Foundations Subjected to Vibration Loads

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1 Foundations Subjected to Vibration Loads A practical design tool for sizing equipment mats By Leonel I. Almanzar Micheli and halid Motiwala 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 machine 1 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 MODEL A 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 100 mm), modulus values were varied from 5 to 00 kip/ft 3 (3.93 to 31.5 MPa/m), and distributed equipment weight values were varied from 0 to 500 lb/ft (0 to 4 kpa). In addition to equipment weight, models included induced vibration forces uniformly distributed over the slab area. MODEL PARAMETERS Slab and subgrade The slab was modeled as an isotropic material with gross section properties. The modulus of elasticity was assumed to be 3605 ksi (4,900 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, 1, must be corrected for the size and configuration of the actual foundation. First, the corrected subgrade modulus for a B x B foundation, where B is the smaller of the actual foundation plan dimensions in ft (m), is determined using one of the following two equations. For sandy soils: B + 1 ft B B m For clays: 1 1 B B 1 3.8B (1) (1) () () 4 July 009 / Concrete international

2 Fig. 1: Finite element model used for dynamic analysis of a mat foundation s response to a vibrating load The corrected subgrade modulus for a B x L foundation, where L is the larger of the actual foundation plan dimensions in ft (m), is then determined from the following equation Fig. : Example of natural frequency results from finite element analysis of a 10 x ft (3 x 6.7 m) mat foundation with varying slab thickness and subgrade modulus (1 in. 5.4 mm; 1 kip/ft MPa/m) B L 1.5 The spring stiffness in the model was equal to the tributary area of a node multiplied by. Dynamic load and damping The system was subjected to a vibrating load F that followed the function 3 (3) F F sin (ω t) (4) o where F o 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 used 4 F W g r o eω (5) where W r is the weight of equipment engine in lb (N), g is the acceleration due to gravity taken as in./s (9.81 m/s ), 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 5,000 rpm (60 rad/s), e can be approximated by 0.5/ω in inches ( /ω in meters). The internal damping of the soil was approximated using a conservative value of % of critical damping. 5 NATURAL FREQUENCY Only 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. Fig. 3: Natural frequency results for a subgrade modulus of 50 kip/ft 3 (7.85 MPa/m) as a function of foundation pressure and slab thickness h (1 in. 5.4 mm; 1 lb/ft kn/m ) As shown by the natural frequency results from the model presented in Fig., 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/ft 3 (7.85 MPa/m), Fig. 3 gives the natural frequency for a specific foundation pressure. This value is then corrected for the actual -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 Concrete international / July

3 Fig. 4: Natural frequency correction factor for mat foundations when subgrade modulus is different than 50 kip/ft 3 (7.85 MPa/m) (1 kip/ft MPa/m) Fig. 5: Example of vertical displacement time history results (1 in. 5.4 mm) DYNAMIC RESPONSE The 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. Therefore, 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/ft 3 (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 CIRCLE READER CARD #1 A f A C C C (6) o July 009 / Concrete international

4 Fig. 6: Initial vertical displacement amplitude for a 50 kip/ft 3 (7.85 MPa/m) subgrade modulus and an 1800 rpm equipment operating speed for various slab thicknesses h (1 in. 5.4 mm; 1 lb/ft kn/m ) 1/3 Vert. where A f is the corrected amplitude in inches (m), A o is the amplitude obtained from Fig. 6 in inches (m), C 1 is the operating speed correction factor from Fig. 7, C is the force amplitude correction factor from Fig. 8, and C 3 is the modulus of subgrade reaction correction factor from Fig. 9. EXAMPLE The following example demonstrates the use of the method and figures presented in this article for the design of a 000 kw generator foundation. The following equipment and soil data are known: Total generator weight W 38,155 lb (169.7 kn); Engine weight W r 17,885 lb (79.6 kn); Operating frequency 1800 rpm 30 Hz rad/s; Foundation rests on sandy soil with 1x1 5 kip/ft 3 (35.3 MPa/m); Eccentricity e in. ( m); Total generator length 1 ft (6.4 m); and Generator width 7.75 ft (.36 m); Based on the generator dimensions, the foundation is initially estimated to be 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: 3 3 B + 1 ft 11 ft + 1 ft 5 kip/ft kip/ft B (11 ft) B m 3.35 m m 35.3 MPa/m MPa/m B (3.35 m) B 11 ft L kip/ft ft kip/ft B 3.35 m L MPa/m 6.7 m 8.73 MPa/m CIRCLE READER CARD #13 Concrete international / July

5 Fig. 7: Displacement amplitude correction factor C 1 for operating speeds other than 1800 rpm Next, the natural frequency is found from Fig. 3. The foundation area is 11 ft ft 4 ft (3.35 m 6.7 m.45 m ), and the total equipment weight/foundation area is 38,155 lb/4 ft lb/ft (169.7 kn/.45 m 7.56 kn/m ). 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 50 kip/ft 3 (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 of kip/ft 3 (8.73 MPa/m). Therefore, the natural frequency for the system is 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 Therefore, the selected foundation size meets the criteria. Now, the displacement amplitude can be determined. Using Fig. 6, the initial displacement amplitude A o is in. ( m) for an equipment weight/foundation area of lb/ft (7.56 kn/m ). This amplitude must be corrected using Fig. 7, 8, and 9. For an operating speed of 1800 rpm, Fig. 7 gives a correction factor C To find correction factor C, the force amplitude F o must be calculated using Eq. (5) ω Fig. 8: Displacement amplitude correction factor C for force amplitudes other than 5% of the total equipment weight ω Fig. 9: Displacement amplitude correction factor C 3 for subgrade moduli other than 50 kip/ft 3 (7.85 MPa/m) (1 kip/ft MPa/m) From Fig. 8, the force amplitude correction factor C From Fig. 9, the modulus of subgrade reaction factor C for kip/ft 3 (8.73 MPa/m). The corrected initial displacement amplitude A f A o C 1 C C in in. ( m m). This vibration amplitude is compared to the allowable vibration amplitudes shown in Fig 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 46 July 009 / Concrete international

6 References 1. ACI Committee 351, Foundations for Dynamic Equipment (ACI 351.3R-04), American Concrete Institute, Farmington Hills, MI, 004, 63 pp.. Das, B.M., Principles of Foundation Engineering, fifth edition, Thomson/Brooks/Cole, Pacific Grove, CA, 004, pp Wolf, J.P., Foundation Vibration Analysis Using Simple Physical Models, PTR Prentice Hall, Englewood Cliffs, NJ, 1994, 43 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-ent Publishing Co., Boston, 1993, pp Selected for reader interest by the editors after independent expert evaluation and recommendation. Fig. 10: Suggested allowable combinations of foundation vibration amplitude and vibration frequency (after Reference 5) (1 in. 5.4 mm) on the damping of the system, but can be estimated to be about 1/5 to 1/3 the initial amplitude when % of critical damping is used. Suggestions During design of mat foundations subjected to vibrating load, the following recommendations should be taken into consideration: To avoid resonance in vibrating machinery with low operating speed, the natural frequency should be at least two times the operating frequency; To avoid resonance in vibrating machinery with high operating speed, the natural frequency be no more than half the operating frequency; When vibrating force amplitude cannot be obtained, at least 10% of the weight of the rotating machine should be considered 5 ; When data are not available, the internal damping of soil can be approximated using % of critical damping; and 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. 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 halid 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. Concrete international / July

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