Design of Machine Foundations and Foundation Engineering Study Notes

Introduction to Machine Foundations

• Foundations serving under various types of machinery typically support two distinct categories of loading:

  • Static Loads: These encompass the self-weight of the specific machine coupled with the weight of the foundation itself.

  • Dynamic Loads: These are characterized by rapid changes in both magnitude and direction. Such fluctuations occur due to the movement of unbalanced machine components during the operation of the equipment.

• The methodology for designing machine foundations subjected to static loading aligns with the principles of ordinary foundation design. The standard procedure involves:

  • Establishing preliminary dimensions based on assumptions.

  • Evaluating the stresses induced within the soil.

  • Verifying these stresses against the allowable bearing capacity of the soil.

• For dynamic loading scenarios, the vibration behavior of the system is a critical consideration. In these instances, the machine and its foundation are treated as an integrated vibrating system.

Fundamental Concepts of Oscillation and Vibration

• Oscillation refers to the repeating motion associated with a rigid body or a point mass. Illustrative examples of oscillation include:

  • The movement of a pendulum.

  • The motion of a ship navigating through waves.

• Vibration is defined as the repeating motion or deformation occurring within an elastic structure. Common examples include:

  • Rotating fans.

  • Electric motors.

  • Machine foundations.

• The pivot point of distinction between these two concepts is that vibration involves the physical deformation of the body, whereas the pure oscillation of a rigid body does not involve deformation.

Comparative Analysis of Static and Dynamic Loads

• The essential difference between static and dynamic loading is defined by the presence of acceleration.

• Static Analysis: Loads are assumed to be applied incrementally and gradually until a state of equilibrium is achieved. Acceleration is considered negligible.

• Dynamic Analysis: Acceleration is a factor that cannot be ignored. Consequently, the effects of inertia must be explicitly integrated into the analytical model. This is governed by Newton's Second Law:

  • $F = m \times a$

D'Alembert's Principle

• This principle suggests that the effects of acceleration can be represented as an inertia force.

• This inertia force acts in a direction diametrically opposite to the direction of the acceleration.

• Formula for Inertia Force:

  • $\text{Inertia Force} = m \times a$

• Implementation: By introducing these inertia forces, a complex dynamic problem can be simplified and analyzed as an equivalent equilibrium problem.

Classification of Vibrations: Natural vs. Forced

• Natural Vibrations:

  • These occur after a body experiences an initial disturbance from its equilibrium state and is subsequently released.

  • The resulting vibration is sustained through the continuous exchange of kinetic energy and strain energy.

  • Properties: The frequency of natural vibration is an inherent characteristic of the system, depending on factors such as mass and stiffness. It is independent of the magnitude of the initial displacement.

• Forced Vibrations:

  • These occur when an external exciting force acts on the system either continuously or periodically.

  • Most industrial machines generate this type of periodic excitation during their operational cycles.

Degrees of Freedom in Vibratory Systems

• Single Degree of Freedom (SDOF) Systems:

  • These systems require only one independent coordinate to fully describe their motion.

  • Examples include mass-spring systems, simple pendulums, and rotating systems characterized by a single generalized coordinate.

  • SDOF systems are frequently utilized as idealized models in analysis to simplify the derivation of equations of motion.

• Multiple Degree of Freedom (MDOF) Systems:

  • These systems are composed of several discrete masses or represent a continuous distribution of mass.

  • Example: Multi-storey buildings, where every floor level possesses its own distinct mass.

  • Description: Multiple coordinates are necessary for a complete description of motion. MDOF systems are characterized by having multiple natural frequencies and various vibration modes.

Categories and Sources of Excitation

• Excitation acting on SDOF and MDOF systems is classified into two broad categories:

  • Excitation caused by a force:

  • Sinusoidal forces: Typical of unbalanced rotating machinery.

  • Non-sinusoidal forces: For example, the force exerted by a person walking.

  • Excitation caused by support displacement:

  • Sinusoidal displacement.

  • Non-sinusoidal displacement: A primary example is ground motion resulting from an earthquake.

Principal Forces in Vibrating Systems

• Four primary forces are analyzed within the context of vibration:

  • 1. Inertia Force: Defined as $m \times \ddot{x}$.

  • 2. Spring Force: Defined as $k \times x$. This force is a result of elasticity and operates according to Hooke's Law.

  • 3. Damping Force: Defined as $c \times \dot{x}$. This force opposes motion and typically acts in opposition to velocity. The most prevalent model used is viscous damping, where the force is directly proportional to the velocity.

  • 4. External Exciting Force: The specific force that initiates and maintains the vibration.

Industrial Classifications of Machinery

• 1. Reciprocating Machines:

  • These produce periodic unbalanced forces. Examples include steam engines. They generally operate at relatively low speeds.

• 2. Impact Machines:

  • These generate impact loads. A classic example is a forging hammer. In these machines, the load reaches its peak intensity over a very short duration of time.

• 3. Rotary Machines:

  • These are high-speed systems such as turbo-generators and rotary compressors. The operating speeds of these machines often exceed several thousand revolutions per minute (RPM).

Foundation Engineering Definitions and Bearing Capacity

• Shallow Foundation:

  • A foundation where the depth below the surface ($Z$) is less than or equal to its least dimension ($B$).

  • Criterion: $Z \leq B$

• Deep Foundation:

  • A foundation where the depth below the surface is greater than its least dimension.

  • Criterion: Z > B

• Ultimate Bearing Capacity ($q_u$):

  • The average contact pressure between the foundation and the soil that triggers a shear failure within the soil mass.

• Safe Bearing Capacity ($q_a$):

  • The allowable contact pressure that can be safely applied to the soil without the risk of shear failure.

  • Relation to Ultimate Bearing Capacity:

  • $q_a = \frac{q_u}{F}$

  • where $F$ represents the factor of safety.

Terzaghi's Bearing Capacity Theory

• Developed by Karl Terzaghi, this equation provides the classical framework for determining bearing capacity.

• General formulation for strip footings:

  • $q_u = c'N_c + qN_q + 0.5 \gamma B N_\gamma$

• Variable Definitions:

  • $c'$ = Cohesion of the soil.

  • $\gamma$ = Unit weight of the soil.

  • $q$ = Surcharge pressure, calculated as $\gamma \times \text{depth of the foundation (D)}$.

  • $B$ = Width of the footing.

  • $N_c, N_q, N_\gamma$ = Dimensionless bearing capacity factors that are functions of the angle of internal friction ($\phi$).

• Note: Modified versions of this equation are applied when dealing with square or circular footing geometries.

Technical Worked Example for Footing Design

• Provided Parameters:

  • Applied Load = $30,000\,kg$

  • Soil Density ($\rho$) = $1850\,kg/m^3$

  • Angle of internal friction ($\phi$) = $35^\circ$

  • Cohesion ($c'$) = $0$

  • Foundation Depth ($D$) = $1\,m$

• Unit Conversions and Preliminary Calculations:

  • Unit weight ($\gamma$) = $\frac{1850 \times 9.81}{1000} \approx 18.15\,kN/m^3$

  • Total Load in Force units = $\frac{30000 \times 9.81}{1000} \approx 294.3\,kN$

• Bearing Capacity Factors for $\phi = 35^\circ$:

  • $N_c = 57.8$

  • $N_q = 41.4$

  • $N_\gamma = 42.4$

• Design Calculation Steps:

  • The square footing form of the equation is utilized.

  • A Factor of Safety ($F$) of $3$ is applied: $q_a = \frac{q_u}{3}$.

  • Substituting the known values into the equation and iterating through a trial-and-error process to find the width ($B$).

  • Result: The required width for the square footing is approximately $0.75\,m$.