In this paper, the influence of soil-structure interaction (SSI) on a torsionally coupled turbo-generator (TG) machine foundation is studied under earthquake ground motions. The beneficial effects of base isolators in the TG foundation under earthquake ground motions are also studied duly, considering the effects of SSI. A typical TG foundation is analyzed using a three-dimensional finite element (FE) model. Two superstructure eccentricity ratios are considered to represent the torsional coupling. Soft soil properties are considered to study the effects of SSI. This research concludes that the effects of torsional coupling alter the natural frequencies, if ignored, could lead to unsafe design. The deck accelerations and displacements are increased with an increase in superstructure eccentricity. On the other hand, the deck accelerations and displacements are greatly reduced with the help of base isolators, thus confirming the beneficial use of base isolators in machine foundations to protect the sensitive equipment from the strong earthquake ground motions. However, the effects of SSI reduce the natural frequencies of the TG foundation resting on soft soil conditions and activate the higher mode participation, resulting in amplifying the response.

Machine foundations are used to help in the distribution of the machine loads and mitigate the vibrations developed due to the rotating machine parts. Turbo-generator foundations also known as framed-type machine foundations are the common supporting system for rotary machines of medium to high-speed [

The effects of SSI on block type machine foundation are well established using simplified models due to its predictable modes of vibration such as vertical, horizontal, and rocking modes [

Currently, either simplified methods without considering the effects of soil or rigorous finite element method (FEM) considering detailed soil-structure interaction (SSI) are being adopted for TG foundation in practice. From the above studies, it is evident that ignoring the effects of the SSI may lead to the unsafe design of the machine foundations, especially under earthquake forces. Therefore, it will be interesting to study the dynamic response of TG foundation under different earthquake ground motions while duly considering the effects of SSI, which is attempted in this work.

For protecting structures from the damaging effects of vibrations, base isolation is considered one of the most ingenious ways. Base-isolated structures are found effective in mitigating the damaging consequences of earthquake excitation by decoupling the structure from the ground motions [

In this study, an attempt is made by implementing elastomeric type isolators in TG foundation and time-history analysis is carried out for different earthquake ground motions to quantify the effects of SSI on base-isolated machine foundation.

Observations of damages on structures in the recent past have shown the importance of torsional vibrations induced by earthquake ground motions. The eccentricity between the center of mass (CM) and the center of stiffness (CS) may lead to torsional coupling in framed structures. Though CM and CS coincide in the symmetrical buildings, it has been observed that there exists some uncertainty, and it is always safe to design the structures considering some degree of accidental torsion. Most of the codes prescribed the need to include 5–10% accidental torsion by assuming eccentricity in different directions in the framed structure while analyzing it [

Dynamic analysis of the TG foundation is carried out in a finite element (FE) software

Schematic diagram of base-isolated machine foundation; (a) three-dimensional view, (b) plan view at deck level, and (c) plan view at base raft level.

The present study investigates the effects of elastomeric type isolators on machine foundation against earthquake ground motion. The lead-rubber bearing (LRB) is the commonly used elastomeric type isolator. The LRB consists of rubber and thin steel plates in alternate layers, which are mounted at the top and bottom mounting plates as shown in Figure _{b}), damping of isolation system (_{b}), and normalized yield strength (_{0}) [

Schematic representation of lead rubber bearing (LRB); (a) actual model and (d) idealized

Three-dimensional FE model of framed-type machine foundation resting on a soil medium.

The selection of proper constitutive models to define material behavior is essential in numerical analysis. The RC frame is assumed to remain within the elastic limit during the transient vibration response under earthquake excitation and hence the material properties considered are Young’s modulus of concrete, _{s} = 30 GPa; mass density, _{c} = 2500 kg/m^{3}; Poisson’s ratio, _{c} = 0.15. Most of the past research on soil-structure interaction had used elastic material models to simulate the soil behavior. In this study, the elastic material model is used to simulate the soil behavior. The soil properties considered are shear wave velocity, _{s} = 1600 kg/m^{3}; Poisson’s ratio, _{s} = 0.40 [

The critical damping ratio of the superstructure is taken as 5%. The damping coefficient of 5% is considered for the base isolator. A constant value of 5% equivalent damping ratio is used for the foundation-soil system as per the recommendation of ACI 351.3R-04 (2004). Few researchers used this type of damping in dynamic soil-structure interaction analysis [_{0}, where _{0} is the predominant frequency of loading and

Twenty-node cubic solid continuum (C3D20) elements are used for modeling the TG foundation. Eight-node cubic solid continuum (C3D8) elements are used for modeling the soil medium. Connector element is used to model the base isolators. The column is considered as a cantilever beam (shape = square cross section; length = 10 m; width = 0.5 m) fixed at the base for the purpose of conducting mesh sensitivity test. A transverse load is applied at the top and transverse deflections are obtained from the FE analysis, and then compared with the theoretical values for different mesh sizes. The mesh size yielding results that match more closely with the theoretical prediction are selected for further analysis.

In the dynamic analysis of soil-structure interaction, the surrounding soil strata are considered as infinite in the horizontal direction. Therefore, it is important to avoid wave reflection at the vertical boundaries. Kelvin elements (spring-dashpot) are widely used to simulate the boundaries. The solution of the constants of Kelvin elements is developed by Novak and Mitwally [

In soil-structure interaction analysis, it is important to simulate the in-situ conditions before applying any dynamic loads in the model. In this model, loading is applied in two consecutive steps: geostatic step and dynamic loading step. In the geostatic step, the geostatic stress condition is simulated by applying gravity load to the system and a predefined stress field applied to the soil mesh. Then the program creates a force equilibrium system where the in-situ stresses are calculated. They are in equilibrium with the external forces under the prescribed boundary conditions and produce negligible deformations. Then the dynamic analysis of the TG foundation is performed in two steps, i.e., free-vibration analysis and forced-vibration analysis. In the free-vibration analysis, the natural frequencies and the mode shapes are extracted from the FE analysis. In the second step, the machine foundation is analyzed under various earthquake ground motions.

To ensure the validity of the FE model, the following validation studies are conducted.

The dynamic analysis of base-isolated single-story shear frame is carried out using FE software and the results are compared with the standard literature [

Comparison of deck acceleration for base-isolated structures.

In order to verify the boundary conditions and soil-structure interaction under dynamic loads, FE analysis has been carried out in two conditions: (i) for predicting the free-field motion of a ground considering the data reported in the literature [

Free-field response obtained from the FE analysis

Now, the validation has been carried out for the soil-pile system. In this case, the base of the soil-pile system has been given a horizontal sinusoidal excitation as studied by Fan et al. [_{u} (=│_{p}│/_{ff}, where _{p} is pile response at top, and _{ff} is the amplitude of free-field motion) and dimensionless frequency _{0} (= _{m}_{m} is the operating frequency, _{u} is close to the value of 1 for the low frequency region (i.e., 0 < _{0} < _{01}). However, when _{0} reaches a factor, _{01} (≈0.2–0.3), _{u} declines rapidly and then fluctuates around a value of about 0.4 after the factor, _{02} (≈5 to 10 times the natural frequency of the soil deposit for homogeneous soil profile). From Figure

Pile head response under a base excitation

The response quantities considered in the present study are the natural frequency (_{n}), peak displacement (lateral and torsional), and absolute acceleration at deck level. These response quantities are of importance because the natural frequencies are the key parameter to choose the structural arrangements of a suitable machine foundation. Acceleration and displacement at deck level are the direct measures to check the limiting amplitude of vibration in the machine foundation. And also, the deck acceleration developed in the superstructure is proportional to the force exerted due to earthquake ground motion. For efficient operation of the machine, the amplitude of vibration also needs to be within the prescribed permissible limits. Otherwise, the design becomes unacceptable. Therefore, to limit the response of machine foundation within the permissible limits, it is essential to understand the different parameters affecting the response of machine foundation against earthquake ground motion.

In general, the superstructure frequency ratio, _{s} (the ratio between the uncoupled torsional frequency to the uncoupled lateral frequency) for the framed structures will be in the range of 0.5 to 1.5. For the present study, the superstructure frequency ratio of the typical TG foundation considered is _{s} = 1.37 (i.e. torsional frequency of 4.48 Hz and lateral frequency of 3.26 Hz as given in Table _{r} = 0.05 and 0.15, are considered including the accidental eccentricity of 0.05 as specified by the UBC [_{r} = 0).

Natural frequency obtained from the FE analysis.

Case | Natural frequency in lateral mode (Hz) | Natural frequency in torsional mode (Hz) | |||||
---|---|---|---|---|---|---|---|

Without SSI | With SSI | % change | Without SSI | With SSI | % change | ||

Torsionally uncoupled | 3.26 | 2.30 | 29.40 | 4.48 | 3.31 | 26.20 | |

Torsionally coupled | _{r} = 0.05 | 3.11 | 2.21 | 28.91 | 4.32 | 3.35 | 22.53 |

_{r} = 0.15 | 2.75 | 2.00 | 27.52 | 4.05 | 3.24 | 19.93 |

In order to study the effect of torsional coupling on natural frequencies, TG foundations of different frequencies (by assuming different machine masses) are considered with different eccentricity ratios (_{r}). The total mass of the machine (machine mass and deck masses) is varied such that the superstructure frequency ratio comes in the range of 1.40, which is typically adopted in practice while designing the TG foundation. The variations in natural frequencies in the first mode (_{n1}) and the second mode (_{n2}) are shown in Figure _{n1}) and the second mode (_{n2}) are shown in Figure _{r} = 0.05) as observed by past studies [

Effect of torsional coupling on natural frequency.

The natural frequencies of the TG foundation duly considering the SSI obtained from the FE analysis are presented in Table

To estimate the SSI effects on torsionally coupled TG foundation under earthquake ground motions, the dynamic response parameters such as deck accelerations and displacements are obtained from the FE analysis. Three different earthquake ground motions are considered in the present study (Table _{r} = 0.05 and 0.15) along with the torsionally uncoupled condition are summarized in Table

Input earthquake ground motions.

Name of the earthquake | Peak acceleration ( |
---|---|

October 15, 1979 Imperial Valley, California (array # 5) | 0.36 |

January 17, 1994 Northridge, California (Sylmar Station) | 0.72 |

January 1, 1995 Kobe, KJM station | 0.83 |

Peak deck acceleration and displacement obtained from the FE analysis of the nonisolated machine foundation.

Case | Peak deck acceleration, | Peak deck displacement, | Peak deck displacement, _{x} (mm) | ||||
---|---|---|---|---|---|---|---|

Without SSI | With SSI | Without SSI | With SSI | Without SSI | With SSI | ||

Torsionally uncoupled | 1.19 | 3.31 | 28.26 | 138.54 | 0 | 0 | |

Torsionally coupled | _{r} = 0.05 | 1.36 | 3.21 | 35.63 | 142.69 | 1.60 | 5.31 |

_{r} = 0.15 | 1.63 | 2.80 | 54.06 | 141.92 | 6.15 | 16.25 | |

Torsionally uncoupled | 2.39 | 4.84 | 56.35 | 189.12 | 0 | 0 | |

Torsionally coupled | _{r} = 0.05 | 2.60 | 4.67 | 67.80 | 188.24 | 2.99 | 8.22 |

_{r} = 0.15 | 2.73 | 3.99 | 89.03 | 174.78 | 10.58 | 24.67 | |

Torsionally uncoupled | 2.04 | 5.41 | 48.58 | 220.71 | 0 | 0 | |

Torsionally coupled | _{r} = 0.05 | 2.50 | 5.40 | 64.65 | 236.42 | 2.96 | 8.82 |

_{r} = 0.15 | 3.25 | 4.32 | 106.30 | 263.25 | 12.47 | 20.75 |

Acceleration time-history response of torsionally uncoupled and coupled nonisolated framed-type machine foundation under the Imperial Valley, Array # 5, 1979 earthquake.

Acceleration time-history response of torsionally uncoupled and coupled nonisolated framed-type machine foundation under the Northridge, 1994 earthquake.

Acceleration time-history response of torsionally uncoupled and coupled nonisolated framed-type machine foundation under the Kobe, 1995 earthquake.

Displacement time-history response of torsionally uncoupled and coupled nonisolated framed-type machine foundation under the Imperial Valley, Array # 5, 1979 earthquake.

Displacement time-history response of torsionally uncoupled and coupled nonisolated framed-type machine foundation under the Northridge, 1994 earthquake

Displacement time-history response of torsionally uncoupled and coupled nonisolated framed-type machine foundation under the Kobe, 1995 earthquake.

Displacement time-history response of torsionally uncoupled and coupled nonisolated framed-type machine foundation under the Imperial Valley, Array # 5, 1979 earthquake.

Displacement time-history response of torsionally uncoupled and coupled nonisolated framed-type machine foundation under the Northridge, 1994 earthquake.

Displacement time-history response of torsionally uncoupled and coupled nonisolated framed-type machine foundation under the Kobe, 1995 earthquake.

A similar response is observed for the other two earthquake motions, as shown in Figures

In the present study, the beneficial effects of base isolators in machine foundations under the influence of torsional coupling are studied. Also, the SSI effects in base-isolated machine foundations are also investigated. The LRB type base isolator is considered for the present study and the input parameters considered are isolator time period, _{b} = 2.5 sec, isolator damping, _{b} = 0.05, and yield displacement, _{r} = 0.05 and 0.15) along with the torsionally uncoupled condition and the results are summarized in Table

Peak deck acceleration and displacement obtained from the FE analysis of base-isolated machine foundation.

Case | Peak deck acceleration, | Peak deck displacement, | |||
---|---|---|---|---|---|

Without SSI | With SSI | Without SSI | With SSI | ||

Torsionally uncoupled | 0.47 | 0.37 | 14.16 | 28.93 | |

Torsionally coupled | _{r} = 0.05 | 0.49 | 0.96 | 16.15 | 28.69 |

_{r} = 0.15 | 0.52 | 0.97 | 21.01 | 27.60 | |

Torsionally uncoupled | 0.66 | 1.73 | 19.92 | 49.73 | |

Torsionally coupled | _{r} = 0.05 | 0.70 | 1.73 | 22.85 | 49.08 |

_{r} = 0.15 | 0.72 | 1.74 | 27.83 | 47.47 | |

Torsionally uncoupled | 0.25 | 1.66 | 7.62 | 61.43 | |

Torsionally coupled | _{r} = 0.05 | 0.25 | 1.66 | 8.13 | 61.58 |

_{r} = 0.15 | 0.24 | 1.65 | 9.26 | 61.86 |

Acceleration time-history response of torsionally uncoupled and coupled base-isolated framed-type machine foundation under the Imperial Valley, Array # 5, 1979 earthquake.

Displacement time-history response of torsionally uncoupled and coupled base-isolated framed-type machine foundation under the Kobe, 1995, earthquake.

Similar behavior is observed for the other earthquakes (1994 Northridge and 1995 Kobe earthquake motion) as shown in Figures

Acceleration time-history response of torsionally uncoupled and coupled base-isolated framed-type machine foundation under the Northridge, 1994 earthquake.

Acceleration time-history response of torsionally uncoupled and coupled base-isolated framed-type machine foundation under the Kobe, 1995, earthquake.

Displacement time-history response of torsionally uncoupled and coupled base-isolated framed-type machine foundation under the Imperial Valley, Array # 5, 1979, earthquake.

Displacement time-history response of torsionally uncoupled and coupled base-isolated framed-type machine foundation under the Northridge, 1994, earthquake.

FE analysis is carried out to evaluate the dynamic response of a typical base-isolated TG foundation subjected to different earthquake ground motions. The effects of torsional coupling in machine foundation are also studied and the results are compared with the torsionally uncoupled case. Soft soil conditions are considered to analyze the effects of SSI and the results are compared with the fixed-base condition. From the results of the present study, the following conclusions are drawn:

The SSI effects decrease the natural frequency of the entire structure-foundation-soil system, which is significant in higher modes, especially for TG foundation resting on soft soil strata. Also, the dynamic response of such machine foundations is greatly affected by the presence of superstructure eccentricities. The natural frequencies in lower modes are further reduced by the superstructure eccentricities.

The SSI effects increase the deck acceleration and lateral displacement in TG foundation resting on soft soil strata. In addition, the deck acceleration and lateral displacement are also increased in such machine foundations when the superstructure eccentricities are considered. Due to the excitation in torsional modes, the horizontal displacement in the other direction is also increased significantly with an increase in eccentricity ratio.

Since the TG foundations are rigid as compared to the conventional low-rise buildings, the forces exerted on the superstructure are severe under earthquake ground motions. Hence, the base isolators are beneficial in TG foundations, by which the superstructure accelerations are greatly reduced. In addition, the relative deck displacements are also greatly reduced with the help of base isolators. Hence, base isolators are beneficial to protect sensitive equipment from damaging earthquake events.

Though the deck accelerations are significantly reduced by using the base isolators, in the case of the soft soil, the involvement of higher mode participation amplifies the seismic response. The amplification is found to increase further when the effects of torsional coupling and SSI are combined. Hence, neglecting the effects of SSI and torsional coupling in the dynamic response of base-isolated TG foundations may lead to poor performance of machine foundations.

In this research, the effects of soil-structure interaction on the dynamic response of machine foundations are investigated by considering elastic soil parameters. However, it would be interesting to study the dynamic response by considering nonlinear soil behavior, especially under earthquake ground motions.

Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.

The authors declare that they have no conflicts of interest.