OPTIMAL DESIGN OF MACHINERY SHALLOW FOUNDATIONS WITH CLAY SOILS

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 91-103, © IAEME
91
OPTIMAL DESIGN OF MACHINERY SHALLOW FOUNDATIONS WITH
CLAY SOILS
Galal A. Hassaan
Emeritus Professor, Mechanical Design & Production Department, Faculty of Engineering, Cairo
University, Giza, Egypt
ABSTRACT
Optimal design of shallow foundations used in supporting various machinery is of vital
importance. Badly designed foundations result in serious problems such as soil failure, large soil
settlement, low vibration efficiency and resonance of structures near by the foundation.
An optimal design approach based on using MATLAB optimization tool box is used to
provide the optimal design of a machinery foundation. To minimize the foundation cost the
foundation mass is used as an objective function. To control the high technical quality of the
foundation from soil mechanics and vibrations point of view, 10 functional constraints are used.
The soil isolation efficiency of the soil is responsible about the vibration transmitted to the
surrounding. Therefore, the effect of the minimum isolation efficiency in the range of 80 – 96 % on
the foundation optimal design is investigated in details. The performance of the foundation against
the minimum isolation efficiency is outlined so that the civil engineer can compromise between cost
and performance.
Keywords: Shallow Foundations – Optimal Design – Foundation _ Soil Interaction – Clay Soils.
1. INTRODUCTION
Machinery foundations have to be precisely designed to avoid the side effects of the dynamic
loads encountered with all the machinery. The difficulty in this respect is due to the interaction
between the machine, foundation and soil. The reason for this is the great variability in soil type ,
properties and water content.
Design techniques range from following local national codes to optimal approaches. There is
a little work done in behalf of optimal design of machinery foundations and this research work
comes to fill this gap and introduces an approach for the optimal design of machinery foundations
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING
AND TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 5, Issue 3, March (2014), pp. 91-103
© IAEME: www.iaeme.com/ijmet.asp
Journal Impact Factor (2014): 7.5377 (Calculated by GISI)
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IJMET
© I A E M E

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taking into consideration the cost (minimum mass objective function) and performance during
operation (through using nine functional constraints). Most of research activities are focused on soil
properties required for the foundation design and the foundation-soil interface models.
Gazetas (1983) reviewed the state of art of analyzing the dynamic response of foundations
subjected to machine-type loading. He presented his results in the form of formulae and
dimensionless graphs [1]. Pecker (1996) reviewed the evaluation of seismic bearing capacity of
shallow foundations resting on cohesive and dry cohesionless soils [2]. Adel et al. (2000) applied the
expert system approach to foundation design providing the user with means to assess the effect of
various foundation alternatives and ground conditions [3]. Pender (2000) explained a model for the
cyclic stress-strain behavior of cohesive soil and the application of the model in estimating the
dynamic compliance of footing under vertical cyclic loading [4]. Tripathy et al. (2002) studied the
void ratio and water content of specimens at several intermediate stages during swelling. They
studied the void ratio and water content characteristics of the soil during swelling and shrinkage [5].
Anteneh (2003) presented the foundation performance requirements and the basic steps employed in
the design of a machine foundation. He reviewed all the soil parameters required for the foundation
design and the basic concepts in foundation vibrations [6]. Ostadan et al (2004) discussed a series of
parametric studies for structure-soil interaction damping and presented an effective approach to
estimate the system damping for such systems [7]. Park and Hashash (2004) proposed formulations
used in nonlinear site response analysis which showed that the equivalent linear frequency domain
solution used to approximate nonlinear site response underestimated the surface ground motion
within a period range relevant to engineering applications [8].
Dewookar and Huzjak (2005) studied the effect of the effective normal stress, liquid limit and
plasticity index on the soil friction angle. They viewed the available models relating soil friction
angle and plasticity index [9]. Prakash and Puri (2006) discussed the analysis methods used in
determining the foundation response when subjected to vibrating loads. They considered the machine
foundation as a mass-spring-damper model with one or two degree of freedom [10]. Ahmed et. al.
(2006) studied the nature, structural features, engineering behavior and properties of soil samples
extracted from Penana Island. The studied the effect of the soil depth on particle size, moisture
content, liquid limit, plastic limit, specific gravity, cohesion and friction angle [11]. Anggraini (2006)
evaluated the increase of shear strength of fibrous peat due to the application of consolidation
pressure between 50 and 200 kPa [12]. Steinhausen (2007) tried to solve the problem of installing
large and heavy reciprocating machines implying high dynamic loads at the substructure of offshore
systems [13]. Chandrakaran et al. (2007) presented a simplified design procedure for the design of
machine foundations subjected to vertical dynamic loading using the half-space theory. They
presented characteristic curves that can be used by the practicing engineers to reduce the number of
trials in the design process and to arrive at the optimal design [14]. Simoes et al (2007) showed how
to predict in a simple way the effect of vibration due to machine operation on its surroundings
considering the foundation as a vibration source on the surface of an elastic medium [15].
Chakravarti (2008) studied the design of rectangular block foundation for vibrating machines
considering both the effect of damping and coupled modes using time-history analysis [16]. Bhatia
(2008) highlighted the need for a better interaction between foundation designers and machine
manufacturers to ensure improved machine performance. He discussed the design methodology for
foundation design and the vibration isolation system for heavy-duty machines [17]. Gunduz (2008)
studied the time dependant settlements calculated with numerical methods and subsequent
comparisons with field measurements [18]. Michaels (2008) proposed an alternative to the traditional
representation of damping. He considered that the imaginary part of the shear modulus will vary
directly with frequency which is useful in cases where the water table may change affecting the
foundation design [19]. Karray and Lefebvre (2008) studied the effect of Poisson’s ratio on Rayleigh
wave phase velocities leading to more accurate soil characterization [20].

93
Jain et. al. (2010) used an artificial neural network technique to predict the shear strength
parameters of medium compressibility soil. They studied the variation of soil cohesion and internal
friction angle with soil dry density and degree of saturation [21]. Kenarangi and Rofooein (2010)
investigated the performance of tuned mass dampers in reducing the nonlinear response of irregular
buildings under bi-directional horizontal ground motions considering soil-structure interaction[22].
Puri and Rao (2010) evaluated the modulus of elasticity and the resilient modulus of subgrade clayey
sand soils by laboratory testing. They used the California Bearing Ratio Test and the Unconfined
Cyclic Triaxial Test which are useful in studying soil settlement and the mechanistic pavement
design [23]. Davidovic et al. (2010) compared the settlements calculated using different settlements
prediction techniques and those of experiments in order to test techniques accuracy and reliability
[24]. Olsson (2010) focused on how to calculate long-term settlement in soft clays and discussed
ways of evaluating some of the important parameters used [25]. Romerco et al. (2010) presented
motion formation to study the coupled response of rigid foundations modeled by springs and
dampers. Their use of matrix manipulation has led to an expression providing closer approximation
to the real phenomenon [26].
Aziz and Ma (2011) focused on the design and analysis of bridge foundation using four
codes. They found which code is better for design and control of the high settlement problem due to
loading [27]. Moghaddasi et al. (2011) investigated the effects of soil – foundation – structure
interaction on seismic response of structures through a robust Monte Carlo simulation using a wide
range of soil – foundation systems and earthquake input motions in time history analysis [28].
Clayton (2011) studied the factors affecting the stiffness of soils and weak rocks. He described the
impact of a number of stiffness parameters on the displacements around a retaining structure [29].
Kim et al. (2011) used an electrical resistivity cone probe to determine the electrical resistivity and
void ratio of seashore soft soils. They concluded that their technique may be an effective one for the
estimation of the offshore soft soil void ratio [30].
Kaptan (2012) proposed an empirical formulation for the rapid determination of the allowable
bearing pressure of shallow foundations in soils and rocks. The proposed expression was consistent
with the results of the classical theory and proved to be rapid and reliable [31]. Miyata and Bathurst
(2012) used the results of load measurements and that reported in the literature to examine the
prediction accuracy of the Coherent Gravity Method [32]. Palacios et al. (2012) studied the
geotechnical characteristics of foundation design under seismic conditions in towns in the Granada
province of Spain. They recommended using deep foundations [33]. Tafreshi and Dawson (2012)
presented the results of laboratory-model tests on strip footings supported on unreinforced and
geocell-reinforced sand beds under combination of static and dynamic loads. They studied the
influence of various parameters including the amplitude of the dynamic load [34]. Zamani and El-
Shamy (2012) presented a 3-dimensional micro-scale framework utilizing the discrete element
method to analyze the seismic response of soil-foundation-structure systems. They examined the
impact of structure and damping, foundation embedment, exciting amplitude and frequency on the
system response [35]. Hassaan, Lashin and Al-Gamil (2012) collected a large number of soil and
foundation parameters available in the literature and molded them in the form of mathematical
models suitable for the computer-aided analysis and design of foundations. The models used had
high multiple correlation coefficient for accurate representation of the data [36].
Pan, Tsai and Lin (2013) presented the ultimate bearing capacity of shallow foundations.
They utilized the weighted genetic programming and soft computing polynomials for the accurate
prediction and visible formulas for the ultimate bearing capacity [37]. Karamitros, Bouckovalos and
Chaloulos (2013) presented a simplified analytical methodology for the computation of the seismic
settlements of stripped rectangle foundations resting on liquefiable soil with clay crust [38]. Nagah
and Nwankwoala (2013) recommended appropriate foundation design and construction for large-
scale ground and elevated water storage in the Onne area, Rivers State, Nigeria [39].

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Shahnazari, Shahin and Tutunchran (2014) utilized the evolutionary polynomial regression, classical
genetic programming and gene expression programming for the accurate prediction of shallow
foundation settlements on cohesionless soils [40]. Cunha and Albuquerque (2014) summarized the
historical development of foundation engineering in Brazil since its early beginning illustrating how
foundation engineering technologies were improved and approached [41]. Fellenius (2014) studied in
details aspects of bearing capacity of shallow foundations considering inclined and eccentric loading,
inclination and shape factors, overturning and sliding [42].
2. ANALYSIS
Objective function:
The foundation mass, Mf is used as the objective function of the optimization problem to
reduce its cost. It is related to the foundation length L, width B and height Hf through:
Mf = ρ LBHf (1)
Where ρ is the foundation density.
Functional constraints:
(i) Length / width ratio constraint, C1:
C1 = B / L ( ≤ 1) (2)
This constraint means that the foundation has a rectangular or square shape.
(ii) Hight / Width ratio constraint, C2:
C2 = Hf / B ( ≤ 2 ) (3)
This constraint may be less than 1 for shallow foundations or the limit may go up to 4 [43].
(iii) Soil working stress constraint, C3:
C3 = Mt g / (BL) ( ≤ Qall ) (4)
where: Mt = total foundation-machine mass = Mf + Mm
Mm = machine mass.
Qall = allowable bearing capacity of the soil = Qu / FS
Qu = ultimate bearing capacity of the soil.
FS = design factor of safety
The soil ultimate bearing capacity Qu is related to the soil properties parameters through [44]:
Qu = [(c/tanφ) + 0.5γB√Kp] [Kp eπtanφ
– 1] (5)
Where:
c = soil cohesion
φ = soil internal friction angle
Kp = Rankin's coefficient = (1 + sinφ)/(1-sinφ)

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(iv) Soil elastic settlement constraint, C4:
The soil settlement is the resultant of 3 types of settlements: elastic, primary consolidation
and secondary consolidation. Here, we are going to consider only the elastic type with setting the
upper bound the lowest range of the allowable settlement which is 25 mm [45] .
The elastic settlement as given by Meyerhof depends on the foundation width B as follows
[45]:
C4 = 2C3/N60 for B ≤ 1.22 m (6)
And
C4 = (2C3/N60) [B/(B+0.3)] for B > 1.22 m (7)
Where N60 = standard penetration number.
N60 is function of the soil internal friction angle φ. Carter and Bentley gave this relation in a
graphical form [46]. The following polynomial model if fitted by the authors to the graphical data
with 0.9998 correlation coefficient:
N60 = 42.422122955322 – 4.69926404953φ + 0.120638884604φ2
Where φ is in degrees.
(v) Maximum vibration amplitude in lateral direction, C5:
According to the work of Prakash and Puri, the foundation vibration in the horizontal and vertical
directions can be considered as uncoupled and defined by a SDOF dynamic model [10]. The
maximum vibration amplitude of a SDOF system excited by a rotating unbalance is function of the
specific unbalance me/Mt and the system damping ratio. That is:
C5 = (me/Mt) / [2ζx √(1-ζx
2
)] (8)
Where me is the machine unbalance, ζx is the damping ratio of the soil in the lateral direction.
According to Gazetas, the damping ratio in the lateral direction ζx is given by [1]:
ζx = 0.29/√ Bx (9)
where Bx is the mass ratio in the lateral direction:
Bx = M(2 – ν)/(8ρR3
)
ν = soil Poisson’s ratio
M = foundation mass
R = equivalent radius of the equivalent circular foundation
= √ (BL/)
(vi) Maximum vibration amplitude in vertical direction, C6:
The maximum vibration amplitude of a SDOF system excited by a rotating unbalance is function of
the specific unbalance me/Mt and the system damping ratio in the z-direction ζz. That is:
C6 = (me/Mt) / [2ζz √(1-ζz
2
)] (10)

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Where according to Gazetas [1]:
ζz = 0.425/√ Bz (11)
Bz is the mass ratio in the vertical direction:
Bz = M(1 – ν)/(4ρR3
)
(vii) Soil isolation efficiency in the lateral direction, C7:
The isolation efficiency in the lateral direction is defined as:
C7 = 100(1-TRx) (12)
where TRx is the vibration transmissibility in the lateral direction given by:
1 + (2ζxrx)2
0.5
TRx = ------------------------ (13)
(1 – rx
2
)2
+ (2ζxrx)2
rx is the frequency ratio in the lateral direction:
rx = ω/ωnx (14)
where ω is the angular exciting frequency of the forced vibrations of the foundation, and ωnx is the
lateral natural frequency of the foundation-soil dynamic system given by [10]:
ωnx = √ (kx / Mt) rad/s (15)
kx is the soil stiffness in the lateral direction given according to Gazetas by [1]:
kx = 8GRCx(L/B) / (2-ν) (16)
where:
G = soil modulus of rigidity
Cx is a correction factor in the lateral direction depending only on the foundation ratio L/B as
indicated by Barken [47]. The authors fitted the following third order polynomial to Barken’s data
with an 0.99977 correlation coefficient:
Cx=1.026129841805-0.04249420017(L/B)+ 0.011028882116(L/B)2
- 0.000512405066(L/B)3
(17)
(viii) Soil isolation efficiency in the vertical direction, C8:
The isolation efficiency in the vertical direction is defined as:
C8 = 100(1-TRz) (18)
where TRz is the vibration transmissibility in the vertical direction given by:

97
1 + (2ζzrz)2
0.5
TRx = ------------------------ (19)
(1 – rz
2
)2
+ (2ζzrz)2
Rz is the frequency ratio in the lateral direction:
Rz = ω/ωnz (20)
ωnz is the vertical natural frequency of the foundation-soil dynamic system given by [10]:
ωnz = √ (kz / Mt) rad/s (21)
kz is the soil stiffness in the vertical direction given according to Gazetas by [1]:
kz = 4GRCz(L/B) / (1-ν) (22)
Cz is a correction factor in the vertical direction. The authors fitted the following fourth order
polynomial to Barken’s data with an 0.9998 correlation coefficient:
Cz=0.968339145184-0.044979844242(L/B)+ 0.033221840858(L/B)2
- 0.004676759709(L/B)3
+
0.000208628044(L/B)4
(24)
(ix) Vibration velocity in the lateral direction, C9:
It is recommended by the Building Department of the Government of Hong Kong that the
vibration velocity in lateral or vertical directions not to exceed 15 mm/s [48]. Therefore, this and the
next constraint is set on vibration velocity of the foundation.
The vibration velocity in the lateral direction for the rotating unbalance excited vibrations is
given by:
C9 = (ωmerx
2
/Mt) / √ {(1 – rx
2
)2
+ (2ζxrx)2
} (25)
(x) Vibration velocity in the vertical direction, C10:
The vibration velocity in the vertical direction for the rotating unbalance excited vibrations is
given by:
C10 = (ωmerz
2
/Mt) / √ {(1 – rz
2
)2
+ (2ζzrz)2
} (26)
3. CONSTRAINTS LIMITS3
The optimization problem in hand is a constrained one on both foundation dimensions and
functional constraints. The upper and lower limits used in this optimal design problem are as follows:
(i) Foundation dimensions limits (depend on machine dimensions for L and B):
2 ≤ L ≤ 10 m
1 ≤ B ≤ 5 m (27)
0.2 ≤ Hf ≤ 2 m

98
(ii) Functional constraints limits:
1 ≤ C1 ≤ 10
0 ≤ C2 ≤ 2
0 ≤ C3 ≤ allowable bearing capacity
0 ≤ C4 ≤ 25 mm
0 ≤ C5 ≤ Allowable vibration amplitude mm
0 ≤ C6 ≤ Allowable vibration amplitude mm
Minimum isolation efficiency ≤ C7 ≤ 100 %
Minimum isolation efficiency ≤ C8 ≤ 100 %
0 ≤ C9 ≤ 15 mm/s
0 ≤ C10 ≤ 15 mm/s
4. ALLOWABLE VIBRATION AMPLITUDES
The limit of the peak vibration amplitude for machinery foundations is function of the
vibration frequency [44]. The vibration limit – frequency is presented in a graphical form. To suit
computer-aided application of the foundation design, this relation is defined by a power model in the
form [36]:
A = 112.429489135742 f(-1.969963312149)
mm
Where f is the vibration frequency in Hz.
5. MINIMUM ISOLATION EFFICIENCY
The isolation efficiency is a key factor in controlling the vibrations transmitted to
surrounding during machine operation. This is a famous known problem facing contracting
companies. To help in solving this problem, the lower limit of the isolation efficiency constraint is
left adjustable to examine the effect of its level of the foundation design.
The following limits are used for the minimum isolation efficiency: 80, 82, 84, 86, 88, 90, 92, 94 and
96 %.
6. OPTIMAL FOUNDATION DESIGN
The objective function given by Eq.1 has to be minimized subject to 10 functional constraints
given by Eqs.2, 3, 4, (6 and 7), 8, 10, 12, 18, 25 and 26, and the design variables constraints in
Eq.27. The MATLAB optimization toolbox is used to perform this task [49, 50].
7. CASE STUDY
A 150 kW motor-centrifugal pump unit has the parameters:
- Speed: 1800 rev/min.
- Total mass: 500 kg
- Rotor mass: 300 kg
- Overall length: 2 m
- Overall width: 0.60 m
- Residual unbalance: 12 kgmm

99
The clay soil has the properties:
- Plasticity index: 30 %
- Poisson’s ratio: 0.2
- Water content: 20 %
- Cohesion: 29.03 kN/m2
- Unit weight: 18 kN/m2
- Friction angle: 29 degrees
- Shear modulus: 0.517 MN/m2
- Shear strength: 23.3 kN/m2
Foundation density: 2400 kg/m3
Design factor of safety: 3
Requirements: The foundation dimensions supporting the motor-pump unit for the stated clay soil
properties.
Minimum foundation dimensions: The machine dimensions set the minimum length and width of
the foundation at 2 and 1 m respectively.
Optimization results: The optimization results is presented graphically against the minimum
isolation efficiency:
- Foundation dimensions and mass: Figs.1 and 2.
Fig.1: Optimal foundation dimensions Fig.2: Optimal foundation mass
- Soil stress at foundation interface and elastic settlement: Figs.3 and 4.
Fig.3: Optimal soil stress at foundation Fig.4: Optimal soil elastic settlement
Interface

100
- Vibration maximum peak amplitudes and isolation efficiencies: Figs.5 and 6.
Fig.5: Optimal vibration peak amplitude Fig.6: Optimal vibration isolation
of the foundation Efficiencies
- Vibration peak velocities: Fig.7.
Fig.7: Optimal vibration peak velocities
8. DISCUSSIONS
- Optimization is a powerful technique which leads to successful machinery foundation design
fulfilling all the required objectives in case of machinery foundations subjected dynamic loads.
- All the design constraints are simultaneously considered without any trial work.
- Recommendations on soil bearing capacity, foundation relative dimensions, foundation
maximum vibrations and isolation efficiency are all considered.
- Isolation efficiency is a very important parameter in foundation design since it controls the
vibration and noise induced vibrations of surrounding structures.
- 10 functional constraints are considered increasing the level of the foundation performance and
effectiveness associated with a specific dynamic machine.
- The minimum isolation efficiency is used to direct the design giving the structural engineer a
chance to compromise between cost and performance.
- The isolation efficiency range considered is from 80 % to 96 % (lower limit).
- The foundation mass ranges is from 3.12 to 20.96 ton.
- The foundation height/width ratio ranges from 0.65 to 2.

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- The vertical soil stress at the foundation interface ranges from 17.76 to 71.21 kN/m2
.
- The elastic settlement ranges from 4.67 to 19.42 mm.
- The maximum peak vibration amplitude in the lateral direction ranges from 1.46 to 4.99 µm.
- The maximum peak vibration amplitude in the vertical direction ranges from 0.96 to 3.60 µm.
- The isolation efficiency in the lateral direction ranges from 87.56 to 97.52 %.
- The isolation efficiency in the vertical direction ranges from 80 to 96 %.
- The peak vibration velocity in the lateral direction ranges from 0.106 to 0.637 mm/s.
- The peak vibration velocity in the vertical direction ranges from 0.105 to 0.632 mm/s.
- All the functional constraints were within the pre-assigned limits.
- When trying to increase the minimum limit of the isolation efficiency than 0.96, non-consistent
values for the foundation dimensions were obtained.
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