42 July 2009 / Concrete international

By leonel I. AlmAnzAr mIchelI And KhAlId motIwAlA

Foundations Subjected to Vibration Loads

A practical design tool for sizing equipment mats

The structural design of a foundation system supporting dynamic equipment is complicated by the need to

model the periodic load, foundation, and soil conditions; calculate the structural response of the system; and interpret the resulting displacements. Designers therefore may follow historical rules of thumb—for example, ensuring that the total weight of the mat foundation (slab) is three times the weight of a rotating machine or five times the weight of a reciprocating machine1—but these rules may lead to oversized foundations.

To obtain an optimum slab size while avoiding resonance with the supported machine and ensuring minimal human perception of vibrations, it’s important to track a slab’s natural frequency and its displacement amplitudes. In this article, we present practical guidelines and recommendations for design of slabs subject to induced vibration forces from steady-state frequency equipment without vibration isolators. Using graphs and tables, we also show the effects of changes in dimensions of the foundation, operational speed of the equipment, and modulus of subgrade reaction on a slab’s natural frequency and peak displacement.

FOUNDATION MODELA finite element model incorporating 4-node plate

elements with linear spring supports located at each node was used to find the natural frequencies and vibration amplitudes for slabs (Fig. 1). Slab thickness, corrected modulus of subgrade reaction, and distributed equipment weight values were varied to develop parametric curves. Thickness values were varied from 6 to 48 in. (150 to 1200 mm), modulus values were varied from 25 to 200 kip/ft3 (3.93 to 31.5 MPa/m), and distributed equipment

weight values were varied from 0 to 500 lb/ft2 (0 to 24 kPa). In addition to equipment weight, models included induced vibration forces uniformly distributed over the slab area.

MODEL PARAMETERSSlab and subgrade

The slab was modeled as an isotropic material with gross section properties. The modulus of elasticity was assumed to be 3605 ksi (24,900 MPa). Elements sizes were constant at 1 x 1 ft (0.3 x 0.3 m) except when varied to verify convergence of the results.

The modulus of subgrade reaction from a 1 x 1 ft (0.3 x 0.3 m) reaction plate test, K1, must be corrected for the size and configuration of the actual foundation. First, the corrected subgrade modulus for a B x B foundation KBxB, where B is the smaller of the actual foundation plan dimensions in ft (m), is determined using one of the following two equations.2

For sandy soils:2

11 ft

2BxBBK KB

+ = (1) (in.-lb units)

2

10.3 m2BxB

BK KB

+ = (1) (SI units)

For clays:

1

1BxBK K

B =

(2) (in.-lb units)

11

3.28BxBK KB

= (2) (SI units)

Concrete international / July 2009 43

Fig. 2: Example of natural frequency results from finite element analysis of a 10 x 22 ft (3 x 6.7 m) mat foundation with varying slab thickness and subgrade modulus K (1 in. = 25.4 mm; 1 kip/ft3 = 0.157 MPa/m)

The corrected subgrade modulus for a B x L foundation K, where L is the larger of the actual foundation plan dimensions in ft (m), is then determined from the following equation

1 0.5

1.5BxB

BLK K

+ =

(3)

The spring stiffness in the model was equal to the tributary area of a node multiplied by K.

Dynamic load and dampingThe system was subjected to a vibrating load F that

followed the function3

(ω )oF F sin t= (4)

where Fo is the amplitude of the vibrating force in lb (kN), ω is the operating speed of the equipment in rad/s, and t is time in seconds.

To compute the amplitude of the vibrating force to be transmitted to the foundation, the following equation can be used4

2roWF eg

= ω

(5)

where Wr is the weight of equipment engine in lb (N), g is the acceleration due to gravity taken as 386.4 in./s2 (9.81 m/s2), ω is the operating speed of the equipment in rad/s, and e is the eccentricity of unbalanced mass in inches (m). When e is not known and the operating speed of the equipment is less than 25,000 rpm (2620 rad/s), e can be approximated by 0.25/ω in inches (0.00635/ω in meters).

The internal damping of the soil was approximated using a conservative value of 2% of critical damping.5

NATURAL FREQUENCYOnly the first three mode shapes and frequencies were

evaluated for the slab models. The lowest frequency value is taken as the natural frequency of the foundation.

As shown by the natural frequency results from the model presented in Fig. 2, the natural frequency of the system increases as the modulus of subgrade reaction increases and as the thickness of the slab decreases.

Other parameters also affect the natural frequency of the system, such as the slab dimensions and the equipment weight. Therefore, two graphs have been developed to approximate the natural frequency of the actual system.

For a modulus of subgrade reaction of 50 kip/ft3 (7.85 MPa/m), Fig. 3 gives the natural frequency for a specific foundation pressure. This value is then corrected for the actual K-value using the factor obtained from Fig. 4. The natural frequency of the foundation-soil system should be either less than half or at least two times the operating frequency of the equipment.6

Fig. 1: Finite element model used for dynamic analysis of a mat foundation’s response to a vibrating load

Fig. 3: Natural frequency results for a subgrade modulus of 50 kip/ft3 (7.85 MPa/m) as a function of foundation pressure and slab thickness h (1 in. = 25.4 mm; 1 lb/ft2 = 0.048 kN/m2)

44 July 2009 / Concrete international

DYNAMIC RESPONSEThe displacement time history

was obtained from the finite element model. An example is shown in Fig. 5. The initial amplitude, as well as the steady-state amplitude during operation, should be recorded or observed, although the initial amplitude was always found to be larger in our results. From these values, a set of curves was developed by changing the operational speed of the equipment and the ratio of equipment weight to foundation area. Each curve represents a different slab thickness. It should be noted that the vibrating force amplitude also changes with equipment type, weight, and operating speed. There-fore, to set up typical curves, the vibrating force amplitude was fixed at 5% of the total equipment weight for the development of the curves.

Figure 6 shows initial amplitude curves for a modulus of subgrade reaction of 50 kip/ft3 (7.85 MPa/m) and an equipment operating speed of 1800 rpm.

The displacement amplitude from Fig. 6 is corrected for the actual operating speed, force amplitude, and subgrade modulus using the following equation

1 2 3f oA A C C C= (6)

Fig. 4: Natural frequency correction factor for mat foundations when subgrade modulus is different than 50 kip/ft3 (7.85 MPa/m) (1 kip/ft3 = 0.157 MPa/m)

Fig. 5: Example of vertical displacement time history results (1 in. = 25.4 mm)

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Concrete international / July 2009 45

Fig. 6: Initial vertical displacement amplitude for a 50 kip/ft3 (7.85 MPa/m) subgrade modulus and an 1800 rpm equipment operating speed for various slab thicknesses h (1 in. = 25.4 mm; 1 lb/ft2 = 0.048 kN/m2)

where Af is the corrected amplitude in inches (m), Ao is the amplitude obtained from Fig. 6 in inches (m), C1 is the operating speed correction factor from Fig. 7, C2 is the force amplitude correction factor from Fig. 8, and C3 is the modulus of subgrade reaction correction factor from Fig. 9.

EXAMPLEThe following example demonstrates the use of the method and figures

presented in this article for the design of a 2000 kW generator foundation. The following equipment and soil data are known:n Total generator weight W = 38,155 lb (169.7 kN);n Engine weight Wr = 17,885 lb (79.6 kN);n Operating frequency = 1800 rpm = 30 Hz = 188.5 rad/s;n Foundation rests on sandy soil with K1x1 = 225 kip/ft3 (35.3 MPa/m);n Eccentricity e = 0.001 in. (0.000025 m); n Total generator length = 21 ft (6.4 m); andn Generator width = 7.75 ft (2.36 m);

Based on the generator dimensions, the foundation is initially estimated to be 22 x 11 ft (6.7 x 3.35 m).

Using Eq. (1) and (3), the modulus of subgrade reaction is corrected for the actual foundation size:

23 3

11 ft 11 ft 1 ft225 kip/ft 66.94 kip/ft

2 2(11 ft)BxBBK KB

+ + = = = (in.-lb units)

2

10.3 m 3.35 m 0.3 m35.3 MPa/m 10.48 MPa/m2 2(3.35 m)BxB

BK KB

+ + = = = (SI units)

3 3

11 ft1 0.51 0.522 ft66.94 kip/ft 55.78 kip/ft

1.5 1.5BxB

BLK K

++ = = =

(in.-lb units)

3.35 m1 0.51 0.56.7 m10.48 MPa/m 8.73 MPa/m

1.5 1.5BxB

BLK K

++ = = =

(SI units)

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1/3 VERT.

46 July 2009 / Concrete international

Next, the natural frequency is found from Fig. 3. The foundation area is 11 ft × 22 ft = 242 ft2 (3.35 m × 6.7 m = 22.45 m2), and the total equipment weight/foundation area is 38,155 lb/242 ft2 = 157.67 lb/ft2 (169.7 kN/22.45 m2 = 7.56 kN/m2). The authors recommend that the initial trial analysis foundation weight be at least 1.3 to 1.5 times the total equipment weight. Therefore, an initial thickness of 30 in. (0.76 m) is used. From Fig. 3, the natural frequency for K = 50 kip/ft3 (7.85 MPa/m) is 8.8 Hz.

The natural frequency from Fig. 3 must be corrected using the factor from Fig. 4, which is 1.04 for an actual K of 55.78 kip/ft3 (8.73 MPa/m). Therefore, the natural frequency for the system is 1.04 × 8.8 Hz = 9.15 Hz. As mentioned, it’s recommended that the natural frequency be less than half or more than twice the equipment operating frequency. For this example, the ratio of natural to operating frequency is 9.15 Hz/30 Hz = 0.31. Therefore, the selected foundation size meets the criteria.

Now, the displacement amplitude can be determined. Using Fig. 6, the initial displacement amplitude Ao is 0.00075 in. (0.000019 m) for an equipment weight/foundation area of 157.67 lb/ft2 (7.56 kN/m2). This amplitude must be corrected using Fig. 7, 8, and 9. For an operating speed of 1800 rpm, Fig. 7 gives a correction factor C1 = 1.0.

To find correction factor C2, the force amplitude Fo must be calculated using Eq. (5)

ω

(in.-lb units)

ω

(SI units)

(in.-lb units)

(SI units)

From Fig. 8, the force amplitude correction factor C2 = 0.85.From Fig. 9, the modulus of subgrade reaction factor C3

= 0.95 for K = 55.78 kip/ft3 (8.73 MPa/m).The corrected initial displacement amplitude Af =

AoC1C2C3 = 0.00075 in. × 1.0 × 0.85 × 0.95 = 0.00061 in. (0.0000195 m × 1.0 × 0.85 × 0.95 = 0.000016 m). This vibration amplitude is compared to the allowable vibration amplitudes shown in Fig. 105 to determine the adequacy of the foundation in terms of vibration perception. The initial vibrating amplitude of the generator foundation will be noticeable to people. This occurs, however, only for a short time, and it’s not considered to be transmitted to nearby structures. The operating amplitude depends

Fig. 9: Displacement amplitude correction factor C3 for subgrade moduli other than 50 kip/ft3 (7.85 MPa/m) (1 kip/ft3 = 0.157 MPa/m)

Fig. 8: Displacement amplitude correction factor C2 for force amplitudes other than 5% of the total equipment weight

Fig. 7: Displacement amplitude correction factor C1 for operating speeds other than 1800 rpm

Concrete international / July 2009 47

Leonel I. Almanzar Micheli is a Senior Structural engineer in Aecom’s orlando, Fl, office. he received a BS in civil engineering from technological Institute of Santo domingo and an mS in structural engineering and a Phd in structural engineering from the university of Puerto rico. he is a member of the ASce-7 main committee and Special Project chair of the SeI-central

Florida chapter. he has experience in analysis, design, and preparation of preliminary plans and final contract plans, and the management of construction services for water and wastewater treatment facilities, dams, and transportation structures.

AcI member Khalid Motiwala is a Principal Structural engineer in Aecom’s orlando, Fl, office. he received a BS in civil engineering from the university of central Florida. he has served as a professional engineer in the areas of structural design of water and wastewater treatment facilities, pump stations, box culverts, retaining walls, industrial, and manufacturing facilities,

as well as assessment of existing infrastructure and facility engineering. he is a member of AcI committee 350, environmental engineering concrete Structures, and subcommittees 350A, General and concrete, and 350d, Structural. In addition, he is the current President of the AcI central Florida chapter.

on the damping of the system, but can be estimated to be about 1/5 to 1/3 the initial amplitude when 2% of critical damping is used.

SUggESTIONSDuring design of mat foundations subjected to vibrating

load, the following recommendations should be taken into consideration:n To avoid resonance in vibrating machinery with low

operating speed, the natural frequency should be at least two times the operating frequency;

n To avoid resonance in vibrating machinery with high operating speed, the natural frequency be no more than half the operating frequency;

n When vibrating force amplitude cannot be obtained, at least 10% of the weight of the rotating machine should be considered5;

n When data are not available, the internal damping of soil can be approximated using 2% of critical damping; and

n The system should be designed to have a maximum displacement amplitude response in the range of barely to easily noticeable to people to avoid human discomfort.

Fig. 10: Suggested allowable combinations of foundation vibration amplitude and vibration frequency (after Reference 5) (1 in. = 25.4 mm)

References1. ACI Committee 351, “Foundations for Dynamic Equipment

(ACI 351.3R-04),” American Concrete Institute, Farmington Hills, MI,

2004, 63 pp.

2. Das, B.M., Principles of Foundation Engineering, fifth edition,

Thomson/Brooks/Cole, Pacific Grove, CA, 2004, pp. 277-278.

3. Wolf, J.P., Foundation Vibration Analysis Using Simple Physical

Models, PTR Prentice Hall, Englewood Cliffs, NJ, 1994, 423 pp.

4. Arya, S.C.; O’Neill, M.W.; and Pincus, G., Design of Structures

and Foundations for Vibrating Machines, Gulf Publishing Co.,

Houston, TX, 1979, 191 pp.

5. Whitman, R.V., and Richart, F.E. Jr., Design Procedures for

Dynamically Loaded Foundations, University of Michigan, Ann Arbor,

MI, 1967, 53 pp.

6. Das, B.M., Principles of Soil Dynamics, PWS-Kent Publishing Co.,

Boston, 1993, pp. 182-255.

Selected for reader interest by the editors after independent expert

evaluation and recommendation.