static analysis the material and geometric properties needed are density, Young’s modulus of elasticity, stiffness, thermal coefficient and Poisson’s ratio.

For foundations supporting large vibratory a.k.a. “dynamic” loadings, a dynamic analysis is required to determine the dynamic response of the structure. The response of the structure is measured in the form of vibration amplitudes (e.g. displacements, velocity or accelerations); these are done by satisfying the equation of motions based on Newton’s second law of motion (i.e. F = m a). In dynamic or vibration analysis, in addition to material and geometric properties needed in static analysis; damping coefficient and natural frequency of the machine-foundation system are required. 2.2 Theory of Vibrations

In the analysis and design of machine-foundation system, a designer should be first familiar with fundamentals of theory of vibration. As mentioned previously, traditional approach in controlling vibrations in design of foundations was to increasing the mass (or weight). It was only in the 1950s where foundation engineers begin to use vibration analyses which was based on a theory of a surface load on elastic half-space (i.e. the portion of supporting soil supporting the foundation which is assumed to behave elastically), see Figure 1.

Figure 1. Foundation base in equilibrium position just prior to being displaced slightly downward by a quick push(Source: J.E. Bowles, 1997)

According to Bowles (1997), Figure 1 is similar to a beam-on-elastic-foundation case except the beam uses several static soil springs in the analysis but here the foundation uses only one spring and it is a dynamic soil spring i.e. dependent on time. Note that since the usual value of unbalance loads is

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relatively small compared to the weight of the machine-foundation system the assumption that the soil behaves elastically is acceptable. The soil spring in (a) has been compressed in an amount of zs = W/Kz wherein zs and Kz are both static values; in (b) the footing experiences a quick push in the z direction and a quick release which makes the block moves up and down (i.e. it vibrates), hence, the z s

and Kz at this time are dynamic values.

Displaced or compressed value:

In Figure 1a, we can write the differential equation to describe the motion in a form of F=ma from Newton’s second law of motion to be:

Solving by the methods given in differential equation textbooks after dividing through by the mass term m and defining 2

n = Kz / m, also known as the angular frequency, we can obtain the period of vibration T as:

And the natural frequency fn as any one of the following:

From Eq. (b) it would appear that the vibration will continue forever; we know from experience that this is not so. There must be some damping present, so we will add a damping device termed a dashpot (analog = automobile shock absorbers) to the idealized model. To maintain symmetry we will add half the dashpot to each edge of the base as shown in Figure 1b. Dashpots are commonly described as developing a restoring force that is proportional to the velocity (ź) of the moving mass being damped (i.e. being restrained). Knowing this concept of the dashpot (also damping or viscous force) cz, a vertical force summation gives the following differential equation:

Equation (c) is the equation of motion for a Single Degree of Freedom System (SDOF) with damping. The solution of this equation yields the dynamic characteristic of the system e.g. natural frequency of the system.

From the solution of the differential equation (c), the critical damping (in units of force/velocity) is defined as:

Critically damped system happens when,

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When a forcing function, F, is introduced to the system, the equation becomes

According to Arya (1979), vibrations developed by operating machinery produces several effects that needs to be considered in the design aside from the usual static loads. The usual procedure in doing a vibration analysis is to establish a mathematical model of the real structure. The structural configuration of machine foundation is generally determined by geotechnical consultant and machine manufacturer, these initial configurations may change to suit design criteria or avoid interference of other fixed objects such as pipelines and other ancillary items. The common configurations or types of machine foundations are shown in Figures 2a and 2b.

Figure 2a. Types of machine foundations (a) Block foundations (b) Box or caisson foundations (c) Complex foundations

(Source: S. Prakash, 1991)

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F (d)


Figure 2b. Types of foundations for vibrating machines (Source: S. Arya, 1979)

The following basic definitions are important in the vibration (dynamic) analysis:

Vibration - an oscillation of the parts of a fluid or an elastic solid whose equilibrium has been disturbed

Period (T) - if motion repeat itself in equal intervals of time, it is called a periodic motion and the time elapsed in repeating the motion is called its period of vibrations. It is the time needed for one complete cycle of vibration to pass a given point. As the frequency of a wave increases the period of the wave decreases, i.e. frequency is the reciprocal of period

Figure 3. Graph of Period of vibration along time, t

Cycle - motion completed during a period is referred to as a cycle

Frequency (f) - the number of cycles of motion in a unit of time is called the frequency of vibrations

Natural frequency (fn) - if an elastic system vibrates under the action of forces inherent in the system and in the absence of any externally applied force, the frequency with which it vibrates is its natural frequency

Forced vibrations - vibrations that occur under the excitation of external force are termed forced vibrations

Degrees of freedom (n) - number of independent coordinates necessary to describe the motion of a system specifies the degrees of freedom of the system. When a system have several degrees of

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freedom, the system is called multidegree of freedom system (MDOF). Figure 4a shows degrees of freedom a rigid block foundation. Figure 5 shows systems with different degrees of freedom.

Figure 4a. Degrees of freedom of a rigid block foundation (Source: S. Prakash, 1991)

The rigid block foundation has six degrees of freedom (also called modes of vibrations)1. Translation along Z axis2. Translation along X axis3. Translation along Y axis4. Rotation along Z axis5. Rotation along X axis6. Rotation along Y axis

(Note: Each degrees of freedom of the system will produce different natural frequencies)

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Figure 4b. Types of motion of a rigid foundation due to unbalanced forces of reciprocating machines: (a) pure vertical translation; (b) pure rocking; (c) simultaneous

horizontal sliding and rocking (d) pure torsional oscillations (Source: S. Prakash, 1981)

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Figure 5. Systems with different degrees of freedom (a) One degree of freedom (n=1); (b) Two degrees of freedom (n =2); (c) Three degrees of freedom (n=3); (d) Infinite

degrees of freedom (n=∞)(Source: S. Prakash, 1991)

Resonance - if the frequency of excitation force coincides with any one of the natural frequencies of the system, resonance will occur. This is a phenomenon that is being avoided in machine foundation design because it will the amplitudes (e.g. displacement, velocity or acceleration) of vibrations to be excessive and cause damage to the system.

Normal mode of vibrations - when the amplitude of some point of the system vibrating in one of the principal modes is made equal to unity, the motion is called the normal of vibrations

Damping - damping is associated with energy dissipation and opposes the free vibrations of a system. If the force of damping is proportional to its velocity, it is called viscous damping. If it is not

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n = 1

Simple pendulum

(a) (b)

n = 2

Equilibrium

Position

(c)

(d)

n = 3

m1

m2

m3

n = ∞


dependent on its material property and it is contributed by geometry of the system it is called geometrical damping.

2.3 Harmonic Motion

The simplest form of motion is harmonic motion, which is represented by sine or cosine functions. The excitation in structures or foundations caused by the unbalanced force in the machines are generally in the form of harmonics under steady-state condition.

Rewriting equation (d) with a harmonic force;

where; Fo = unbalanced mass of the machine; t = time

In dynamic system, the excitation force present arises out of unbalances in the rotating masses. Shown below are the forces generated for reciprocating and centrifugal machines.

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Fo sin (t) (e)


Figure 6. (a) Crank mechanism of a reciprocating machine; (b) Forces from a centrifugal machine (rotating mass excitation)

(Source: S. Arya, 1979)

For reciprocating machines:

For centrifugal machines:

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Figure 7a. Rotating vector representation of a harmonic function x = A sin t(Source: S. Arya, 1979)

Figure 7b. Harmonic motion representation of displacement (x), velocity (ẋ) and acceleration (ẍ)


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3. VIBRATING (DYNAMIC) MACHINES

There are many kinds of machines that generates periodic forces; the three most important categories are:

a. Reciprocating machines: machines that produced unbalanced force (such as compressor and reciprocating engines). The operating speeds of such machines are usually less than 600 rpm. For the analysis of their foundations, the unbalanced forces can be considered to vary sinusoidally.

Figure 8a. Single cylinder crank mechanism (a simple reciprocating machine)(Source: Prakash and Puri, 1988)

Figure 8b. Reciprocating machine diagram(Source: ACI 351.3R, 2004)

b. Rotary machines: high-speed machines like turbogenerators or rotary compressors may have speed of more than 3,000 rpm and up to 10,000 rpm. High speed rotary machines are well balances and the eccentricity e is generally very small. However, due to their high speed of rotation, the magnitude of exciting loads may be significant and may cause the effective eccentricity to increase due to wear and tear.

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Figure 8. Unbalanced forces due to rotary machines, (a) single rotor; (b) two rotors with equal unbalanced forces in phase; (c) two rotors with equal unbalanced forces but with

a phase difference of 180o; (d) two rotors with equal unbalanced forces at any phase(Source: Prakash and Puri, 1988)

Dynamics loads during operations are caused by unbalanced loads (created when the mass centroid of rotating part does not coincide with the center of rotation) with a frequency corresponding to machine speed. These loads are usually given by the machine manufacturer. If not, they may be calculated via the balance quality grade G of the rotor.

G = e where; e is in (mm) and is in (rad/s)

From ISO 1940/1 specification the usual value of G is 2.5 mm/s

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Figure 9. Rotating machine diagram(Source: ACI 351.3R, 2004)

Figure 10. Simple cycle gas turbine(Source: General Electric)

c. Impact machines: machines that produce impact loads like forging hammers. Their speeds are from 60 to 150 blows per minute. Their dynamic loads attain a peak in a very short interval and then practically die out.

3.1 Typical assembly of machine

Machine would necessarily include:- A drive machine- A driven machine- A coupling device

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3.2 Machine - Foundation Loading Data

Loadings due to self-weight of machine and unbalanced (dynamic) loading is provided by machine manufacturer. This is sometimes called vendor data. Other loadings due to machine operation are listed including loading due to temperature operation, accidental loads due to malfunctioning, loss of blades, emergency torques, etc. A sample of foundation loading data is shown below.

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3.3 Machine - Mechanical Outline

A designer should review the mechanical outline (drawings) of the machine to properly model later the machine in the structural analysis. Shown below is a sample of machine drawings from vendor.

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3.4 Machine - Anchor bolt layout and details

The layout of anchor bolts should be carefully reviewed by the foundation designer to make sure all the location of loadings will match on what is considered in the analysis. Shown below is a sample of anchor bolt layout and details of a machine.

3.5 Machine - Dynamic Loads from Vendor Data

The rated (operating) and critical speeds of the machine and unbalance loadings are taken from vendor data.

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3.6 Machine - Suggested Load Combinations from Vendor => there are specific load combinations usually recommended by vendor to protect the machine during operation and accidental situation. The designer should include this in his analysis and design.

3.7 Machine - Dynamic Design Criteria

Sample of dynamic design criteria for rotating machine is shown below

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Allowable amplitude of vertical vibration (note: 1 cps = 1 Hertz)

4. SOIL DYNAMICS

DefinitionsSoil Mechanics - coined by Dr. Karl Terzhagi who is recognized as the “father of soil mechanics”. Soil mechanics deals with engineering properties of soil under stress.Soil dynamics - is that branch of soil mechanics which deals with engineering properties of and behavior of soil under dynamic stress.

Types of dynamic loads that may act on foundations and structures:- earthquakes- bomb blasts- operation of reciprocating and rotating machines and hammers- construction operation (such as pile driving)- quarrying- fast-moving traffic

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- wind- wave action of water

Earthquake constitutes to the single most important sources of dynamic loads on structures and foundations. Every earthquake is associated with a certain amount of energy released at its source and can be assigned a magnitude (M) which is just a number (Richter, 1958).

Definition of earthquake terms

Shown below is a sample of soil behavior during the Canterbury earthquake in NZ.

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Figure 11. Sample Damage of Niagata earthquake 1964(Source: S. Prakash, 1991)

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Automobile sunk in during Niagata earthquake 1964

Sewage treatment tank floated to surface during Niagata earthquake 1964 (after Idriss, 1967)

Tilting of building during Niagata earthquake 1964 (After Seed and Idriss, 1967


Due to ground motion during an earthquake, footings may settle, building may tilt, soils may liquefy and loses ability to support structures, and light structures may float.

Below are the sample ground acceleration readings of past earthquakes captured in accelerogram.

Compared to the graph of dynamic loading for reciprocating and rotary machines, the machine’s dynamic loadings are sinusoidal in nature. Although in reality it is not perfectly sinusoidal e.g. the peaks in any two cycles may be different.

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Figure 12. Trace of vertical acceleration of ground due to pile driving(Source: S. Prakash, 1991)

Natural undamped frequency of point bearing piles on rigid rock

4.1 Soil behavior during dynamic loading

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During earthquake, the ground may be susceptible two categories:

4.2 Dynamic Soil parameters needed for dynamic analysis of machine foundation

Dynamic loading of soils are divided into small and large strain amplitude responses. In a machine foundation, the amplitudes of dynamic motion and, consequently, the strains in the soil are usually low, whereas a structure that is subjected to an earthquake may undergo large deformations and thus induce large strains in soil.

The “elastic halfspace model” is usually used in determining the response of the soil that is subjected to harmonic loading. In this model, it presumes that a circular footing rests upon the surface of an elastic halfspace (the soil) extending to an infinite depth, which is homogeneous and isotropic and whose stress-strain properties can be defined by two elastic constants, usually shear modulus (G) and Poisson’s ratio (v).Dynamic soil parameters to be taken from field and/or laboratory test or from published correlations of static properties of soil:

- Dynamic shear modulus, G- Poisson’s ratio, v- Soil Damping ratios, D

4.3 Geotechnical investigation to determine dynamic soil properties

Note: all of these soil investigation should be reported in detailed in geotechnical report.

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Sample schematic of cross-hole test on field

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Figure 13. Crosshole Test(Source: S. Arya, 1979)

Sample schematic of down-hole test on field

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Sample schematic of spectral analysis of shear wave

Figure 13. Surface Oscillator Test(Source: S. Arya, 1979)

Sample of Downhole Test Results

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Borehole plan

4.4 Computation of Equivalent Soil Springs

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Marl

Marl

Sandstone

Sandstone

Sandstone

Sandstone

Sandstone

Silty Sand and Gravel

Silty Sand and Gravel

KdEdGd

Dynamic Soil Properties (MPa)

Poi-sson Ratio

VSVP

Seismic Wave VelocityN-ValueType of

SoilSoil

Depth (m)

Bore Hole No.


Table 1. Equivalent Spring Constants (Source: S. Arya, 1979)

Figure 14. Coefficients z, x and


Table 2. Equivalent damping ratios (Source: S. Arya, 1979)

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5. STRUCTURAL ANALYSIS AND DESIGN (for block foundation)

Two Stages:Static Analysis includes check for strength of foundation, stability of foundation and check for soil bearing capacity (actual soil bearing should be less than 50~75% of allowable bearing capacity)Dynamic Analysis includes determination of natural frequencies of the machine-foundation system and checking of response vibration amplitudes

5.1 Foundation Trial Sizing Check

- Footing thickness shall be greater than the maximum of ( 0.60 m ; the rigid criterion 0.60 + L / 30) , provide additional 0.60m on both sides for maintenance purpose

- Footing width shall be greater than 1.0~1.5 x vertical distance (foundation base to machine C.G.) + 0.60 m on both sides for maintenance purpose

- Mass ratio check (for rotary machines: foundation mass = 2 to 3 times of machine mass; for reciprocating machines: 3 to 5 times of machine mass)

- Foundation eccentricity check shall not be greater the 5% on both plan dimensions (i.e. the centroid of mass of machine-foundation system wrt base contact area)

Note: Table-top type foundation will have different requirement in trial sizing.

5.2 Structural Modeling

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5.2.1 Finite element modeling (Note: Modeling is a critical part in the structural analysis)

5.2.2 Modeling of supports (Note: Static soil spring is different from dynamic soil spring; dynamic modulus of elasticity, Ec’ of concrete = 1.60* static modulus of elasticity, Ec)

5.3 Time-History Analysis

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5.3.1 Dynamic input in analysis software

Sample of input in analysis software (operating speed)

Sample of input in analysis software (emergency speed)

5.3.2 Unbalanced loads model

5.3.3 Determine machine-foundation system natural frequencies, fn

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5.3.4 Investigate the shapes of observed critical modes

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5.3.5 Resonance check

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Sample of resonance check

5.3.6 Vibration amplitudes check

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Sample of velocity amplitude check

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5.3.7 Show plots of governing amplitudes

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6. SUMMARY OF DESIGN CHECKING

Static Design check for serviceability and strength requirements, update accordingly; follow best practices in reinforced concrete detailing; check seismic requirements if project is located in high seismic zone area.

Dynamic Design verify if vibration amplitudes are below allowable values, review and/or adjust accordingly; check mode shapes of the system, check if natural frequencies are outside the resonance range, check other requirement checking required by vendor e.g. deflection of bearing points, misalignment of rotor.

Below is a schematic flow of dynamic design process.

Figure 15. Schematic diagram of a machine-foundation system subjected to dynamic loads

(Source: K. Bhatia, 2008)

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REFERENCES

J.E. Bowles (1997) “Foundation Analysis and Design 5 th Ed. - Chapter 20 Design of foundation for vibration controls”

S. Prakash (1991) “Soil Dynamics - Chapter 9 Machine Foundation”

S. Prakash (1991) “Soil Dynamics - Chapter 1 Introduction and 2 Theory of Vibration”

Prakash and Puri (1988) “Foundation for Machines - Analysis and Design”

Prakash and Puri (2006) “Journal on Foundation for Vibrating Machines”

S. Arya et al. (1979) “Design of structures and foundations for vibrating machines”

K.G. Bhatia (2008) “Journal on Foundations for industrial machines and earthquake effects”

American Concrete Institute ACI 351.3R (2004) “Foundations for Dynamic Equipment”

Technical Paper on “Seismic Design of Buried Structures in PH and NZ” presented by Lawrence Galvez in 17th ASEP International Convention” (2015)

Lecture notes on Advance Foundation Engineering and Design Subject in University of the City of Manila - Prof. Rolando D. Rabot, ASEAN Engr, M.Engg, Ph.D-Te

OTHER SUGGESTED REFERENCES FOR READING

German StandardsDIN 4024 Part 1 Machine Foundations; flexible structures that support machines rotating machinesDIN 4024 Part 2 Machine Foundations; rigid structures that support machines periodic excitation

British StandardsCP 2012 Code of Practice for Foundations for machinery

American StandardsAmerican Society of Civil Engineers (ASCE) Task Committee on “Design of large steam turbine- generator foundations”

ABOUT THE AUTHOR

Lawrence “LG” Galvez is a licensed civil engineer professional involved in structural engineering and design of industrial and infrastructure projects for local and international clients. His experience in civil and structural design are from oil & gas, petrochemicals, mining, telecommunications, water & wastewater, power generation, bridge & railway and land development industries. LG is a PICE Life Member and a Regular Member of ASEP. He can be reach thru email: [email protected].

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mailto:[email protected]

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Machine Foundation Design - An Introduction