082975

8/2/2019 082975

1/10

Hydrocarbon Processing / August 2008 Supplied by the NIOC Central Library

Designing rectangular block foundations for vibratingequipment

Method considers both the effect of damping and coupled modes using time-history analysis

K. Chakravarti, Petrofac International Ltd., Sharjah, UAE

Unless there is a process/ functional requirement for placing rotating equipment foundations atcertain elevations, they are normally reinforced concrete block foundations on the ground orelevated framed structures.Normal practice. In general, civil engineers choose one of the two methods for designingreinforced concrete block foundations for vibrating equipment, each having its own advantagesand disadvantages:

Barkan's method1

Arya, O'Neil and Pincus' method.2

Barkan's method is simpler and is used in most countries, especially third-world and middle-eastcountries. It accounts for the coupled mode of vibration, i.e., longitudinal translation with pitchingand lateral translation with rocking, but does not account for material and soil damping of thefoundation system. The effect of damping for soil-supported foundations is often substantial andshould not be neglected. Ignoring damping makes the design conservative.Ayra, O'Neil and Pincus' method accounts for material and soil damping, which makes it morecost-effective and popular with engineering companies, but it does not account for the coupledmode of vibration. Thus, the actual vibration scenario can not be correctly idealized with the helpof this method. At the end of the analysis, amplitudes in all modes are suitably added by othermethods to get resultant amplitude.Suggested method. Due to the disadvantages of the existing methods, a third method ispresented that offers the advantages of both methods without the disadvantages. Steps of theproposed method are:

8/2/2019 082975

2/10


1. Required soil data: Weight density,

Mass density, (May be calculated based on weight density) Shear wave velocity, Vs Dynamic shear modulus, Gdyn(May be calculated based on shear wave velocity, Gdyn=

Vs2 ) Poisson's ratio, .

2. Required foundation dimensions and shape parameters: Foundation base length in plan, flx Foundation base breadth in plan, fl z Foundation embedment depth, h Height of combined foundation with equipment from the foundation base, S.

3. Foundation and equipment weight, mass and mass moment of inertia about the centroid at thebase level:

Weight, W Mass, m= W/g Mass moment of inertia about longitudinal (X) axis at base level, Mass moment of inertia about transverse (Z) axis at base level, Mass moment of inertia about vertical (Y ) axis at base level,

4. Calculating effective footing radius:

5. Calculating embedment coefficients:

8/2/2019 082975

3/10

8/2/2019 082975

4/10


Values of nwith respect to values of Bfor rocking and pitching modes:

For equations under sections 4 to 9, see reference 2.10. Total damping:Total damping ratio, , (subject to a maximum of 0.2 or 20% as per standard industrial practice) =geometric damping + internal damping(may be assumed to range from 2% to 5%).So far, whatever calculation steps have been shown are common with the method suggested byArya, O'Neill and Pincus. Henceforth, a different analysis procedure will be adopted that is "time-

history analysis" by the Wilson- method.11. Formation of mass, stiffness and damping matrices:Vibration in "vertical" and "yawing" modes are comparatively simpler, since they are normally notcoupled with any other mode. Under normal conditions, "rocking" mode is coupled with"transverse translation" and "pitching" mode is coupled with "longitudinal translation" mode.For simplicity, consider a foundation model that is subjected to horizontal periodic force in one

direction and periodic moment in the same direction (about an axis perpendicular to the directionof force, Fig. 1).

1

8/2/2019 082975

5/10


Fig. 1 A foundation model that is subjected tohorizontal periodic force in one direction andperiodic moment in the same direction.

Now, the equilibrium equation for translation is:

The equilibrium equation for rotation is:

Hence, matrix notation of the simplified case in two degrees of freedom is:

8/2/2019 082975

6/10


Similarly, for six degrees of freedom:Mass matrix, M:

Stiffness matrix, K:

Since C=2(mK )1/2,

Hence,

8/2/2019 082975

7/10


Damping matrix, C :

12. Force vector formation:The unbalanced "force vectors" for drive, gearbox and driven equipment need to be formed

separately with frequencies and phase differences as:For drive:

For gear box:

For driven equipment:

8/2/2019 082975

8/10


Total force vector:

where is the phase difference.Once M, C, Kand {F(t)} are available, the dynamic equilibrium equation may be formed as: [M]{x} + [C]{x} + [K ]{x} = {F(t)}This equation may be solved by the Wilson- method of time-history analysis, steps of whichfollow:

313. Time step, t, is selected preferably as 60/Highest rpm 814. Considered:

15. The effective stiffness matrix is formed as:

16. For each time step: For the first step,

and for the next steps

and of previous step

8/2/2019 082975

9/10


Effective load at time t+ tis:

Calculated displacements, velocities and accelerations at time, t + t, are:

These steps are to be repeated up to the desired number to get governing values ofdeflection or velocity amplitude.

Absolute maximum amplitude of the nozzle connection point or any other point of interestis calculated at a certain instant combining the effects of all the modes.

All the steps may be programmed in software to get the displacement, velocity and accelerationfor all the six degrees of freedom in tabular form. Graphs may be plotted from such tabular resultsof dynamic analysis.Figs. 2 and 3 show displacement and velocity amplitudes of the nozzle connection point of asample case.

Fig. 2 Displacement amplitude.

Interested readers may obtain the Excel spreadsheet (with macro) for the method from theauthor (e-mail: [email protected]). HP

LITERATURE CITED1 Barkan D. D., Dynamics of Bases and Foundations.2 Arya, O'Neill & Pincus, Design of Structures and Foundations for Vibrating Machines.3 Klaus-Jurgen Bathe, Finite Element Procedures.

8/2/2019 082975

10/10


The authorKoushik Chakravarti is a lead civil and structural engineer in Petrofac InternationalLtd, Sharjah. Previously he was with Bechtel, New Delhi. He focuses on theunresolved problems faced in day-to-day structural engineering in the hydrocarbonindustry.

082975

Documents

Transcript of 082975