Introduction:
Design of machine foundations come under the topic of soil dynamics in geotechnical engineering. Dynamic forces in soil could originate from sources such as:
- Earthquakes
- Blasts engineered by man
- Pile driving
- Landing of aircraft,
- Action of wind and running water
- Operation of heavy machines etc
Generally, soil dynamics could be applied in the following fields:
- Vibration and settlement of structure, and of foundations of machinery
- Densification of soil by dynamic compaction and vibration
- Penetration of piles and sheet piles by vibration or impact
- Dynamic and geophysical methods of exploration
- Effects of blasting on soils and rock materials
- Effects of earthquakes and earthquake-resistant design of foundations
Machine Foundations
Machines generate dynamic forces and these dynamic forces subject soil on which they are located to dynamic conditions and consequently alter their behaviour. Thus, the behaviour of soil under static load is different from its behaviour under dynamic loads. This variation is as a result of the characteristics of soil under dynamic loading. ‘‘Vibrations caused by dynamic loading impact energy to the soil particles. The soil grains slip into and fill up corresponding void spaces (densification of soil), pore water tends to escape, the modulus of elasticity tends to change, and so does its bearing capacity. The shock tends to reduce the internal friction and adhesion considerably. Loose granular soils may be densified by vibration, while it will have relatively smaller effect on cohesive soils. saturated fine sand or silt may undergo a phenomenon of ‘liquefaction’ as they tend to become ‘quick’ under the action of dynamic forces under certain conditions’’. All these changes in the soil natural state necessitate special considerations in the design of such foundations.
Operation of Machines
Forces from the operation of machines create different kinds of motions in soil such as rectilinear, curvilinear, steady. transient, periodic, aperiodic etc. The changes impacted to the soil by these dynamic forces alter the soil’s structure, internal friction and adhesion thus, affection its performance as a foundation. As noted earlier, there are different areas where design of foundation should consider dynamic forces. In this post I would focus on the design of machine foundations.
Classification of Machine Foundations
Machine foundations can be classified based on dynamic effects and structural forms:
Based on dynamic effects we have;
- Those producing periodical forces e;g compressor
- Those producing impact forces e.g forge hammers and presses
- High speed machines such as turbines and rotary compressors
Based on structural form we have;
- I-Block type foundations
- II-Box or caisson type foundations
- III-Wall type foundations
- IV-Framed-type foundations
Design of Machine Foundations
Machine foundations require special care for the analysis and design of such because of the dynamic nature of the forces acting on it. The method adopted for the design of machine foundation depends on the type of machine. This requires a significant synergy between the machine designer, the mechanical engineer and the foundation designer, the geotechnical engineer from planning stage till installation and commissioning. Accurate design of machine foundation ensures satisfactory performance of the machine.
Criteria for the Design of Machine Foundations
The criteria for the design of machine foundations varies according to type of machine by the criteria laid down here apply to all types of machines. For special criteria peculiar to each machine, consult specialist textbooks.
1. Like ordinary foundation, it should be safe against shear failure caused by superimposed loads and also the settlement should be within safe limits. The soil pressure should normally not exceed 80% of the allowable pressure for static loading.
2. There should be no possibility of resonance. The natural frequency of the foundation should be either greater or smaller than the operating frequency of the machine. Values less than 0.5 times or more than 1.5 times is normally accepted.
3. The amplitude under service conditions should be within the permissible limits for the machine. This is generally prescribed by the manufacturer.
4. The combined centre of gravity of the machine and the foundation should be on the vertical line passing through the centre of gravity of the base plane.
5. Machine foundation should be taken to a level lower than the level of the foundation of the adjacent buildings and should be properly separated.
6. The vibrations induced should neither be annoying to the persons nor detrimental to other structures. Richard (1962) developed a plot for vertical vibration, which is generally taken as a guide for various limits of frequency and amplitude.
7. The depth of the groundwater table should be at least ¼ of the width of the foundation below the base plane since groundwater is a good conductor of waves. This limits the propagation of vibration.
8. Machine foundations should be separated from adjacent buildings components by means of expansion joints.
9. Any pipes carrying hot fluids, if embedded in the foundation must be properly isolated.
10. The foundation should be protected from machine oil by means of suitable chemical treatment, which is acid-resistant.
11. The bearing capacity under dynamic loading conditions is generally considered to be less than for static loading, the reduction factor ranging from 0.25 to 1.0.
12. The settlement should be within permissible limits.
13. All rotating and reciprocating parts of the machine should be so balanced that he unbalanced forces and moments are minimized.
14. The foundation should be so planned so as to permit subsequent alteration of natural frequency by changing the base area or mass of the foundation, if found necessary.
Design Approach for Machine Foundations
1. The dimensions of the machine foundations are fixed according to the operational requirements of the machine. The overall dimensions of the foundation are generally specified by the manufacturers of the machine. If there is choice to the foundation designer, the minimum possible dimensions satisfying the design criteria should be chosen.
2. Once the dimensions of the foundation are decided upon, and site conditions are known, the natural frequency of the foundation-soil system and the amplitudes of motion under operating conditions have to be determined.
3. The design procedure is usually by ‘trial and error’.
4. The specific data required for design vary for different types of machines. However, certain general requirements of the data may be given as follows:
– loading diagram, showing the magnitude and positions of static and dynamic loads exerted by the machine.
– power and operating speed of the machine.
– line diagram showing openings, grooves for foundation bolts, details of embedded parts, and so on.
– nature of soil and its static and dynamic properties, and the soil parameters required for the design.
Vibration Analysis of Machine Foundation
The problem of machine foundation is due to the effect of vibration. Hence, the design of machine foundation cannot proceed without vibration analysis. The following fundamentals of vibration requires special consideration.
Fundamentals of Vibration
- Degree of freedom
- Modes of vibration
- Free vibrations and forced vibrations
- Resonance
- Damping including negative damping
- Free vibrations without damping
- Forced vibrations without damping
- Free vibrations with damping
- Forced vibrations with damping
Procedure for Vibration Analysis of Machine Foundations
Vibration analysis of machine foundation is done to obtain information for the design of the foundation. The information which include the mass (m), the damping constant (c) and the stiffness constant (k) are employed in the determination of the natural frequency of the system which are in turn employed in the structural design of the system.
Reasons for the analysis
A. Mass (m): When a machine vibrates, some portion of the supporting soil mass also vibrates. The vibrating soil mass is known as the participating soil mass or in-phase soil mass. Thus, the total mass of the system, ‘m’ is equal to the mass of the foundation block and machine (mf) and the mass of the participating soil (ms),
Mathematically, m = mf + ms
The total mass ‘m’ usually varies between mf and 2mf while ms varies between 0 and mf
B. Damping constant (c): Damping constant is due to dissipation of vibration energy, which occurs mainly because of the following reasons:
- Internal friction loss due to hysteresis and viscous effects
- Radiational loss due to propagation of waves through soil
The damping factor D for an undamped system can be determined in the laboratory. Vibration response is plotted and the logarithmic decrement ‘δ’ is found from the plot as,
δ = log (z2/z1) (1)
Where, z1 and z2 are amplitudes of two consecutive cycles of an amplitude-response curve. The damping factor, D and the logarithmic decrement ‘δ’ are related as,
δ = 2πD/ √ (1 – D2) (2)
D is approximately equal to δ/2π
Or
The damping factor may also be obtained from the area of the hysteresis loop of the load displacement curve as,
D = ΔW/W (3)
Where, W = total work done, ΔW = work lost in hysteresis
Note: D for most soils varies between 0.01 and 0.1.
C. Spring constant (k): The spring stiffness depends upon the type of soil, embedment of the foundation block, the contact area and the contact pressure distribution. The common methods to determine ‘k’ are outlined below:
a. Laboratory test: a trial test with vertical vibrations is conducted to determine Young’s modulus, E. Alternatively, the modulus of rigidity (G) is determined by conducting the test under torsional vibration, and E is obtained indirectly from the relation,
E = 2 G (1 + μ) (4)
Where, μ is the Poisson’s ratio.
Thus, k = AE/L (5)
Where, A = cross sectional area of the specimen, L = length of the specimen
b. Barkan’s method: The stiffness can also be obtained from the value of ‘E’ using the following relation given by Barkan.
k = [1.13 E/ (1 – μ2)] √A (6)
Where, A = the base area of the machine i.e. the contact area.
c. Plate load test: A repeated plate load test is conducted and the stiffness of the soil in the test (kp) is found as the slope of the load-decrement curve. The spring constant ‘k’ of the foundation is determined as under;
For cohesive soils,
k = kp (B/Bp) (7)
For cohesionless soils,
k = kp [(B + 0.3)/ (Bp + 0.3)]2 (8)
Where, B is the width of the foundation and Bp is the diameter of the plate.
Alternatively, the spring constant can be obtained from the subgrade modulus (ks), as k = ks . A, where, A = area of foundation.
d. Resonance test: The resonance frequency (fn) is obtained using a vibrator of mass ‘m’ set up on a steel plate supported on the ground. The spring stiffness is obtained from the relation.
fn = ωn/2π (9)
Or
fn = (1/2π) √ (k/m) (10)
Or
k = 4 π2 fn m (11)
How to Determine Natural Frequency of Foundation Soil System
The natural frequency of foundation-soil system can be determined using the theory of vibrations
The equation of motion neglecting damping is
m(d2z/dt2) + kz = Fosinωt (12)
Where, m = mass of the machine foundation and the participation soil; k = equivalent spring constant of the soil
The natural frequency of the system is given by
ωn = √(k/m) (13)
Where, ωn is in radians per second
Also, fn = (1/2π) √(k/m) (14)
Where, fn is in cycles per second
Thus,
fn = (1/2π) √(k/ (mf + mc)) (15)
Where, mf = mass of machine and foundation and ms = mass of the participating soil mass
Parken (1962) gave the following expression for the natural frequency,
ωn = √ CuA/m (16)
Where, Cu = coefficient of elastic uniform compression; A = contact area of foundation with the soil.
Comparing Eqn (16) with Eqn (13),
k = Cu x A (17)
The maximum amplitude is given by,
zmax = Fo/ m (1 – r2) (18)
Where, Fo is the exciting force
The coefficient of elastic uniform compression (Cu) depends upon the type of soil. It can be obtained from the following relation.
Cu = 1.13 [E/1 – μ2) . 1/√A (19)
As it is evident, the coefficient of elastic compression varies inversely proportional to the square root of the base area of the foundation. Thus,
(Cu)2/ (Cu)1 = (A1/ A2)1/2 (20)
Table 1 below gives the recommended values of Cu for A = 10 m2 for different soils (Barkan, 1962).
How to Isolate Vibrations from Machine Foundations
Vibrations can cause harmful effects on the adjoining structures and machines. Besides, these vibrations cause annoyance to the persons working in the area around the machine. However, if the frequency ratio is kept outside the critical range of 0.4 and 1.50, and the amplitude is within permissible limits, the harmful effects are considerably reduced, especially if the system is damped. Transmission of vibrations can be controlled and the detrimental effects considerably reduced by isolating either the source (active isolation) or by protecting the receiver (passive isolation). The following measures are generally adopted:
A. Using heavy mass of foundations: These are often provided by the manufacturers of the foundations. See Table below as provided by Couzens (1938) that show the ratios of foundation mass to engine mass suitable for various types of machines. The ratios are used for rough estimate.
B. Geometrical isolation: Locating machine foundation away from the adjoining structures. This is because the amplitude of surface waves (R-waves) reduces with an increase in distance. A considerable reduction in the amplitude is achieved by locating the foundation at great depth as the R-waves also reduces with an increase in depth.
C. Use of additional masses called dampers which are attached to the foundations of high frequency machines to make it a multiple degree freedom system and to change the natural frequency.
D. Placing absorbers such as rubber mountings, felts and corks between the machine and the base.
E. Attaching an auxiliary mass with a spring to the machine foundation to change the system to two-degree freedom system. The method is usually effective for system in resonance.
F. The natural frequency of the system can be increased by increasing the strength of soil by chemical or cement stabilization. This method is effective for machines of low operating frequency.
G. Modifying the natural frequency of the system by making structural changes to the foundation such as connecting the adjoining foundations, changing the base area or mass of foundation or use of attached slabs.
H. Reduction of the propagation of waves by the use of sheet piles, screens or trenches.
Construction principles of machine foundations
Concrete
Concrete strength should be C 150 for block foundation and C 200 for framed foundations. The concreting should be done in ONE operation and construction joints should be properly located especially for thick blocks exceeding 5 m. The construction joints should be treated with dowels and shear keys. Cement grout with no shrinkable additive should be used under the machine bed plate and for pockets of anchor bolts.
Reinforcement
1. Reinforcement should be used in all surfaces, openings, cavities required to be provided in the machine foundation
2. Reinforcement in the concrete block should not be less than 25 kg/m3 or 250 N/m3 of concrete.
3. For machines requiring special design consideration of foundations, such as machine pumping explosive gases, the minimum reinforcement is 40 kg/m3.
4. The reinforcement should be used in all three directions especially in block-type foundation.
5. Steel reinforcement around all pits and openings shall be at least equal to 0.5 to 0.75% of the cross-sectional area of the pit or openings in form of a cage.
6. The minimum reinforcement shall usually consist of 12 mm at 200 to 250 mm spacing or 16/ 25 mm bars at 200 mm to 300 mm spacing extending both vertically and horizontally near all faces of the foundation blocks. The ends of the all bars should always be hooked.
7. If the height of the foundation block exceeds 1m, shrinkage reinforcement shall be placed at suitable spacing in all the three directions.
8. The concrete cover should be a minimum of 75 mm at the bottom and 50 mm on sides and the top.
