1.6 Foundation and Soil Design Parameters

To perform the analysis and obtain the response for a vibrating foundation, certian geometrical and material properties for both the foundation and the supporting medium will be required.

In this section, the definitions of these properties will be introduced to make us ready for the uncoming sections.

Design parameters can be typically categorized into three main catergories, the machine properties which define the load that shakes the foundation, the soil and foundation parameters which define how the system resists and dissipates that load, and the environmental requirements which define what the surroundings can tolerate.

It is worth spending time on collecting this data before starting any calculation. The analysis we will develop in the upcoming chapters is only as reliable as the numbers we feed into it, and most of these numbers come from the machine vendor and from the geotechnical investigation, not from the structural engineer.

1.6.1 Machine Properties

These properties are supplied by the machine manufacturer and they are used to build the loading function, i.e. the force that the machine applies to the foundation at every instant of time.

- Outline drawing of the machine assembly. This drawing fixes the geometry of the problem. It tells us the plan area the block has to cover, where the machine feet and anchor bolts land, and where the auxiliary components sit. Without it, the foundation dimensions cannot even be started.

- Weight of the machine and of its rotating components (or the weight of the falling head for hammers). The machine weight is added to the block weight to form the total vibrating mass, and the total mass is what controls the natural frequency of the system. The rotor weight has a second role, it scales the magnitude of the unbalanced force as we will see below.

- Location of the center of gravity, both vertically and horizontally. If the center of gravity of the machine is not aligned with the center of gravity of the block and with the center of resistance of the soil, the vertical load produces a moment and the horizontal and rocking motions become coupled. Keeping these centers aligned is one of the cheapest ways to simplify the behaviour of the foundation.

- Speed range of the machine and of its components, or directly the frequency of the primary and secondary unbalanced forces. This gives us the operating (excitation) frequency $\omega$, which is compared against the natural frequency of the foundation-soil system to check how far we are from resonance. It also decides whether the foundation is tuned below or above the machine speed.

- Magnitude, direction and point of application of the unbalanced forces, both vertically and horizontally. The magnitude sets the size of the response, the direction decides which degrees of freedom from Section 1.3 are excited, and the point of application decides the moment arm and therefore how much rocking or torsion is generated.

- Tolerance limits on deflection and on vibration amplitudes. These are the acceptance criteria of the design. They are the limits that the machine needs in order to function properly, and without them the computed amplitudes cannot be judged as acceptable or not.

To obtain the magnitude of the unbalanced force we need the eccentricity of the rotating parts, i.e. the small offset between the center of mass of the rotor and its axis of rotation. For a rotor of mass $m_r$ spinning with an eccentricity $e$ at a circular frequency $\omega$, the unbalanced force amplitude is,

The eccentricity is rarely measured on site, so design values are normally taken from published guidelines. Arya et al., 1979 provide recommended design eccentricities for the different types of machines.

Observations

1.6.2 Soil and Foundation Parameters

This group describes the system that has to carry the load, the soil below and the concrete block itself.

1.6.2.1 Soil Parameters

Knowledge of the soil profile and of its properties is required for the dynamic analysis. The information comes from field borings or soundings together with laboratory tests, and the geotechnical report should state clearly the following quantities.

- The soil profile (soil formation). The layering matters as much as the properties of the individual layers. A stiff layer or bedrock at shallow depth reflects the waves back toward the foundation instead of letting them travel away, which removes a large part of the radiation damping and can make the response much larger than a half-space calculation would suggest.

- Density of the soil $\gamma$, or mass density $\rho$. It combines with the shear modulus to give the shear wave velocity $V_s = \sqrt{G / \rho}$, which is the speed at which energy leaves the foundation. The soil mass that vibrates with the block also comes from $\rho$, and this is the mass behind the added mass concept mentioned in Section 1.5.

- Poisson’s ratio $\nu$. It fixes the ratio between the compression and the shear wave velocities and therefore affects the vertical and rocking impedances more than the horizontal ones. A saturated clay behaves almost undrained under the fast loading of a machine and takes a value close to 0.5, while a dry sand is closer to 0.3.

- Shear modulus of the soil $G$, at several levels of strain. This is the most influential single parameter in the whole analysis, since the stiffness of the impedance functions is directly proportional to it. $G$ is not a constant, it decays as the shear strain increases, so it must be quoted at the strain level that the machine actually produces. Machine foundations usually work at very small strains, so the small strain value obtained from shear wave velocity measurements is normally the appropriate one.

- Material damping ratio $D$, at several levels of strain. This is the energy dissipated inside the soil itself through hysteresis, and it is added to the radiation damping that comes from the waves travelling away. Its contribution is small when radiation damping is fully developed, but it becomes the governing source of damping when a shallow rigid layer blocks the radiation, and it is felt most strongly near resonance.

1.6.2.2 Foundation Parameters

- Minimum depth of the foundation. It is usually governed by the frost depth or by the level of a suitable bearing stratum. It is not only a geotechnical requirement, the embedded part of the block mobilizes the side soil and adds both stiffness and damping to the system.

- Base dimensions required by the machine and by the other components attached to it. These dimensions are the lower bound of the block size before any dynamic consideration is applied.

- Type of the foundation system to be used, normally recommended by the geotechnical consultant. A block resting directly on the soil and a pile supported block behave in a completely different way, so this choice comes before any impedance calculation.

- Configuration and layout of the foundation, its width, length and depth. For piled foundations we also need the number of piles, the pile geometry (diameter or width and cross-sectional area), the pile length and the spacing between the piles, in addition to the configuration of the block itself. The spacing is particularly important because closely spaced piles interact with each other through the soil, and the group does not behave as the simple sum of the individual piles.

- Material properties of the foundation, the unit weight of the concrete or the steel, Poisson’s ratio and the elastic modulus. The unit weight gives the mass and the mass moments of inertia of the block, while the elastic modulus compared against the soil modulus tells us whether the block is stiff enough to be treated as a rigid body, which is the assumption behind most of the impedance solutions we will use.

1.6.3 Environmental Requirements

The first two groups look inward at the machine and its foundation. This third group looks outward, in both directions.

- Vibration leaving the site. The machine produces vibrations that travel to the neighboring vicinity. If the amplitudes are significant, measures have to be taken to minimize the environmental impact of the machine. This is a major concern for shock producing equipment such as hammers and presses, where a single impact sends a strong pulse into the ground.

- Vibration arriving at the site. The opposite situation is equally possible, the machine may be installed close to a source of vibration such as quarry blasting or vehicular traffic, or in a seismically active area. In this case we need to know the character of the incoming vibration and how much it attenuates before reaching the installation site, because it adds to the motion produced by the machine itself and may violate the tolerance limits of Section 1.6.1.

- Seismic effects. These are not covered by the ordinary machine foundation calculation. They are addressed with special techniques that deal with wave propagation and ground response analyses.

Observations